CN116964596A - Reconfigurable qubit entanglement system - Google Patents

Abstract

According to some embodiments, a system includes a first input coupled to a first qubit and a first switch, wherein the first switch includes a first output, a second output, and a third output. The system also includes a first single-qubit measurement device coupled to the first output of the first switch and a second single-qubit measurement device coupled to the first output of the second switch. The system also includes a first two-qubit measurement device coupled to the second output of the first switch and the second output of the second switch and a second two-qubit measurement device coupled to the third output of the first switch and the third output of the second switch.

Description

Reconfigurable qubit entanglement system

Cross Reference to Related Applications

The present application claims priority from U.S. provisional application No. 63/140784 entitled "fusion-based quantum computing (Fusion Based Quantum Computing)" filed on month 22 of 2021 and U.S. provisional application No. 63/293592 entitled "reconfigurable quantum fusion system (Reconfigurable Qubit Fusion System)" filed on month 23 of 2021, the entire contents of both U.S. provisional applications being incorporated herein by reference.

Technical Field

One or more embodiments relate generally to quantum technology devices (e.g., hybrid electronic/photonic devices), and more particularly, to quantum technology devices for generating entangled states of quantum bits (e.g., entangled states that may be used as quantum computing, quantum communication, quantum metering, and other quantum information processing tasks), and systems and methods for generating syndrome (syndrome) graphics data that may be used for quantum error correction within fault tolerant quantum computing systems. One or more embodiments of the present disclosure relate generally to quantum computing devices and methods, and more particularly, to fault tolerant quantum computing devices and methods.

Background

In fault tolerant quantum computing, quantum error correction is required to avoid accumulation of qubit errors that would lead to erroneous computed results. One way to achieve fault tolerance is to use error correction codes (e.g., topology codes) for quantum error correction. More specifically, a set of physical qubits (also referred to herein as an error correction code) may be generated in an entangled state that encodes a single logical qubit that is protected from errors.

In some quantum computing systems, a cluster of multiple qubits (or more generally, a pattern) may be used as an error correction code. A figure state is a highly entangled multi-qubit state that can be visually represented as a graph with nodes representing qubits and entangled edges between the representing qubits. However, various problems of inhibiting the generation of entangled states or destroying the entanglement once generated have hampered advances in quantum technology that rely on the use of highly entangled quantum states.

Furthermore, in some qubit architectures (e.g., photonic architectures), generating entangled states of multiple qubits is an inherently probabilistic process that will have a low probability of success.

Thus, there remains a need for improved systems and methods for quantum computing that do not necessarily rely on large clusters of qubits.

Disclosure of Invention

Described herein are embodiments of a reconfigurable qubit entanglement system according to one or more embodiments.

According to some embodiments, a method may include: receiving a plurality of quantum systems, wherein each of the plurality of quantum systems comprises a plurality of quantum systems in an entangled state, and wherein respective ones of the plurality of quantum systems are independent quantum systems that are not entangled with each other; performing a plurality of destructive joint measurements (e.g., fusion operations) on different ones of the quantum systems from respective ones of the plurality of quantum systems, wherein the destructive joint measurements destroy the different ones of the quantum systems and generate joint measurement result data and transmit quantum state information from the different ones of the quantum systems to other unmeasured ones of the quantum systems; and determining a logical qubit state based on the joint measurement data. The logical qubit states may be determined in a fault tolerant manner.

According to some embodiments, a method may include: receiving a plurality of quantum systems, wherein each of the plurality of quantum systems comprises a plurality of quantum systems in an entangled state, and wherein respective ones of the plurality of quantum systems are independent quantum systems that are not entangled with each other; performing a logical qubit gate by performing a plurality of destructive joint measurements (e.g., fusion operations) on different quantum subsystems from respective ones of the plurality of quantum systems, wherein the destructive joint measurements destroy the different quantum subsystems and generate joint measurement result data and transmit quantum state information from the different quantum subsystems to other unmeasured quantum subsystems from the plurality of quantum systems; and determining a result of the logic qubit gate based on the joint measurement result data. The result of the logic qubit gate may be determined in a fault tolerant manner.

According to some embodiments, a quantum computing device may include: a qubit entanglement system that generates a plurality of quantum systems, wherein each of the plurality of quantum systems includes a plurality of quantum systems in an entangled state, and wherein respective ones of the plurality of quantum systems are independent quantum systems that are not entangled with each other; a qubit fusion system that performs a plurality of destructive joint measurements on different quantum subsystems from respective ones of the plurality of quantum systems, wherein the destructive joint measurements destroy the different quantum subsystems and generate joint measurement result data and transmit quantum state information from the different quantum subsystems to other unmeasured quantum subsystems from the plurality of quantum systems; and a typical computing system that determines a logical qubit state based on the joint measurement data.

According to some embodiments, a quantum computing device may include: a qubit entanglement system that generates a plurality of quantum systems, wherein each of the plurality of quantum systems includes a plurality of quantum systems in an entangled state, and wherein respective ones of the plurality of quantum systems are independent quantum systems that are not entangled with each other; a qubit fusion system that performs a logical qubit gate by performing a plurality of destructive joint measurements on different ones of the quantum systems from respective ones of the plurality of quantum systems, wherein the destructive joint measurements destroy the different ones of the quantum systems and generate joint measurement result data and transmit quantum state information from the different ones of the quantum systems to other unmeasured ones of the quantum systems; and a typical computing system that determines the result of the logical qubit gate based on the joint measurement result data.

According to some embodiments, a system includes a first input coupled to a first qubit and a first switch, wherein the first switch includes a first output, a second output, and a third output. The system also includes a first single-qubit measurement device coupled to the first output of the first switch and a second single-qubit measurement device coupled to the first output of the second switch. The system also includes a first two-qubit measurement device coupled to the second output of the first switch and the second output of the second switch and a second two-qubit measurement device coupled to the third output of the first switch and the third output of the second switch.

In some embodiments, the system further comprises a converged network controller circuit coupled to the first and second switches.

In some embodiments, the system further comprises a decoder coupled to the output of the first single-qubit measurement device, the output of the second single-qubit measurement device, the output of the first two-qubit measurement device, and the output of the second two-qubit measurement device.

In some embodiments, the first qubit is entangled with one or more other qubits as part of a first resource state and the second qubit is entangled with one or more other qubits as part of a second resource state, and none of the qubits from the first resource state are entangled with any of the qubits from the second resource state.

In some embodiments, the first and second two-qubit measurement devices are configured to perform destructive joint measurements on the first and second qubits and output representative information representative of the joint measurements.

In some embodiments, the first and second qubits are optical qubits.

In some embodiments, the coupling between the first and second qubits and the first and second switches comprises a plurality of photonic waveguides.

In some embodiments, the first single-qubit measurement device is configured to measure the first qubit on a Z-basis.

In some embodiments, the second single qubit measurement device is configured to measure the second qubit on a Z basis.

In some embodiments, the first two-qubit measurement device is configured to perform a projected bell measurement between the first qubit and the second qubit.

In some embodiments, the second two-qubit measurement device is configured to perform a projected bell measurement between the first qubit and the second qubit.

In some embodiments, the projected bell measurement is a linear optical type II fusion measurement.

In some embodiments, the projected bell measurement is a linear optical type II fusion measurement.

The following detailed description and the accompanying drawings will provide a better understanding of the nature and advantages of the claimed invention.

Drawings

Various aspects of the present disclosure are shown by way of example. Non-limiting and non-exhaustive aspects are described with reference to the following figures, wherein like reference numerals refer to like parts throughout the various views unless otherwise specified.

Fig. 1A-1C are schematic diagrams illustrating clustered states and corresponding syndrome diagrams for physical qubit entangled states, according to some embodiments.

FIG. 2 illustrates a quantum computing system in accordance with one or more embodiments.

Fig. 3A-3D illustrate quantum computing systems according to some embodiments.

Fig. 4A illustrates an example of a converged network in accordance with some embodiments.

Fig. 4B illustrates a quantum computing system according to some embodiments.

Fig. 5 illustrates one example of a qubit fusion system according to some embodiments.

FIG. 6 illustrates resource states according to some embodiments.

Fig. 7 illustrates one example of a qubit fusion system according to some embodiments.

Fig. 8A-8C illustrate examples of converged networks and explicit definitions of certain groups in accordance with some embodiments.

Fig. 9A-9E illustrate examples of converged networks and explicit definitions of certain groups in accordance with some embodiments.

Fig. 10A-10E illustrate examples of converged networks in accordance with some embodiments.

FIG. 11 illustrates fault tolerance of numerical calculations for various converged network and resource states in accordance with some embodiments.

FIG. 12 illustrates fault tolerance of numerical calculations for various converged network and resource states in accordance with some embodiments.

Fig. 13A-13D illustrate a simplified example of how original and double boundaries may be created by measuring certain qubits in the Z-base, according to some embodiments.

Fig. 14A-14E illustrate examples of a converged router providing routing using photonic resource state generators, optical routing, and linear optical fusion to generate a 6-ring converged network, in accordance with some embodiments.

Fig. 15A and 15B illustrate two other embodiments of a networked RSG circuit that may be used, for example, in a substance-based qubit architecture (e.g., trapped ions, superconducting qubits, etc.), in accordance with some embodiments.

Fig. 16 illustrates an example of a quantum circuit employing logic feed forward in accordance with some embodiments.

FIG. 17 illustrates a schematic diagram of typical information flowing into and out of a quantum computing system, according to some embodiments.

Fig. 18 illustrates an exemplary configuration of a decoding system including buffering and decoder parallelization according to some embodiments.

Fig. 19 illustrates photonic hardware components within a linear optical quantum computer according to some embodiments.

FIG. 20 illustrates an example of a multiplexed single photon source in accordance with one or more embodiments.

Fig. 21 illustrates one possible example of a fusion site configured to operate with a fusion controller to provide measurement results for fault tolerant quantum computing to a decoder, in accordance with some embodiments.

Fig. 22A-22C illustrate a fusion-based quantum computing scheme for fault tolerant quantum computing in accordance with one or more embodiments.

Fig. 23A-23C illustrate one example of a lattice preparation protocol for fusion-based quantum computing, according to some embodiments.

Fig. 24A-24B illustrate one example of a lattice preparation protocol for fusion-based quantum computing, according to some embodiments.

Fig. 25A-25E illustrate a flow diagram and an exemplary lattice preparation protocol for illustrating a method of fusion-based quantum computation in accordance with one or more embodiments.

26A-26E illustrate representations of a dual-rail encoded optical qubit and photonic circuit for performing a unitary operation on the optical qubit according to some embodiments.

27A-27B illustrate representations of a dual-rail encoded optical qubit and photonic circuit for performing a unitary operation on the optical qubit according to some embodiments.

Fig. 28 illustrates a photon implementation of a beam splitter according to some embodiments, which may be used to implement one or more diffusers, such as Hadamard (Hadamard) gates.

Fig. 29 illustrates a photon implementation of a beam splitter that may be used to implement one or more diffusers, such as hadamard gates, in accordance with some embodiments.

Fig. 30 shows one example of a bell state generator circuit that may be used in some dual-rail encoded photon embodiments.

Fig. 31 illustrates an example of a type II fusion circuit for polarization encoding in accordance with some embodiments.

Fig. 32 illustrates an example of a type II fusion circuit for path encoding in accordance with some embodiments.

Fig. 33A-33D illustrate the effect of fusion in generating cluster states according to some embodiments.

Fig. 34 illustrates an example of a type II fusion gate that is enhanced once in polarization encoding and path encoding, in accordance with some embodiments.

Fig. 35 shows a graph of the variation of type II fusion gate for different measurement references in polarization encoding.

Fig. 36 illustrates an example of a photonic circuit variation for a type II fusion gate for different selections of measurement references in path coding, in accordance with some embodiments.

Detailed Description

Reference will now be made in detail to the various embodiments, examples of which are illustrated in the accompanying drawings. In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the various embodiments described. It will be apparent, however, to one of ordinary skill in the art that the various embodiments described may be practiced without these specific details. In other instances, well-known methods, procedures, components, circuits, and networks have not been described in detail so as not to unnecessarily obscure aspects of the embodiments.

Introduction to Quantum computation

Quantum computing is typically considered in the framework of "circuit-based quantum computing (Circuit Based Quantum Computation, CBQC)", in which operations (or gates) are performed on physical qubits. The gates may be single qubit unitary (rotation), two qubit entangled (e.g., CNOT gates), or other multi-qubit gates (e.g., toffoli gates).

Measurement-based quantum computing (Measurement Based Quantum Computation, MBQC) is another method of achieving quantum computing. In the MBQC method, the calculation is performed by: a particular entangled state of many qubits (commonly referred to as a cluster state) is first prepared, and then a series of single-qubit measurements are made on the cluster state to make quantum calculations. In this approach, the choice of single qubit measurement is dictated by the quantum algorithm running on the quantum computer. In the MBQC method, fault tolerance may be achieved by carefully designing the cluster states and using the topology of the cluster states to encode logical qubits that are protected from any logical errors that may be caused by errors on any of the physical qubits that make up the cluster states. In practice, the value of the logical qubit may be determined (i.e., read out) based on the results of single-event measurements (also referred to herein as measurement results) of the physical qubits in a clustered state as the computation proceeds.

However, the generation and maintenance of remote entanglement across cluster states and subsequent storage of large cluster states would be challenging. For example, for any physical implementation of the MBQC method, a cluster of thousands or more entangled qubits must be prepared and then stored for a period of time before performing a single qubit measurement. For example, to generate a cluster of qubits representing a single logical error correction, one can be at |+>Each of the set of underlying physical qubits is prepared in state and a controlled phase gate (CZ) may be applied between individual pairs of physical qubits to generate an overall cluster state. More specifically, the cluster states of highly entangled qubits can be described by an undirected graph g= (V, E), where V and E represent sets of vertices and edges, respectively, and can be generated as follows: 1) Initializing all physical qubits to | +>A state in which, among other things,and 2) applying a controlled phase gate (CZ) to each qubit pair i, j. Thus, any cluster of states that physically corresponds to a large entangled state of physical qubits can be described as:

wherein CZ _i,j Is a controlled phase gate operator and has V and E as defined above. Graphically, the cluster state defined by equation (1) can also be represented by a graph with vertices V and edges E, where vertex V represents a physical qubit (in |+ >State initialization), edge E represents entanglement between them (i.e., various CZ gates applied). In some cases, e.g., involving fault tolerant MBQC schemes, |ψ> _graph May take the form of a 3-dimensional graph. Similar to that shown in figure 1Examples are shown. In fig. 1A and 22, such a diagram may have a regular structure formed of repeating unit cells, and thus is generally referred to as a "lattice". When represented as a three-dimensional lattice, two-dimensional boundaries of the lattice may be identified. Qubits belonging to these boundaries are called "boundary qubits" and all other qubits are called "block qubits".

Generating |ψ> _graph The large state of such inter-entangled qubits must then be maintained long enough to perform a stabilizer measurement, for example, by taking X measurements on all physical qubits in the lattice block and Z measurements on the boundary qubits.

FIG. 1A shows One example of a Fault-Tolerant cluster that may be used in MBQC, which is introduced by Lawsendorf (Raussendorf) et al and is commonly referred to as a Lawsendorf lattice, as described in further detail in Robert Raussendorf, jim Harrington and "Fault-Tolerant One-Way Quantum Computer, annals of Physics" (321 (9): 2242-2270, 2006) by Kovid Goyal.A. The clustered state is in the form of a recombinant cell (e.g., cell 120) in which physical qubits (e.g., physical qubits 116) are arranged on the sides and edges of the cell (unit cell). Entanglement between physical qubits is represented by edges (e.g., edge 118) connecting physical qubits, where each edge represents the application of a CZ gate, as described above with reference to equation (1). The cluster states shown here are just one example of many, and other topology error correction codes may be used without departing from the scope of the invention. For example, a volume code may be used, such as that disclosed in International patent application publication No. WO/2019/173651, the contents of which are incorporated herein by reference in their entirety for all purposes. Non-cubic cell based codes described in International patent application publication No. WO/2019/178009, the contents of which are incorporated herein by reference in their entirety for all purposes, may also be used without departing from the scope of the present invention. Furthermore, while the examples shown here are represented in three spatial dimensions, the same structure may be obtained from other implementations of code that are not based on pure spatial entanglement clusters, but may include entanglement in 2D space and temporal entanglement, e.g., 2+1d surface code implementations or any other leaf-like code may be used. For clustered implementations of such codes, all quantum gates required for fault tolerant quantum computation can be constructed by making a series of single-event measurements on the physical qubits that make up the lattice.

Returning to fig. 1A, blocks of the lawsonde lattice are shown. Such entangled states may be used to encode one or more logical qubits (i.e., one or more error correction qubits) using a number of entangled physical qubits. A set of single-event measurements of multiple physical qubits (e.g., physical qubit 116) may be used to correct errors and perform fault-tolerant calculations on logical qubits using a decoder. Many decoders are available, one example of which is the decoder described in International patent application publication No. WO2019/002934A1, the disclosure of which is incorporated herein by reference in its entirety for all purposes. Those of ordinary skill in the art will appreciate that the number of physical qubits required to encode a single logical qubit may vary depending on the exact nature of physical errors, noise, etc. experienced by the physical qubit, but to achieve fault tolerance all schemes so far require entangled states of thousands of physical qubits to encode a single logical qubit. The generation and maintenance of such large entangled states is a critical challenge for any practical implementation of the MBQC method.

Fig. 1B-1C illustrate how decoding of logical qubits will proceed for a cluster state based on the lawsonian lattice. As can be seen in fig. 1A, the geometry of the cluster states is related to the geometry of the cubic lattice (unit cell 120) shown superimposed on the cluster states in fig. 1A. Fig. 1B shows the single particle measurement (also superimposed on the cubic lattice) after the states of the individual physical qubits of the cluster state have been measured, with the measurement being placed in the previous position of the physical qubits being measured (only the measurement results from the measurement of these surface qubits are shown for clarity).

In some embodiments, after all qubits have been measured, e.g., in the x-base, the measured qubit state may be represented by a digital bit value of 1 or 0, where a 1 bit value corresponds to a +x measurement and a 0 bit value corresponds to a-x measurement (or vice versa). There are two types of qubits: a qubit located on the side of the unit cell (e.g., at side qubit 122) and a qubit located on the face of the unit cell (e.g., at face qubit 124). In some cases, a measurement of the qubit may not be obtained, or the result of the qubit measurement may be invalid. In these cases, no bit value is assigned to the location of the corresponding measured qubit, but the result is referred to herein as erasure, e.g., illustrated herein as bold line 126. These known lost measurements can be reconstructed during the decoding process.

To identify errors in the physical qubits, a syndrome graph (syndrome graph) may be generated from a set of measurement results generated by the measurement of the physical qubits. For example, the bit values associated with the respective edge qubits may be combined to create a syndrome value associated with a vertex (e.g., vertex 128 as shown in fig. 1B) resulting from the intersection of the respective edges. A set of syndrome values (also referred to herein as parity) are associated with the respective vertices of the syndrome, as shown in fig. 1C. More specifically, in fig. 1C, calculated values of some vertex parities of the correction chart are shown. In some embodiments, the parity calculation entails determining whether the sum of the edge values generated at a given vertex is even or odd, where the parity result for that vertex is defined as the result of the sum divided by the remainder of 2 (mod 2). If no errors occur in the quantum state or in the qubit measurement, then all syndrome values should be even (or 0). Conversely, if an error occurs, some odd (or 1) syndrome values will result. Only half of the bit values from the qubit measurement are associated with the illustrated syndrome (bits aligned with the edges of the syndrome). There is another syndrome graph containing all bit values associated with the faces of the lattice shown. This leads to equivalent decoding problems on these bits.

As mentioned above, the generation and subsequent storage of large clusters of qubits would be a challenge. However, some embodiments, methods, and systems described herein provide for generating a typical set of measured data (e.g., a typical set of data corresponding to the syndrome values of a syndrome) that includes the necessary correlation for performing quantum error correction without first generating large entangled states of quantum bits in the error correction code. For example, embodiments disclosed herein describe systems and methods whereby double-qubit (i.e., joint) measurements (also referred to herein as "fusion measurements" or "fusion gates") can be performed on a much smaller set of entangled states to generate a typical set of data that includes the remote correlations necessary to generate and decode a subgraph for a particular selected cluster state, without actually generating the cluster state. In other words, in some systems and methods described herein, only a relatively small set of entangled states (referred to herein as resource states) were once generated, and then joint measurements are performed directly on these resource states to generate the syndrome data without first generating (and then measuring) large cluster states that form a quantum error correction code (e.g., a topological code, such as a lawsonian lattice).

For example, as will be described in further detail below, in the case of linear light quantum computation using a Lawson doffer lattice code structure, to generate the syndrome data, a fusion gate may be applied to a collection of small entangled states (e.g., 4-GHZ states) that are themselves not entangled with each other and, therefore, are not part of a larger Lawson doffer lattice cluster state. Although the qubits from the individual resource states do not intertwine prior to the fusion measurement, the joint measurement results produced by the fusion measurement generate a syndrome graph that includes all the necessary dependencies to perform quantum error correction. Such systems and methods are referred to herein as fusion-based quantum computing (Fusion Based Quantum Computing, FBQC). Advantageously, the resource states have a size independent of the calculation performed or the code distance used, in sharp contrast to the cluster states of MBQC. This allows the resource states for FBQC to be generated by a constant number of sequential operations. Thus, in FBQC, errors in the resource state are limited, which is important for fault tolerance.

II. System for FBQC

FIG. 2 illustrates a quantum computing system in accordance with one or more embodiments. Quantum computing system 201 includes a user interface device 204 communicatively coupled to a Quantum Computing (QC) subsystem 206, described in more detail below in fig. 3. The user interface device 204 may be any type of user interface device, such as a terminal including a display, keyboard, mouse, touch screen, etc. In addition, the user interface device itself may be a computer, such as a Personal Computer (PC), a laptop computer, a tablet computer, or the like. In some embodiments, the user interface device 204 provides an interface that may be utilized by a user to interact with the QC subsystem 206 directly or via a local area network, a wide area network, or via the internet. For example, the user interface device 204 may run software, such as a text editor, an Interactive Development Environment (IDE), command prompts, a graphical user interface, etc., so that a user may program or otherwise interact with the QC subsystem to run one or more quantum algorithms. In other embodiments, the QC subsystem 206 may be preprogrammed and the user interface device 204 may be an interface at which a user may initiate quantum computation, monitor progress, and receive results from the QC subsystem 206. QC subsystem 206 may also include a typical computing system 208 coupled to one or more quantum computing chips 210. In some examples, the exemplary computing system 208 and quantum computing chip 210 may be coupled to other electronic components 212, such as pulsed pump lasers, microwave oscillators, power supplies, networking hardware, and the like. In some embodiments employing cryogenic operation, quantum computing system 201 may be housed within a cryostat (e.g., cryostat 214). In some embodiments, quantum computing chip 210 may include one or more constituent chips, e.g., an integration (direct or heterogeneous) of electronic chip 216 and integrated photonic chip 218. Signals may be routed on-chip and off-chip in any number of ways, for example, via optical interconnect 220 and via other electronic interconnects 222. In addition, the computing system 201 may employ a quantum computing process, for example, a fusion-based quantum computing process as described in further detail below.

Fig. 3A shows a block diagram of a QC system 301, in accordance with some embodiments. Such a system may be associated with computing system 201 described above with reference to fig. 2. In fig. 3, solid lines represent quantum information channels, and double solid lines represent typical information channels. QC system 301 includes a resource state generator 303, a qubit fusion system 305, and a typical computing system 307. In some embodiments, resource state generator 303 may take as input a set of N physical quantum bits (also referred to herein as a "quantum subsystem") (e.g., physical quantum bits 309 (also schematically represented as inputs 311a, 311b, 311c,..311N)), and may generate quantum entanglement between two or more of them to generate entangled resource states 315 (also referred to herein as a "quantum system," which itself is comprised of entangled states of the quantum subsystem). For example, in the case of an optical qubit, the resource state generator 303 may be a linear optical system (e.g., an integrated photonic circuit) that includes a waveguide, beam splitter, photon detector, delay line, and the like. In some examples, entangled resource state 315 may be a relatively small entangled state of qubits (e.g., a qubit entangled state having between 3 and 30 qubits). In some embodiments, resource states may be selected such that fusion operations applied to certain qubits of these states produce syndrome data that includes the correlation required for quantum error correction. Advantageously, the system shown in FIG. 3 provides fault-tolerant quantum computing using relatively small resource states without requiring the resource states to become entangled with each other to form the typical lattice cluster states required for MBQCs.

In some embodiments, input qubits 309 may be a collection of quantum systems (also referred to herein as quantum subsystems) and/or particles, and may be formed using any qubit architecture. For example, the quantum system may be a particle, such as an atom, ion, nucleus, and/or photon. In other examples, the quantum system may be other engineered quantum systems, such as flux qubits, phase qubits, or charge qubits (e.g., formed by superconducting Josephson (Josephson) junctions), topology qubits (e.g., majorana (Majorana) fermi), spin qubits formed by vacancy centers (e.g., nitrogen vacancies in diamond), or quantum bits otherwise encoded in multiple quantum systems (e.g., goltmann-Kitaev-Preskill (GKP) encoded qubits, etc.). Furthermore, for clarity of description, the term "qubit" is used herein, but the system may also employ a quantum information carrier that encodes information in a manner that is not necessarily associated with a binary bit. For example, according to some embodiments, four qubits (i.e., a quantum system that can encode information in more than two quantum states) may be used.

According to some embodiments, QC system 301 may be a fusion-based quantum computer, which may run one or more quantum algorithms or software programs. For example, a software program (e.g., a set of machine-readable instructions) representing a quantum algorithm to be run on QC system 301 may be passed to a typical computing system 307 (e.g., corresponding to system 208 in fig. 2 above). Typical computing system 307 may be any type of computing device (e.g., a PC, one or more blade servers, etc.), or even a high-performance computing system such as a supercomputer, server farm, etc. Such a system may include one or more processors (not shown) coupled to one or more computer memories (e.g., memory 306). Such a computing system will be referred to herein as a "typical computer". In some examples, the logic processor 308 may take as input a software program and calculate a corresponding set of logic gates to be applied to run the software program on specific hardware available within the QC control system 301. In some embodiments, the software program may be received by other or even more than one module (e.g., by the fusion pattern generator 313). One function of the fusion pattern generator 313 is to generate a set of machine-level fusion instructions (e.g., a set of fusion operations and/or a single qubit measurement across the physical qubit applications that make up the QC system 301). In this way, the logic processor 308 and the fusion pattern generator 313 are able to receive an input software program (which may be high-level code that can be more easily written by a user to program a quantum computer) and to generate a set of machine-readable instructions to be applied to low-level quantum hardware.

In some embodiments, the fusion pattern generator 313 (alone or in combination with the logic processor 308) may operate as a compiler for a software program to be run on a quantum computer. The fusion pattern generator 313 may be implemented as pure hardware, pure software, or any combination of one or more hardware or software components or modules. In various embodiments, the fusion pattern generator 313 may be run-time or pre-operative; in either case, the machine-level instructions generated by the fusion pattern generator 313 may be stored (e.g., in the memory 306). In some examples, the compiled machine-level instructions take the form of one or more data frames that instruct the qubit fusion system 305 to perform one or more fusions between certain qubits from separate (i.e., un-entangled) resource states 315 at a given clock cycle of the quantum computer. For example, the fusion pattern data frame 317 is one example of a set of fusion measurements (e.g., type II fusion measurements, described in more detail below with reference to fig. 18-21) that should be applied between certain qubit pairs from different entangled resource states 315 during a certain clock cycle when executing a program.

In some embodiments, several fusion mode data frames 317 may be stored as typical data in memory 306. In some embodiments, fusion mode data frame 317 may specify whether XX type II fusion will be applied (or whether any other type of fusion will be applied) for a particular fusion gate within fusion array 321 of qubit fusion system 305. Additionally, the fusion mode data frame 317 may indicate that type II fusion is to be performed in a different base (e.g., performing XX fusion, XY fusion, ZZ fusion, etc.). As used herein, the terms XX type II fusion, YY type II fusion, XY type II fusion, ZZ type II fusion, etc., refer to fusion operations that employ a particular two-particle projection measurement (e.g., bell projection) that can project two qubits onto one of 4 Bell states (Bell states) depending on the Bell base selected. Such projection measurements yield two measurements (also referred to herein as joint measurement data) corresponding to a corresponding pair of observably measured eigenvalues measured in the selected basis. For example, XX fusion is a Bayer projection of measured XX and ZZ observables (each observables may have either a +1 or-1 eigenvalue or 0 or 1, depending on the convention used), and XZ fusion is a Bayer projection of measured observable XZ and ZX, and so on. Fig. 23-36 below illustrate example circuits for performing type II fusion for various base selections in a linear optical system, but in other qubit architectures other bell projection measurements are possible without departing from the scope of the invention. Those of ordinary skill in the art will appreciate that in a linear optical system, type II fusion performs probabilistic bell measurements. Fig. 23-36 discuss the probabilistic nature of linear optical fusion in the context of fusing "success" and "failure" results, and will not be repeated here for the sake of clarity.

Fusion network controller circuitry 319 of qubit fusion system 205 may receive data encoding fusion pattern data frames 317 and based on the data may generate configuration signals, such as analog and/or digital electronic signals, that drive hardware within fusion array 321. For example, in the case of an optical qubit, the fusion gate may include a photon detector coupled to one or more waveguides, beam splitters, interferometers, switches, polarizers, polarization rotators, and the like. More generally, the detector may be any detector that can detect a quantum state of one or more qubits in the resource state 315. Those of ordinary skill in the art will appreciate that many types of detectors may be used depending on the particular qubit architecture employed.

In some embodiments, the result of applying fusion pattern data frame 317 to fusion array 321 is to generate typical data (generated by a fusion gate detector), which is read out and optionally preprocessed, and sent to fusion pattern generator and/or decoder 333, either directly (not shown) or via any other module. More specifically, fusion array 321 (also referred to herein as a "fusion network") may include a set of measurement devices that enable joint measurements between certain qubits from two different resource states and generate a set of measurement results associated with the joint measurements. These measurements (also referred to herein as joint measurement data) may be stored in a measurement data frame (e.g., data frame 322) and transmitted back to a typical computing system for further processing. In some embodiments, passing the measurement data frame 322 directly to the fusion pattern generator may enable a fast adaptive feed forward process that allows the system to change the fusion pattern data frame 317 in future clock cycles (e.g., selecting a reference or selecting a single particle measurement) based on measurement data collected in a previous time step.

In some embodiments, any one of the sub-modules in QC system 301 (e.g., controller 323, quantum gate array 325, fusion array 321, fusion network controller 319, fusion pattern generator 313, decoder 323, AND logic processor 308) may include any number of typical computing components, such as a processor (CPU, GPU, TPU), memory (any form of RAM, ROM), hard-coded logic components (typical logic gates, e.g., AND (OR), exclusive OR (XOR), etc.), AND/OR programmable logic components (e.g., field programmable gate array (FPGA, etc.)). These modules may also include any number of Application Specific Integrated Circuits (ASICs), microcontrollers (MCUs), system-on-a-chip (SOCs), and other similar microelectronic devices. Although fig. 3 shows particular modules exchanging data, signals, and messages to perform the functions described above, those of skill in the art will understand that the particular arrangement of modules shown herein is merely one example, and that many different examples are possible without departing from the scope of the present disclosure. For example, the compilation, feed forward functions, etc. described above may be shared among the modules.

In some embodiments, entangled resource state 315 may be any type of entangled resource state that, when performing a fusion operation, produces a measurement data frame that includes the necessary dependencies for performing fault-tolerant quantum computation. Although fig. 3 shows an example of a set of identical resource states, a system may be employed that generates many different types of resource states and may even dynamically change the types of resource states generated based on the requirements of the running quantum algorithm. As described herein, logical qubit measurements 327 may be fault-tolerant recovered (e.g., via decoder 333) from physical qubit measurements 322. The logic processor 308 may then process the logic results as part of the program execution. As shown, the logic processor may feed information back to the fusion pattern generator 313 to affect downstream gates and/or measurements to ensure that the computation is performed fault-tolerant.

Fig. 3B illustrates an example of a resource state generator 401 in accordance with some embodiments. According to some embodiments, such a system may be used to generate qubits (e.g., photons) in an entangled state (e.g., a resource state used in the illustrative examples shown in fig. 7-9 below). Resource state generator 401 is an example of a system that may be employed in an FBQC system, such as resource state generator 303 shown in fig. 3 above. Those of ordinary skill in the art will appreciate that any resource state generator may be used without departing from the scope of the present invention. Examples OF resource state generators can be found in U.S. patent application Ser. No. 16/621,994, entitled "Generation OF entangled qubit states" (published as U.S. patent application publication No. 20200287631), U.S. patent application Ser. No. 16/691,459, entitled "GENERATION OF ENTANGLED PHOTONIC STATES" (published as U.S. patent No. 11,126,062), and U.S. patent application Ser. No. 16/691,450, entitled "GENERATION OF AN ENTANGLED PHOTONIC STATE FROM PRIMITIVE RESOURCES" (published as U.S. patent application publication No. XXXXX), the disclosures OF which are incorporated herein by reference in their entirety for all purposes. For example, in some embodiments, the photon source may directly generate entangled resource states, or may even generate smaller entangled states, rather than generating single photons, which may undergo additional entanglement operations at entangled state generator 400 to produce a final resource state to be used for FBQC. It will be seen that the scope of the term "photon source" as used herein is intended to include at least a source of single photons, a source of multiple photons in an entangled state, or more generally any photon state. Those of ordinary skill in the art will appreciate that the exact form of the resource state generation hardware is not critical and that any system may be employed without departing from the scope of the present invention.

In the illustrative photonic architecture, the resource state generator 401 may include a photon source system 405 optically connected to the entangled state generator 400. Both photon source system 405 and entangled state generator 400 may be coupled to exemplary processing system 403 such that exemplary processing system 403 may communicate with and/or control (e.g., via exemplary information channels 430 a-b) photon source system 405 and/or entangled state generator 400. Photon source system 405 may include a collection of single photon sources that may provide output photon states (e.g., single photons or other photon states such as bell states, GHZ states, etc.) to entangled state generator 400 through interconnection waveguide 402. Entangled state generator 400 may receive the output photon states and convert them into one or more entangled photon states (or into a larger photon state in the case of the source itself outputting entangled photon states) and then output these entangled photon states into output waveguide 440. In some embodiments, output waveguide 440 may be coupled to some downstream circuitry that may use entangled states to perform quantum computation. For example, the entangled state generated by entangled state generator 400 may be used as a resource for downstream quantum optical circuits (not shown).

In some embodiments, photon source system 405 and entangled state generator 400 may be used in conjunction with a quantum computing system as illustrated in fig. 3. For example, the resource state generator 303 illustrated in fig. 3 may include a photon source system 405 and an entanglement state generator 400, and the exemplary computer system 403 of fig. 4 may include one or more of the various exemplary computing components illustrated in fig. 3 (e.g., exemplary computing system 307). In this case, entangled photons exiting via output waveguide 440 may be fused together by qubit fusion system 305, i.e., they may be input to a detection system that performs a set of joint measurements for use in an FBQC scheme.

In some embodiments, system 401 may include typical channels 430 (e.g., typical channels 430-a through 430-d) for interconnecting and providing typical information between components. It should be noted that the exemplary channels 430-a through 430-d need not all be identical. For example, the exemplary channels 430-a through 430-c may include a bi-directional communication bus carrying one or more reference signals, such as one or more clock signals, one or more control signals, or any other signal carrying exemplary information (e.g., an announcement signal, a photon detector readout signal, etc.).

In some embodiments, the resource state generator 401 includes a typical computer system 403 in communication with and/or controlling the photon source system 405 and/or the entanglement state generator 400. For example, in some embodiments, a typical computer system 403 may be used to configure one or more circuits, e.g., using a system clock that may be provided to photon source 405 and entangled state generator 400, as well as any downstream quantum photonic circuits used to perform quantum computation. In some embodiments, the quantum photonic circuit may include an optical circuit, an electronic circuit, or any other type of circuit. In some embodiments, a typical computer system 403 includes a memory 404, one or more processors 402, a power supply, an input/output (I/O) subsystem, and a communication bus or interconnection of these components. The one or more processors 402 may execute software modules, programs, and/or instructions stored in the memory 404 to perform processing operations.

In some embodiments, memory 404 stores one or more programs (e.g., sets of instructions) and/or data structures. For example, in some embodiments, entanglement state generator 400 may attempt to produce entanglement states on successive stages and/or on separate instances, any of which may successfully produce entanglement states. In some embodiments, memory 404 stores one or more programs for determining whether a respective stage is successful and configuring entangled state generator 400 accordingly (e.g., by configuring entangled state generator 400 to switch photons to output when a stage is successful or to pass photons to a next stage of entangled state generator 400 when a stage is not yet successful). To this end, in some embodiments, the memory 404 stores a detection pattern from which the exemplary computing system 403 can determine whether the stage was successful. In addition, memory 404 may store settings provided to various configurable components (e.g., switches) described herein, which are configured by, for example, setting one or more phase shifts of the components.

In some embodiments, some or all of the above functions may be implemented using hardware circuitry on photon source system 405 and/or entangled state generator 400. For example, in some embodiments, photon source system 405 includes one or more controllers 407-a (e.g., logic controllers) (e.g., which may include Field Programmable Gate Arrays (FPGAs), application Specific Integrated Circuits (ASICs), a "system on a chip" including typical processors and memory, etc.). In some embodiments, controller 407-a determines whether photon source system 405 was successful (e.g., for a given attempt on a given clock cycle) and outputs a reference signal indicating whether photon source system 405 was successful. For example, in some embodiments, controller 407-a outputs a logic high value to representative channel 430-a and/or representative channel 430-c when photon source system 405 is successful and outputs a logic low value to representative channel 430-a and/or representative channel 430-c when photon source system 405 is unsuccessful. In some embodiments, the output of controller 407-a may be used to configure hardware in controller 107-b.

Similarly, in some embodiments, entangled state generator 400 includes one or more controllers 407-b (e.g., logic controllers) (e.g., which may include Field Programmable Gate Arrays (FPGAs), application Specific Integrated Circuits (ASICs), etc.) that determine whether the respective stages of entangled state generator 400 have been successful, execute the switching logic described above, and output reference signals to representative channels 430-b and/or 430-d to inform other components of whether entangled state generator 400 has been successful.

In some embodiments, the system clock signal may be provided to photon source system 405 and entangled state generator 400 via an external source (not shown) or by exemplary computing system 403 via exemplary channels 430-a and/or 430-b. Examples of clock generators that may be used are described in U.S. patent No. 10,379,420, the contents of which are incorporated herein by reference in their entirety for all purposes; other clock generators may be used without departing from the scope of the invention. In some embodiments, a system clock signal provided to photon source system 405 triggers photon source system 405 to attempt to output one photon per waveguide. In some embodiments, the system clock signal provided to entangled state generator 400 triggers or gates groups of detectors in entangled state generator 400 to attempt to detect photons. For example, in some embodiments, triggering a set of detectors in entangled state generator 400 to attempt to detect photons includes gating the set of detectors.

It should be noted that in some embodiments, photon source system 405 and entangled state generator 400 may have internal clocks. For example, photon source system 405 may have an internal clock generated and/or used by controller 407-a and entangled state generator 400 has an internal clock generated and/or used by controller 407-b. In some embodiments, the internal clock of photon source system 405 and/or entangled state generator 400 is synchronized (e.g., by a phase-locked loop) with an external clock (e.g., a system clock provided by typical computer system 403). In some embodiments, any of the internal clocks themselves may be used as the system clock, e.g., the internal clock of the photon source may be distributed to other components in the system and used as the master/system clock.

In some embodiments, photon source system 405 includes multiple probability photon sources that may be spatially and/or temporally multiplexed, so-called multiplexed single photon sources. In one example of such a source, the source is driven by a pump (e.g., an optical pulse) coupled into an optical resonator that can generate zero, one, or more photons through some nonlinear process (e.g., spontaneous four-wave mixing, second harmonic generation, etc.). As used herein, the term "attempt" is used to refer to an action of driving a photon source with some sort of drive signal (e.g., a pump pulse) that can non-deterministically generate an output photon (i.e., the probability that the photon source will generate one or more photons in response to the drive signal can be less than 1). In some embodiments, the respective photon source may be most likely to produce zero photons on the respective attempt (e.g., the probability of producing zero photons each attempt to produce a single photon may be 90%). The second most likely outcome of an attempt may be to produce a single photon (e.g., the probability of producing a single photon each time an attempt is made to produce a single photon may be 9%). The third most likely result of an attempt may be to generate two photons (e.g., the probability of generating two photons per attempt to generate a single photon may be about 1%). In some cases, the probability of generating more than two photons may be less than 1%.

In some embodiments, the apparent efficiency of a photon source may be increased by using multiple single photon sources and multiplexing the outputs of the multiple photon sources. In some embodiments, the photon source may also generate a typical announcement signal announcing (or announcing) successful generation. In some embodiments, such a typical signal is obtained from the output of a detector, where the photon source system always produces photon states in pairs (e.g., in SPDC), and the detection of one photon signal is used to announce the success of the process. The announcement signal may be provided to the multiplexer and used to route the successful generation appropriately to the multiplexer output port, as described in more detail below.

The exact type of photon source used is not critical and any type of source may be used that employs any photon generation process, such as spontaneous four wave mixing (SPFW), spontaneous Parametric Down Conversion (SPDC), or any other process. Other kinds of sources may also be used which do not necessarily require non-linear materials, such as sources employing atomic and/or artificial atomic systems, such as quantum dot sources, color centers in crystals, etc. In some cases, the source may be a photonic cavity or may be coupled to a photonic cavity, as is the case, for example, for an artificial atomic system such as quantum dots coupled to the cavity. Other types of photon sources exist for SPWM and SPDC, such as opto-mechanical systems and the like. In some examples, the photon source may emit a plurality of photons that are already in an entangled state, in which case entangled state generator 400 may not be necessary, or alternatively, may take the entangled state as input and generate an even larger entangled state.

In some embodiments, spatial multiplexing of several non-deterministic photon sources (also referred to as MUX photon sources) may be employed. Many different spatial MUX architectures are possible without departing from the scope of the invention. The temporal MUX may also be implemented instead of or in combination with spatial multiplexing. MUX schemes employing logarithmic trees, generalized Mach-Zehnder interferometers, multimode interferometers, linked sources with dump pumping schemes, asymmetric polycrystalline single photon sources, or any other type of MUX architecture may be used. In some embodiments, the photon source may employ a MUX scheme with quantum feedback control, or the like. An example of an n x m MUX source is disclosed in U.S. patent No. 10,677,985, the contents of which are incorporated herein by reference in their entirety for all purposes.

Fig. 3C illustrates one example of a qubit fusion system 501 in accordance with some embodiments. In some embodiments, the qubit fusion system 501 may be employed in a larger FBQC system (e.g., the qubit fusion system 305 shown in fig. 3A).

The qubit fusion system 501 includes a fusion network controller 519 coupled to a fusion array 521 (also referred to herein as a "fusion network"). The converged network controller 519 is configured to operate as described above and below with reference to the converged network controller circuit 319 of fig. 3 above. The fusion array 521 includes a collection of fusion sites, each of which receives two or more qubits from different resource states (e.g., as shown in fig. 4A) and performs one or more fusion operations (e.g., type II fusion) on selected qubits from the two or more resource states and/or performs selected single particle measurements to implement fault tolerant logic, as described in more detail below with reference to fig. 13-14. Measurement operations performed on the qubits may be controlled by the fusion network controller 519 via typical signals sent from the fusion network controller 519 to each fusion site via control channels 503a, 503b, etc. Based on the measurements performed at each fusion site, typical measurements in the form of typical data are output and then provided to a decoder system, as shown and described above with reference to fig. 3. Examples of photonic circuits that can be used as type II fusion gates are described below with reference to fig. 20 and 23-35.

Fig. 3D illustrates one possible example of a fusion site 341 (one of a plurality of fusion sites forming a fusion array 321) configured to operate with a fusion controller 319 to provide measurements to a decoder for fault tolerant quantum computing, according to some embodiments. In this example, fusion site 341 may be an element of fusion array 321 (shown in fig. 3), and although only one example is shown for purposes of illustration, fusion array 321 may include any number of examples of fusion sites 341. In some embodiments, quantum logic gates may be implemented by modifying fusion measurements. To allow the logic to be implemented (at least), a subset of the fusion devices may be reconfigurable, as indicated in fig. 3B, but others do not require reconfiguration. Boundaries or other topological features in the blocks may be achieved by changing the measurement basis of the fusion or by selecting a single qubit measurement instead of fusion, as described below with reference to fig. 13A-13D.

As described above, the qubit fusion system 305 may receive (at two or more inputs) two or more qubits (qubit 1 and qubit 2) to be measured according to the running quantum application. Qubit 1 incident on input 1 is one qubit entangled with one or more other qubits (not shown) as part of a first resource state, and qubit 2 incident on input 2 is another qubit entangled with one or more other qubits (not shown) as part of a second resource state. Advantageously, in contrast to MBQC, none of the qubits from the first resource state need to be entangled with any of the qubits from the second (or any other) resource state in order to be fault tolerant to quantum computing. It is also advantageous that at the input of the fusion site 341, the sets of resource states do not intertwine to form cluster states in the form of quantum error correction codes, and thus there is no need to store and/or maintain large cluster states with long-range entanglement across the entire cluster state. Also advantageously, in some embodiments, the fusion operation occurring at the fusion site may be a completely destructive joint measurement on qubit 1 and qubit 2, such that all that remains after the measurement is typical information representing the measurement results on the detector, e.g., measurement results 603, 605, 607, 609, etc. At this time, typical information is all that is required for the decoder 333 to perform quantum error correction. This can be contrasted with MBQC systems, which can use fusion sites to fuse resource states into clusters that themselves serve as topological codes, and only then generate the required typical information through the additional step of single particle measurement on each qubit in a large cluster. In such MBQC systems, not only is it necessary to store and maintain large cluster states in the system before single particle measurements are made, but an additional single particle measurement system (in addition to the fusion system used to generate the cluster states) must be present to receive each qubit of the cluster states and perform the necessary single particle measurements in order to generate the typical information needed to calculate the syndrome data needed by the decoder to perform quantum error correction.

FIG. 3D shows an illustrative example of one way to implement a fusion site as part of a fusion-based quantum computer architecture. In this example, qubits 1 and 2 may be double-track encoded optical qubits, but any type of qubit is possible without departing from the scope of the disclosure. A brief description of dual-track encoding of optical qubits is provided below with reference to fig. 26-29. Thus, qubit 1 and qubit 2 may be coupled to switches 621 and 623, respectively. In some embodiments, the coupling may be a waveguide and the switches 621 and 623 may be photonic switches. The various output channels of the switch may be coupled to different quantum bit measuring devices that enable different types of measurements. For example, single-qubit measurement device 625 (635) may implement measurement of the state of qubit 1 (qubit 2) in the X-base, single-qubit measurement device 627 (637) may implement measurement of the qubit in the Y-base, and single-qubit measurement device 629 (639) may implement measurement of the qubit in the Z-base. Likewise, the two qubit measuring devices 631 and 633 may implement two qubit measurements of different types, such as the projection bell measurement referred to herein at type II fusions. For example, measurement device 631 may implement XX-fusion and measurement device 633 may implement ZZ fusion or XZ fusion. Fig. 31-36 illustrate exemplary hardware that may be used to implement such a measurement device in accordance with one or more embodiments. In some embodiments, the state of the switches 621 and 623 references (basic) may be hard coded within the converged network controller 319, or in some embodiments, the references may be selected based on external inputs (e.g., instructions provided by the converged pattern generator 313 depending on the requirements of the running algorithm). The layout shown in fig. 3C is merely illustrative, and any number and combination of switches and single and/or multi-particle measurement devices may be used without departing from the scope of the present disclosure.

In some embodiments, for example, in a linear optical implementation, the fusion may be a probabilistic operation, i.e., it implements probabilistic bell measurements, where the measurements are sometimes successful and sometimes failed, as described below with reference to fig. 35. In some embodiments, the probability of success of such an operation may be increased by using additional quantum systems in addition to the quantum systems on which the operation acts. Embodiments using additional quantum systems are often referred to as "enhanced" fusion. In the example shown in fig. 3C, the fusion site implements an unenhanced type II fusion operation on the input qubit. Those skilled in the art will appreciate that any type of fusion operation (and may be enhanced or not) may be applied without departing from the scope of the present disclosure. Additional examples of type II fusion circuits are shown and described below for polarization encoding and dual track path encoding. In some embodiments, the converged network controller 319 may also provide control signals to the measurement devices 625, 627, 629, 631, 633, 635, 637, 639, etc. Control signals may be used, for example, for gating the measurement hardware (e.g., photon detectors) or for otherwise controlling the operation of the hardware. Each measurement device provides a measurement signal (603, 605, 607, 609, etc.), and the signal may be pre-processed at the fusion site 341 to determine the measurement (e.g., success or failure of fusion, which feature value is measured, how many photons are detected, etc.) or passed directly to the decoder 333 for further processing.

According to some embodiments, a fault tolerant quantum computer architecture is disclosed. In some examples, fault tolerant linear optical quantum computers are described that can be fabricated in silicon photonic platforms. Linear optical methods for quantum computing are advantageous for the following reasons, including: (i) Highly coherent qubits and high fidelity single qubit operations can be implemented using well known quantum optical methods; (ii) Silicon photons are manufacturable, providing a way to scale to a large number of qubits; (iii) All required operations-state preparation, gating and measurement-can be performed quickly, resulting in high gating speeds; and (iv) the main source of noise is optical loss, which allows for more efficient error correction, since the location of the error is known.

In linear optics, two qubit gates cannot be realized deterministically because photons do not interact. Only operations with a success probability of less than 1 can be used to generate entanglement. Moreover, single photon sources used to prepare the qubit states may not operate deterministically. Overcoming this limitation results in overhead relative to schemes with deterministic double-qubit operation. This overhead does not increase with increasing computational scale. In this sense, the overhead associated with non-deterministic operation in linear optics is much smaller than the overhead of quantum error correction, which is a slow growth for larger computations.

According to some embodiments, an architecture is disclosed that is tolerant of relatively frequent failures in entangled operations, thereby greatly reducing the overhead of non-deterministic operations relative to other LOQC architectures.

In some aspects, it is an advantage that photons do not readily interact. This limits the possibility of so-called quantum crosstalk, in which case the qubits may become entangled unintentionally. In many other quantum computing methods, this effect is an important source of noise.

According to some embodiments, systems and methods for fault tolerant quantum computing, referred to herein as fusion-based quantum computing (FBQC), are disclosed. In this method, a specific, relatively small entangled state, referred to as a resource state, is produced. The computation is then performed by selecting measurements to be performed on pairs of qubits from different (i.e., unentangled) resource states. As described in further detail below, the measurement may be a linear optical fusion measurement, hence the name fusion-based quantum computation.

FBQC should not be confused with measurement-based quantum computing (MBQC) methods. The MBQC method involves very large entangled resource states, called cluster states, with the number of clustered qubits increasing with the expected number of logical qubits and with the expected number of gate operations in the computation. In MBQC, a single qubit measurement of the clustered state is used to perform the computation. In FBQC, the size of the resource state is not dependent on the number of logical qubits or the number of gates in the computation. As used herein, we refer to a resource state that has a size that is neither dependent on the number of logical qubits nor on the number of gates in the computation as a resource state having a fixed (or constant) size. In FBQC, computation is performed by performing two qubit measurements on qubits belonging to two different (i.e., unentangled) resource states of fixed size.

In a linear optical implementation, the fusion operations are probabilistic and when they fail means that some results of the fusion measurements are not obtained. In FBQC, quantum error correction may be used to handle these lost measurements, since quantum error correction codes can handle such lost measurements well, referred to herein as "erasure".

The most efficient photonic architecture in the current academic literature is based on MBQC and uses fusion to create clusters. The impact of fusion failure is addressed by using the fact that such failure results in the absence of qubits in the desired cluster state. The results of the percolation theory (percolation theory) are used to ensure that if fusion failure is rare enough, the residual cluster has very large connecting members available for MBQC. This seepage-based architecture has serious drawbacks compared to FBQC, including having to calculate the path through the remaining clusters in real time for each logic gate, which would be very challenging and in practice the threshold for this scheme is very low.

The use of a quantum error correction code to compensate for the unavoidable probabilistic linear optical error correction operation allows for very high quantum error correction thresholds in FBQC without the need for a decoder that can also implement the re-normalization calculation of the calculation requirements necessary for the percolation-based approach. The percolation-based scheme requires more complex decoders that have to find paths in the percolation cluster. According to some embodiments, the FBQC architecture may have a physical size (footprint) many orders of magnitude smaller than a percolation-based photonic architecture or an alternative to process probabilistic linear optical operations via gate transfer with very large additional states or "repeat until success" approaches.

In some embodiments, implementing FBQC involves adaptively selecting the ability of each measurement in response to the results of previous fusion measurements. This adaptation may be achieved using typical logic and appropriate switchable elements to move each qubit to an appropriate measurement device (e.g., as described above with reference to fig. 3D).

FBQC combines many advantages. For example, it is advantageous that each single photon in the FBQC encounters only a small, fixed number of optical elements between the source and detector, regardless of the size of the calculation being performed. Such "constant depth" features result in significantly reduced losses relative to other architectures, as each optical element increases the probability of loss. More specifically, photons comprising resource state qubits are measured immediately in a subsequent fusion. The number of optical elements passing photons in the resource state generator depends on the resource state and the method used to generate it, but not on the calculation to be performed.

Because each photon passes through a small fixed number (which may be, for example, 5 or less) of optical elements, and this number remains unchanged as the calculated size increases, the time scale used to generate and detect photons is completely separated from the longer time scale required to achieve non-trivial logic operation or run the decoder. This means that the decoder does not need to be co-located with the rest of the computer, which is advantageous for architectures employing low temperature operation of the quantum elements, as the decoder does not need to be co-located in the cryostat.

Advantageously, the FBQC is consistent with the planar architecture of the computer. In this architecture, most fusion measurements are between qubits that are adjacent to each other in the chip plane. Advantageously, the planar architecture makes implementation practical in silicon photonic chips or any other planar integrated circuit approach using non-optical qubits.

Advantageously, FBQC is flexible enough to implement many different approaches to quantum error correction and fault tolerant logic gates. According to some embodiments, a number of existing tools for fault tolerant quantum computing using surface codes may be employed in FBQC.

According to some embodiments, qubit encoding may be used, where a qubit is a single photon in a certain time-bin in a given transverse mode of one of the two waveguides. This is called dual-rail coding. Variants that encode qubits in one of two time-entanglement traveling in the same waveguide or fiber are also useful. This is called time entanglement encoding.

In double track encoding, each qubit has one photon, and in FBQC, all qubits are measured. The loss of photons results in less than the expected number of photons detected that provide an error signal that has occurred. Advantageously, for the FBQC method disclosed herein, the tolerance of the surface code to such errors is much higher than for errors that are not predicted in this way.

Another advantage of the optical implementation of FBQC is the ability to use optical fibers to create long delays between the resource state generator and the fusion measurements. This enables qubit fusion of not only nearest neighbors in the photonic chip plane.

III. FBQC architecture examples

According to some embodiments, FBQC may be based on two primitive (primary) operations: small constant-size entanglement resource states and projection entanglement measurements are generated, which we will refer to herein as fusion.

FBQC is applicable to many physical systems and is particularly relevant to architectures where multiple quantum bit projection measurements are a local operation. One or more embodiments implement FBQC in linear optical quantum computing. In the examples disclosed herein, the fault tolerance threshold for fusion failure was demonstrated to be 24% (compared to 14.9% reported previously).

FBQC principle

In FBQC, a converged network defines the configuration of converged measurements to be made on qubits of a set of resource states. The fusion network forms a computational structure upon which an algorithm may be implemented by modifying the basis of at least some of the fusion measurements. The fusion measurements are properly combined to give a calculated output. Fig. 4A shows an example of a two-dimensional fusion network. In general, there is no requirement for any particular architecture in the converged network.

Constructing a converged network involves two basic primitives. The first is resource state generation, which describes creating small entangled states. These states have a fixed size and structure, regardless of the size of the computation they are to be used to implement. The resource states may be of any size, and the particular form of the resource states is not generally critical to FBQC, but rather is a design parameter that a quantum engineer handles given a particular qubit architecture and noise model. In some embodiments, the resource state generator means generates a copy of the resource state over some period of time (referred to herein as a "clock cycle"). The resource state generator may physically take a variety of forms: for example, it may be a device that produces entangled photon states, or it may be a substance-based device.

The second primitive is a fusion measurement, which is a projection entanglement measurement over multiple qubits. In some embodiments, the fusion measurement may be implemented by a fusion device having n input qubits, which outputs n typical bits that give the measurement result. For example, the result X from Bell measurements on two qubits ₁ X ₂ And Z ₁ Z ₂ . At least some of the fusion devices (or generated resource states) must be reconfigurable so that at different time steps, the projection measurements they make can be changed to accommodate the computational intent of the FBQC, i.e. in order to run the quantum application.

The physical implementation of the fusion will depend on the underlying hardware. In a linear optical system, fusion can be achieved locally by performing interferometric photon measurements comprising different resource states, which are equivalent only to a proper configuration of beam splitters and photon detectors, a more subtle implementation is also possible in order to increase the probability of success and robustness to hardware defects.

Other methods for quantum computing also employ entanglement measurements performed throughout the computation. In particular, syndrome extraction can be understood as a joint measurement on an entangled basis in a fault-tolerant circuit diagram. In topological quantum computing, combining arbitrary charge projections is necessary to extract typical results from the system and can be the basis for implementing general quantum computing. Redundancy of fusion measurements can be used to naturally adapt to the constant density of syndrome extraction required to mitigate entropy accumulation.

B. Architecture for a computer system

FBQC provides a natural framework for studying fault tolerance of a given resource state and fused primitives, but its advantages translate into significantly simplified physical architecture requirements. Furthermore, for resource state generation and fusion, we can also explicitly identify the third component, i.e., the fusion network router (router) that allows the former two to work in conjunction, by properly routing qubits from the resource state to the fusion measurements. The converged network router provides the greatest advantage for linear optical implementations because the integrated waveguide and optical fiber allow direct and low loss routing of optical qubits over a large distance, while other substance-based approaches require coherent optical-substance coupling, which proves to be only at relatively low fidelity.

Given a converged network there are many possible architectural implementations, for example for a 3D converged network we can choose to create all resource states simultaneously, or conversely we can create one 2D layer at a time, reusing the resource state generator to create a new copy of the state in each clock cycle. The architecture design is captured by a converged network router that delivers qubits created at different spatial and temporal locations (i.e., from different resource state generators and temporal entanglement) to the corresponding converged locations. Thus, the converged network router includes both spatial and temporal routing in the form of delay lines. Fig. 4B shows an illustrative example of an architecture for FBQC that creates a 2D converged network from a 1D array of resource state generators.

In some fault tolerant converged networks, the converged network router implements a fixed routing configuration. Fixed routing means that the qubits generated from a given resource state generator will always be routed to the same location. Such design features are particularly attractive from a hardware perspective and have many practical implications. That is, it minimizes the need for a possibly error-prone handover and reduces the burden of typical control.

Another key feature of the FBQC architecture, and aspects that distinguish it from other methods, is the time scale separation for typical control. As shown in fig. 4B, feed forward control may be implemented at the logic level to process and decode the measurement results and then affect future logic operations. However, this time scale may be several orders of magnitude longer than the clock period of the resource state generation and fusion, and no typical computation or feedback is required on this shorter time scale. In other words, the physical qubit does not need to wait in memory, but rather the computation runs to decide how it should be measured.

IV, fusion

In FBQC, the initial quantum resource is a small entangled resource state of fixed size. When we measure qubits from different resource states, a large scale quantum correlation required for general purpose computation is generated. For this purpose, long-distance entanglement is produced, and at least some of the measurements need to be entangled, i.e. projected onto a subspace containing at least one entangled state.

In general, the measurement may be any positive operator value measurement (positive operator valued measure, POVM), but to achieve fault tolerance, it is helpful to consider that all results are measurements on steady state projections. This makes it straightforward to use existing stabilizer fault tolerance methods. In the examples herein we focus particularly on the case of two qubit measurements, which are bell-state projections, which we next call bell fusion. Bell fusion is based on X ₁ X ₂ ,Z ₁ Z ₂ The input qubit is measured in the stabilizer.

In general, we will study a fusion network in which the vast majority of fusion measurements required to achieve quantum error correction are identical bell measurements. However, to implement logic gates, some parts of the fusion measurement need to be different from others. There are many variations in how this can be achieved, either using two qubit measurements on a modified stabilizer basis, or by including a single qubit measurement. This will be discussed in detail below.

C. Fusion in linear optics

In linear optical quantum computing (linear optical quantum computing, LOQC), fusion on pairs of optical qubits is performed simply, but entanglement occurs indefinitely. This non-certainty means that sometimes the desired measurement results cannot be obtained, and advantageously one or more embodiments of the architecture of the LOQC find a way to bypass this missing information. In the FBQC scheme, we describe here that these fusion failures are corrected directly by quantum error correction.

In the examples we have studied here we consider in particular a "double-track" qubit consisting of a single photon in two photon modes. Photons in the first mode represent logic |0 >While photons in the other mode represent logic |1>. Such qubit encoding is attractive because it is lost, takes the qubits from the computation subspace, and is therefore predicted. Bell fusion on double-track qubits can be achieved using a linear optical circuit, where all four modes of two qubits are measured. This is commonly referred to as type II fusion. Having 1-p _fail Fusion "success" based on X as measured by expectation ₁ X ₂ ,Z ₁ Z ₂ Input qubits in a bell stabilizer. With probability p _fail In which case it performs a separable single qubit measurement Z ₁ I ₂ ,I ₁ Z ₂ . If there is a possibility of photon loss or other defects, there is a third possible outcome: fusion "erase". In this case, the expected stabilizer results were not measured. Fig. 5 shows different possible measurements of linear optical fusion on two qubits.

Fig. 5 shows the result of linear optical bell fusion. Showing the data from two clustersQubit fusion with expected result X ₁ X ₂ And Z ₁ Z ₂ . In the presence of photon losses, there are three possible outcomes: obtaining a fusion success of the two measurements, obtaining only result Z ₁ Z ₂ And fusion erasure without obtaining a measurement result. Fusion failure is inherent to linear optics and can occur even if all operations are ideal. Fusion erasure occurs only due to errors in the system, most often the loss of one or more photons into the fusion measurement.

The fault in the linear optical fusion is a more benign error than erasure because it is predictive and does not lead to a mixed state because we still obtain a pure stabilizer measurement. Two desired results Z ₁ Z ₂ Can be obtained by multiplying two single qubit measurements. Thus, fusion failure can be considered as where X ₁ X ₂ Bell measurements with erased measurement results.

The simplest method to achieve type II fusion involves only two beam splitters and four detectors, and has p _fail =50% failure probability. With additional bell pairs, fusion can be "enhanced" to suppress failure probability to 25%, and fusion success can be further enhanced by using more auxiliary photons. By performing fusion on the encoded qubits, tolerance to photon loss and physical fusion failure can be improved. This method is used in the following example, where physical qubits are encoded using (2, 2) shell codes, and the encoding fusion is achieved by performing the physical fusion across. Next we describe how to suppress erasures in the code fusion and calculate the erasure probability from the measurements of the code fusion in the presence of photon losses and fusion failures.

To implement logic, the measurement basis of linear optical fusion can be directly changed by placing a single qubit rotation before its input. These single qubit gates can be implemented with high precision using beam splitters and phase shifters, which can be implemented directly in an integrated photonic chip. The small switching network prior to convergence allows reconfigurability between different measurements, as will be discussed further below.

V resource state

In FBQC, the small entanglement state used for excitation computation is called a resource state. Importantly, their size is independent of the calculation performed or the encoding distance used. This allows them to be generated by a constant number of sequential operations. Thus, errors in the resource state are borderline, which is important for fault tolerance.

As with fusion, we will focus on the resource state of the quantum stabilizer. This state can be described by a graph G using a graphical representation up to a local Clifford operation, wherein the described quantum state |g>By placing qubits into | + at each vertex>And performing a controlled-Z gate between the qubits for which the corresponding vertices in the graph are adjacent. Equivalently, n stabilizer generators with graphics states from 1 to n labeled vertices are represented by Give, wherein->Is the set of adjacent vertices of vertex i in G.

FIG. 6 illustrates an example of resource states represented as graph states, in accordance with some embodiments. Fig. 6A shows an example of resource states in the form of 6-ring graph states. With the qubits marked in the figure, the stabilizer for the resource state is Z ₆ X ₁ Z ₂ ,Z ₁ X ₂ Z ₃ ,Z ₂ X ₃ Z ₄ ,Z ₃ X ₄ Z ₅ ,Z ₄ X ₅ Z ₆ And Z ₅ X ₆ Z ₁ . Fig. 6B shows that the qubits in the resource states can be replaced with the depicted transform to the resource states of the (2, 2) shell code. Qubit 1 and qubit 2 both have an AND relationship withAdjacent qubits (drawn as dashed circles) to the left where the uncoded qubits are identical. Qubits with H inside have hadamard applied to them with respect to their graphical representation. Fig. 6C shows the resource state in fig. 6A, where each qubit is encoded with a (2, 2) shell code.

The stabilizer for resource state is Z ₆ X ₁ Z ₂ ,Z ₁ X ₂ Z ₃ ,Z ₂ X ₃ Z ₄ ,Z ₃ X ₄ Z ₅ ,Z ₄ X ₅ Z ₆ ,Z ₅ X ₆ Z ₁ . By following the transformation shown in fig. 6B, the (2, 2) shell code can encode the resource states. Replacing each qubit of the 6-ring with a (2, 2) shell encoded qubit gives the resource state depicted in fig. 6C.

The operation for creating the resource state depends on the physical platform used for the process, which may be different from the physical platform used to implement the converged network, so long as the generated resource state qubits are compatible with the converged network. For example, in solid state qubits, resource states may be generated using a single entanglement gate or dissipation. When using linear optics, the generation of resource states is achieved by performing a series of projection measurements (e.g. fusion) on even smaller entangled states, such as the bell state and the 3-GHZ state (which we sometimes refer to as seed states). Methods for generating seed states are fully covered. Since projection entanglement measurements in linear optics are probabilistically successful, as described above, it is often advantageous to use a switching network between fusions to increase the probability of success of the protocol. Using these networks we try probabilistic operations multiple times and only select the case of success. In this sense, multiplexing is used to effectively approximate the post-selection of entanglement fusion results. Since the size and number of probabilistic operations required to generate resource states are fixed, the overhead from repeating probabilistic operations is also fixed. Many options for implementing such a switching network can be found therein, depending on the efficiency and available devices required. Notably, while the resource states are required to be qubit states, i.e., states with multiple entanglement between well-defined qubits, the states obtained at intermediate stages of generating the resource states need not follow this limitation.

Determining the most appropriate resource state is part of the FBQC scheme design for practical hardware implementations, as the noise distribution of the resource state will depend on the generation protocol used. For a given target resource state, there are a large number of possible preparation protocols, each of which will result in a different noise distribution. However, the fixed size of the resource states means that any generation protocol will require a limited number of operations, and thus the noise accumulated in any state generation will be limited. In addition, any error correlation resulting from the independent state generation will be local to that state, which limits the propagation of errors in the converged network, and is discussed below.

VI, converged network

A Fusion Network (FN) specifies the resource states used in the FBQC protocol and how they are connected through fusion. After fusion measurements are made, two types of information are retained: typical information from the measurement results, and some (potential) quantum correlation corresponding to unmeasured qubits. These measurements contain correlations that are "results" of the converged network, providing a computational output for us, or in the case of a fault tolerant converged network, parity check that can be used for error correction. In this section we describe how to build fusion networks and how to analyze them to identify quanta and typical correlations that exist after fusion measurements are made. We focus particularly on stabilizer fusion networks where the resource state is the stabilizer state and the fusion measurement is the stabilizer projection. This allows us to use existing fault tolerant tools.

The stabilizer fusion network can be characterized by two Pauli subgroups: (1) A stabilizer group R describing ideal resource states and (2) a fusion group F, which is a Pauli subgroup defining fusion measurements, where we includeAssuming complete fusion, we will learn the eigenvalues of all operators in F by implementing a fusion network. Because the individual measurement results of the fusion operator are dependentAnd so-1 is included. A fusion stabilizer, which by definition should have a "+1" result, corresponds to a consistent symbol element of F. After fusion measurements are made, we can describe the remaining system by surviving stabilizer sets.

Which is the centromer (centraliser) of F in R. The stabilizers of F and R are not all interconnected, so after fusion measurements are made, only some subset of the original stabilizers remain: surviving stabilizers. These surviving stabilizers contain both "measured" qubits (where the remaining information is purely typical) and not yet measured qubits (where quantum correlation is preserved). Any remaining qubits are output by the output stabilizer group S _out The output stabilizer group S is described _out Is a limitation on the remaining qubits of S until the sign can be determined from the particular fusion result. That is, calculate S _out The sign of the stabilizer in (c) is such that when the element of the fusion set F is multiplied by a sign compatible with the obtained measurement results, an element of S is produced.

In the fault tolerant converged network example disclosed herein, each qubit in the network is measured and there are no remaining qubits. The computation in FBQC uses correlations between fusion measurements generated by the fusion network structure.

(a) Examples of converged networks with three resource states: two qubit patterns and two copies of three qubit linear patterns. There are two fusions, shown as orange line, both of which measure operators<XZ,ZX>. Specifically, the resource state composed of qubits {1,2,3} is composed of<Z ₁ X ₂ Z ₃ ,X ₁ Z ₂ I ₃ ,I ₁ Z ₂ X ₃ >Stable and similar for {6,7,8 }. If each measurement in the converged network is successful and returns a +1 eigenvalue, then the qubit {4,5} is represented by<X ₄ Z ₅ ,Z ₄ X ₅ >The stability of the product is improved, the product is stable,unmeasured qubits {1,2,7,8}, are represented by<X ₁ Z ₂ ,Z ₁ X ₂ Z ₇ ,Z ₂ X ₇ Z ₈ ,Z ₇ X ₈ >Stable, which corresponds to the 4-line pattern shown in (b).

Fig. 7A shows a simplified example of a converged network, which results in the state in fig. 7B, up to the result dependency stabilizer symbol. The set of resource states R is generated from a combination of stabilizers of different resource states, which can be inferred from their graphical representations. R= <X ₁ Z ₂ ,Z ₁ X ₂ Z ₃ ,X ₄ Z ₅ ,Z ₄ X ₅ ,Z ₂ X ₃ ,X ₆ Z ₇ ,Z ₆ X ₇ Z ₈ ,Z ₇ X ₈ >. The fusion group is generated by the union of all fusion measurement operators, i.e. F =<X ₃ Z ₄ ,Z ₃ X ₄ ,X ₅ Z ₆ ,Z ₅ X ₆ ,-1>. Typical information generated by the converged network is derived from the measurement results of F. Furthermore, the fusion network is leaving quantum information about unmeasured qubits with stabilizers S _out ＝<±X ₁ Z ₂ ,±Z ₁ X ₂ Z ₇ ,±Z ₂ X ₇ Z ₈ ,±Z ₇ X ₈ >Depicted by the pattern in fig. 7B. S is S _out The sign of (a) depends on the fusion measurement.

There are aspects in which some FBQC architectures are not covered by converged networks. In particular, converged networks do not capture converged time sequences, physical qubit routing, or typical processing requirements. The converged network does not specify the order of convergence: they may be executed in the order best suited to the underlying hardware. Furthermore, the resource states involved in the converged network need not all exist at the same time, and their generation may be staggered, so long as the converged measurements can be made on all necessary qubit pairs.

Below we describe how redundancy is added to the converged network for fault tolerance.

VII fault tolerant converged network

The fusion network may be constructed to be fault tolerant such that, as long as errors occur with a sufficiently low probability, errors in the resource state or noise fusion circuit that result in fusion measurements that differ from their ideal values may be corrected. In this section we describe fault tolerance in a converged network.

The fault tolerant converged network (fault tolerant fusion network, FTFN) may be constructed by circuit-based quantum error correction or fault tolerant cluster-based states. This approach may be a useful initial guideline, but direct conversion generally yields an inefficient solution, and a better solution can be found by working more directly in the converged network diagram, as we show in the examples herein.

A. Stabilizer form for FTFN

When modeling a physical error, whether from resource state preparation or from fusion metrics, we interpret it as occurring in space where all resource states are ready and before fusion is performed. In this figure, the key to fault tolerance is the redundancy between Pauli operator F and stabilizer R of resource states measured during fusion. This redundancy is reflected in the existing check operator set

C:＝R∩F.

In other words, the check group C corresponds to a subset of the stabilizers R on the resource state provided by the fused measurement group F. In the absence of errors, the fusion result should be compatible with all operators in C that have positive eigenvalues, i.e. the typical fusion result forms a typical linear binary code that allows correction of them. The error correction process is described below.

The undetectable error group is defined by the centralized sub-definition of check group C over the entire Pauli group

As the name suggests, this subset of Pauli operators, if applied to the qubits prior to fusion of another ideal process, leaves no trace on the check operator result. However, not all undetectable errors are problematic for computation, as some errors do not affect the final correlation. For example, any element in R or F will have no detrimental effect when the element in R leaves the resource state invariant and the element in F leaves the fusion invariant and can be absorbed into the ideal state preparation or fusion measurement, respectively. More generally, the tiny undetectable error group is

Including the definition < R, F >. Thus, undetectable errors can be classified by elements of quotient U/T, where S is a surviving stabilizer group defined below. Two errors are considered to be different if they result in different syndromes or have different logical operations. Instead, if they are distinguished only by the elements of T, they are equivalent in all relevant ways. In this way, the equivalence fault corresponds to an equivalence class P/T.

Fig. 8A-8C illustrate examples of converged networks and explicit definitions of these groups. All fusion measures input qubits on an XX, ZZ basis. When all measurements are successful, the set of output stabilizers S _out ＝<±X ₁ X ₈ ,±Z ₁ Z ₈ >I.e. the converged network generates bell pairs. Fig. 8A shows an example of a converged network of resource state groups R, converged groups F, and check groups C, which may be explicitly defined in fig. 8B. FIG. 8B illustrates a set of resource state group generators that are a union of stabilizers of different resource states from which the stabilizers can be inferred. The green fusion measures the input qubits on an XX, ZZ basis. Fusion groups were generated from all fusion measurements and-1. Generators from different resource states in R and different fusions in F are ordered by column. When all measurements are successful, the set of output stabilizers S _out ＝<±X ₁ X ₁₆ ,±Z ₁ Z ₁₆ >I.e. the converged network generates bell pairs. S is S _out Will depend on the particular measurement. Fig. 8C shows a check sub-graph from the converged network (a), wherein the measured values correspond to each edge of the label. Will be adjacent to checkIs multiplied by the measured value of (C) to obtain the same generator s=as for C in table (b)<X ₁ Z ₄ Z ₁₃ Z ₁₆ ,Z ₁ Z ₂ Z ₃ X ₄ Z ₅ X ₆ Z ₇ X ₈ X ₉ Z ₁₀ X ₁₁ Z ₁₂ X ₁₃ Z ₁₄ Z ₁₅ Z ₁₆ ,C>。

Due toIt is never necessary to distinguish between different errors, which correspond to the elements of F. Therefore, we choose to represent the decoding problem with the elements of P/F (i.e., the quotient of the entire fusion group and Pauli group). While the different elements of P/F may correspond to equivalent errors (equivalent classes of fully reduced P/T according to), such a reduction has the advantage of preserving a large number of locality structures in the error model. In particular, a single qubit Pauli error on a resource state is interpreted as a measurement error on one or more generators corresponding to F. When F consists of a Bell fusion measurement, the quotient P/F identifies a single pair of qubits Pauli in the P that jointly generates the F element. Thus, we can choose to represent the decoding problem for P/F with a direct correspondence to specifying which fusion results are flipped. For example, in the example of FIG. 4, a single qubit error X ₄ And X ₁₃ Is an equivalent error when they are multiplied by X ₄ X ₁₃ When it is an element of F. At Z ₄ Z ₁₃ Until this equivalence, the error can be characterized by the flipped F generator (fusion result).

The most advantageous type of error is the trivial error of the element corresponding to T. However, they may have a non-trivial representation in P or in terms of flipped fusion results. For example, if measurement result X ₂ X ₅ And X ₁₀ X ₁₄ Both are flipped (i.e. possibly due to error Z ₂ Z ₁₀ ) The verification is not affected, thus Z ₂ Z ₁₀ E U is an undetectable error. The error Z ₂ Z ₁₀ Also communicates with the remaining generators of S, which means that itIs also a trivial error (i.e. Z ₂ Z ₁₀ E T) and does not affect the predicted sign of the output stabilizer. Resulting in X ₂ X ₅ The physical error of the flip is insignificant and X ₁₀ X ₁₄ Is Z ₂ Z ₁₀ Or any other operator in T. In fact, in an idealized setting, X ₂ X ₅ And X ₁₀ X ₁₄ Any combination of results is equally possible and is used in extracting the check operator and calculation for the output stabilizer S _out Only their combined parity check (XOR) is used when symbols are taken.

The worst error is a non-detectable, non-trivial error. This is for X ₄ X ₁₃ The case of a flip of results, which may be made up of a single qubit error belonging to U but not to T (e.g. Z ₄ Or Z is ₁₃ ) And (3) generating. That is, while these errors are in communication with the check operator in C and thus undetectable, they form a return communication (anti-communication) with one of the other generators in S. Thus, they will result in + -Z on the output stabilizer ₁ Z ₆ Is a misprediction of the symbols of (a).

For fault tolerant converged networks, the most interesting case is a detectable error. Z is Z ₄ Z ₁₃ Corresponding to a detectable error (i.e) As it would result in inconsistencies in the two check generators in fig. 4B. For such detectable errors, the decoder operates to predict the sign of the external stabilizer based on the most likely type of physical error consistent with the detection pattern. If the decoder correctly guesses the error category, all logical results will be correctly recovered.

For fault tolerance we focus on converged networks where the weight of non-trivial undetectable errors (also called the distance of encoding) increases with the size of the network. Some examples of such networks are discussed below. In large periodic networks, which we handle later, it is inconvenient to explicitly write the full resource state group R and the fused group F as done in fig. 4. Instead, we only specify a single resource state stabilizer and a stabilizer measured in a single fusion; by repeating the same stabilizer for different sets of qubits, resource states and fusion group generators can be obtained. Rather than explicitly writing the parity group, the parity is more conveniently represented graphically using a "parity check graph" discussed below.

B. Local and topology FTFN

We can introduce a localized FTFN concept that is entirely similar to other similar structures in error correction. That is, if it has a local check operator generator (i.e., each generator involves a bounded number of fuses, and each fuse involves a bounded number of generators), then the stabilizer FTFN family is local, with a fuse network in the family for any integer d, such that the nontrivial undetectable errors support at least d qubits.

The usual combination parameters may then be applied to indicate the presence of an error threshold. For example, if we employ an error model (where our fusion may be erased or randomly flipped), for any given local FTFN, there will be sub-threshold regions in the plane defined by the erase and flip rates, so that FT can be achieved. That is, as long as the combination of erasure and flip rates is within the sub-threshold region, we can achieve any desired logical error rate by building a sufficiently large converged network.

An important class of local FTFNs is topology convergence networks. Using the circuit diagram as a reference, a (large part of) topology fusion network can be regarded as an FT error correction process of simulated topology coding. Thus, for 2D topology encoding, a 3D topology fusion network can be obtained, wherein the elements of the surviving stabilizer group S take the form of a film (word lines of string operators) and the undetectable errors in U take the form of closed strings (word lines of topological charge). Although there is said equivalence between these 3D topology fusion networks and 2D topology coding, the 3D topology fusion networks are not constrained to represent layered codes and can support the more general measurement-based fault-tolerant framework described. Examples of toric code based topologies FN are discussed below.

C. Check subgraph

The redundancy of typical codes associated with fault tolerant fusion networks is often well described by a check sub-graph representation, which makes it simple to apply existing decoders (e.g., minimum weight matching and joint lookup decoders) to the FBQC framework. Another advantage of the syndrome graph is that it allows graphically designing fault tolerant schemes with higher thresholds. It should be noted, however, that not all schemes may naturally be represented by a check sub-graph structure.

A check sub-graph is a graphical representation of a typical linear code through multiple graphs, where vertices correspond to check generators and edges correspond to variable nodes. We will use them in such a way that each vertex represents a check operator in C and each edge represents a generator of the fused set F (or more precisely an error therein). If the corresponding generator F is used in the factorization of the corresponding check operator in C, then the edge will be connected to the vertex. Note that given the set of independent generators for fusion group F, each element of check group C has a unique factorization based thereon. Further, note that this depends on the fault tolerant fusion network and the choice of producers for C to ensure that each edge is connected to at most two vertices (i.e., each fusion producer is used in at most two check producers). This is possible in the topology FBQC, which is a representation of the fact that the error chain leaves only non-trivial syndromes at its endpoints.

The parity of the check operator is evaluated by taking the joint parity of all the measurement results that constitute the check. Given a set of fusion measurements, each parity has an associated parity value +1 or-1. All of these parity result configurations are referred to as syndromes. If the fusion result is flipped, then the check incident on its edge in the graph will flip its parity value. If the fusion result is deleted or lost, the two checks incident on the graph edge may be multiplied/combined into a single check operator. The check subgraph may have "dangling edges" where an edge is connected to only one check vertex, in which case the check/vertex is deleted if the fusion result is deleted or lost. In some cases, multiple edges may also exist between two check nodes when multiple fused measurements contribute to the same pair of checks.

Fig. 4C shows how the converged network example is represented as a check sub-graph. In this simplified example there are two check operators, each with four incident edges. One edge is shared by two check operators, connecting two vertices. The other edges are "dangling edges" that connect to only a single check node. In this small example, not all fusion results are included in the check operator, but in the topological fault tolerant fusion network we present in the next section, all fusion results will be part of at least one check operator. In particular, we will consider a topological converged network based on surface codes. As with any surface code construct, these networks may be represented by a partially broken original subgraph and a double subgraph. In most cases, each bell fusion contributes two measurements, corresponding to one edge of the original subgraph and one edge of the dual subgraph, respectively.

VIII fault tolerant converged network examples

In this section we describe two explicit examples of fault tolerant fusion networks implementing surface code error correction. These examples provide a simplified illustration of how fault tolerance is implemented in the FBQC framework. They were chosen as useful teaching examples, not the best FBQC architecture. However, even with these examples, we demonstrate significant performance improvements.

"4 star" converged network

According to some embodiments, a "4 star" converged network is shown in fig. 9. The resource state is with stabilizer X ₁ Z ₂ Z ₁ X ₂ 、Z ₁ Z ₂ Z ₃ Z ₄ ,X ₁ X ₂ I ₃ I ₄ ,I ₁ X ₂ X ₃ I ₄ And I ₁ I ₂ X ₃ X ₄ Greenberger-Horne-Zeilinger (GHZ) state. For clarity of the figure, we represent this resource state as a 5-qubit star-shaped figure with the central qubit masked (FIG. 9A), because of thisThe states are obtained when the central qubit of the 5-star pattern is measured on the X basis of the "+1" result. The 4-GHZ state can be created directly without preparing a 5-qubit physical resource state. For example, using linear optics, the described circuit can be used to produce 4-GHZ states from single photons. Four purple circles, corresponding to qubits in the resource state, are input to the fusion in the network. As shown in fig. 9A and 9D, a fusion network can be constructed from cubic unit cells, with resource states placed on each face and side of the unit cell. The resources are aligned either parallel to the face or perpendicular to the edges. As shown in fig. 9B, fusion measurements are made on pairs of qubits from resource states centered on the unit cell face and pairs of qubits from resource states centered on adjacent edges (if the central qubit in the resource state is not measured, then fusion will result in a cluster state used in the MBQC implementation of the surface code). Each fusion attempts to measure stabilizer operator X ₁ Z ₂ And Z ₁ X ₂ As shown in fig. 9C.

The converged network generates the check sub-graph shown in fig. 9E: a cubic lattice, wherein each side is a four-way polygon corresponding to four measurements. There are a total of 24 fusion measurements, all of which are used to evaluate each check operator. The fusion network is symmetrical in all three dimensions, shifted by half the lattice constant. This means that the original and dual subgraphs are identical.

Since the syndrome is used across hardware implementations of surface codes, it is a useful tool to understand the correspondence between FBQC and circuit-based surface code implementations. In the circuit model, class space edges in the 3D check subgraph correspond to physical qubit errors, while class time edges correspond to measurement errors. In FBQC, class temporal and class spatial edges correspond to fused measurements-there is no distinction between physical and measurement errors in the model. Another comparison point is the interpretation of the original and dual check subgraphs. In the circuit model, the original subgraph captures Pauli-X errors and measurement errors on Z-parity, while the double subgraph captures Pauli-Z errors and measurement errors on X-parity. In this FBQC example, each 2 qubit fusion contributes one measurement to the original graph and the other measurement to the double graph. One observation is that the two fused measurements behave similarly to Pauli-X and Pauli-Z parts of the wrong channel on the physical qubit in the circuit model.

"6 Ring" converged network

Our second example, the 6-ring converged network shown in fig. 10 improves on the 4-star network. It requires fewer resource states and fewer fusion measurements to achieve the same distance code, as we will see in the next section, it provides a significantly improved threshold.

In the converged network, the resource states are in the form of six qubit rings. As shown in fig. 10A and 10D, the fusion network has two resource states per unit cell of cubic unit cells. The fusion measurements connect the qubit pairs at each face and each edge, as shown by the orange line of fig. 10B. Each fusion attempt measures a stabilizer operator X on an input qubit ₁ X ₂ And Z ₁ Z ₂ . Fig. 10 shows a plurality of unit cells, where it can be seen that the resource states form layers in planes perpendicular to the (1, 1) direction. In the following section, a formal definition of a 4-star and 6-ring converged network will be given.

The check sub-graph for this converged network is shown in fig. 10E and is a cubic lattice with additional diagonal edges. Each check vertex has 12 incident measurements, and half of the 24 measurements contribute to each check operator in the 4-star network. The 6-ring network has the same symmetry when translating half the lattice constant in all three dimensions. Thus, as with the 4-star network, the original and dual subgraphs are identical.

The diagonal edges shown in the checkplots here are based on familiar features in surface code circuits, where they will be interpreted as so-called "hook" errors, where a single error event extends to adjacent qubits during the stabilizer measurement circuit. The sources of this type of correlation error are very different in the converged network setting, but the occurrences in the check subgraph are the same.

C. Performance comparison

We studied the performance of the two converged networks by modeling their behavior under the Pauli error and erasure model. We consider two error models:

a model of phenomenological errors in which each fused measurement can be erased and flipped with some probability

A linear optical error model, where each fusion has a probability of failure and each photon in the resource has a probability of loss.

We performed Monte Carlo simulations on fusion networks each with lxlxlx×l unit cells and all three-dimensional periodic boundary conditions. We draw error samples based on the above model and use a simplified version of the joint lookup decoder to perform decoding and count instances of logical errors. To establish a threshold surface in two error parameters, we repeat this simulation over a range L of erasure probabilities, pauli error rates, and system sizes. The original and dual subgraphs are decoded separately.

D. Model of phenomenological errors

Threshold curves for star (blue line) and 6-ring (orange line) fusion networks with two error parameters: fusing erasure probability p _erasure And measuring the error probability p _error . Here a joint look-up decoder is used. The green star depicts the operating point with erasures due to linear optical faults if the qubits in the resource states are encoded in a (2, 2) shell code and all fusions are enhanced to a success probability of 75% with a randomly selected physical fault basis. The effective erasure probability for code fusion with linear optical faults and losses is calculated as follows.

Each fusion in the fusion network produces two measurements, which we call fusion measurements. The phenomenological error model is an independent and identical error model in the fusion measurement: each measurement in the converged network is with probability p _erasure Erases and with probability p _error And (5) turning over. This allows capture of single qubit Pauli errors and erasures from the fusion instrument itself resulting from the generation of the resource states.

In contrast to previously studied fault tolerant MBQC, which focuses on erasure and error thresholds of single qubit measurements on lattices that already have long-distance entanglement, the model captures the errors in joint measurements for creating long-distance entanglement starting from a small resource state. In this way we say that the phenomenological error model is closer to a circuit level error model, where the single resource state and fusion measurements act as the basic gates.

Fig. 5 shows the threshold curves (obtained with a joint look-up decoder) for a 4-star (blue line) and 6-ring (orange line) fusion network with this error model. If the erasure probability p is measured each time _erasure Probability of error p _error Is below the threshold curve, the error is in a correctable region of the corresponding converged network. In the correctable region, the probability of non-trivial undetectable errors (also referred to as logical errors) is exponentially suppressed in the size of the network.

The correctable area of the 4-star network is contained in the correctable area of the 6-ring network, i.e., the area (p) _erasure ，p _error ) Any value of (2) may also be corrected by the 6-ring network. Marginal p of 4-star network _erasure The threshold is 6.9% and the marginal threshold for a 6-ring network is 11.9%. Marginal p of 6-ring network _error The threshold (0.94%) is also higher than the threshold of a 4 star network (0.65%). Therefore, we consider a 6-ring network to be more fault tolerant than a 4-star network.

The general trend we observe is that the phenomenological threshold w.r.t. Fusion errors in fault tolerant fusion networks can be improved by relying on larger resource states. This is because the phenomenological error model captures only errors in the fused graph portion and has no internal errors in the resource state. In general, the complexity of the resource state generator will increase with the size and entanglement of the generated states. In fault tolerant FBQC architecture design, maintaining as low complexity as possible and increasing the trade-off between the phenomenological thresholds of fusion errors is a key objective for optimization.

E. Linear optical error model

Threshold curves for 4-star (blue) and 6-ring (orange) fusion networks have a photon loss probability p _loss Is a linear optical error model with random selectionFusion failure probability p of selected codes and failure bases _fail Is a linear optical error model. The green curve corresponds to a 6-ring fusion network of qubits encoded in a (2, 2) shell code. The error model for evaluating these curves is explained below.

We now examine the performance of these fusion networks under the error model of linear optical drives. Each fusion is a linear optical "type II" fusion, with four photons: two photons from the measured sub-bits and two photons from the bell pair to enhance the probability of fusion success. Each of these four photons (including two enhanced photons) is with probability p _loss The≡l is lost. If any photons in the fusion are lost, fewer than the expected photons are detected and both fusion results are considered erased. As a result, the probability 1-eta ⁴ η=1-l, fusion does not yield information. Even if no photons are lost in the fusion, there is a probability p that linear optical fusion performs separable single qubit measurements instead of the expected bell measurement _fail . As described above, this can be considered as one of two expected fusion measurements that are erased. The linear optical fusion circuits for each fusion are randomly selected so that the erasure probabilities from the two measurements of the fusion are the same. Fusion error model details the erasure probability per physical fusion measure in the network is p using this randomization ₀ ＝1-(1-p _fail /2)η ⁴ 。

To further reduce the erasure probability, we also consider the case where each qubit in the resource state is replaced by a qubit encoded with a (2, 2) shell code, as depicted in fig. 6. With this replacement, the fusion between (uncoded) qubits in the resource state is replaced by the encoded fusion between encoded qubits in the resource state, which consists of a pairwise fusion between the physical qubits that make up the encoded qubits. The fusion measurements in the network are replaced by encoded fusion measurements.

The (2, 2) shell code refers to four qubits [ [4,1,2 ]]]A quantum code, which may be obtained by concatenating repeated codes of X and Z observables. According to the cascade sequence, the generated codes are emptyWill be stabilized by code<XXXX,ZZII,IIZZ>Or (b)<ZZZZ,XXII,IIXX>To describe. For simplicity we assume that this choice is uniformly random for each coded fusion. Encoding fusion attempts at measuring encoded bellyl on two qubits a and B Is encoded into a qubit, wherein +.>And->The coding X and Z operators of the partial (2, 2) shell code are shown, respectively, which are defined explicitly below. The local stabilizer in the encoded qubit allows the logical bell measurement to be performed in a variety of ways that suppress the erasure rate of the encoded fusion. In the following we explicitly calculate the erasure probability for the measurement in the encoding fusion. If the erasure probability of the uncoded fusion measurement is p ₀ The probability of erasure of the coded fusion measurement is +.>For p ₀ <0.5,p _enc <p ₀ I.e. the coding suppresses erasure probabilities.

We model three levels of coding separately: the lowest level is the linear optical specific encoding that represents each physical qubit as a two-rail photon. Local codes such as (2, 2) shell codes can then be used for the resource state qubits to achieve code fusion that is less affected by faults and losses in linear optical fusion. Finally, a converged network similar to a 6-ring network consisting of many resource states and fusions defines topologically protected logical qubits.

The green star in FIG. 11 shows the probability p of failure when the erasure comes only from fusion _fail Coded erasure probability p when =25% _enc Amplitude of =0.043, the fusion failure probability is a value obtained by enhancing with bell pairs. The numbers can be combined by combination P in the preceding paragraph of this section ₀ And p _enc Obtained from the expression of (c) and described in detail below.

Using (2, 2) Shor coding, erasure of fusion failure places us in the correctable region of the 4-star and 6-ring networks. For a 6-ring network, the gap between the baseline operating point and the threshold curve is significantly greater than for a 4-star network. The baseline erasure rate is less than half the erasure margin, leaving room for other errors (e.g., photon losses).

We numerically see the loss margin in the presence of fusion failure in figure 6. Blue and orange lines represent threshold curves of 4-star and 6-ring fusion networks, respectively, without qubit encoding. However, the failure threshold of these networks is below 25%. With (2, 2) shell coding, the 6-loop fusion network provides a significantly larger marginal failure threshold of 43% and marginal loss threshold of 5.9%. With 25% failure probability obtained by bell pair enhanced fusion we have a loss tolerance of 2.7% per photon. In other words, by enhancing the fusion with bell pairs, the fusion network can be in the correctable region even when the probability of losing at least one photon in the fusion is 10.37%.

Quantum computation with fault tolerant fusion network

How to create fault-tolerant blocks (fault tolerant bulk), which behave as structures for topological quantum computing, has been described above. The block created is the most critical part of the architecture because it determines the error correction threshold. But additional features are required to achieve fault tolerant computing. We now discuss how this block is used to implement fault tolerant logic, as well as the impact on typical processing and physical architecture.

A. Logic gate

To perform fault tolerant logic, the systems and methods disclosed herein allow topology features to be created in addition to blocks. Different methods may be used to create the fault tolerant Clifford gate set. The boundary may be used to create perforations (perforations) that may be woven to perform gating. Boundaries may be used to create patches on which lattice surgery may be performed. Alternatively, logical qubits may be encoded as defects and distortions. All of these logic methods are compatible with FBQC. Such topological features may be created by modifying fusion measurements in specific locations or adding single qubit measurements in appropriate configurations. Here we give an example of how to create two types of boundaries to facilitate the encoding and manipulation of logical qubits in a puncture or patch. In some embodiments, these two boundary types correspond to rough and smooth boundaries in the surface code map, but in FBQC they are more naturally referred to as original and dual boundaries, depending on whether they can match the excitation in the original/double check subgraph, respectively. The original boundary corresponds to a rough boundary in the original subgraph and to a smooth boundary in the double subgraph, for example, as shown in fig. 22C.

Fig. 13 shows one example of how the original boundary and the double boundary are created by measuring some qubits in the Z-base. Fig. 13A shows the creation of boundaries, where the qubit layer at the boundary of the unit cell is measured in Z-basis, or simplified to never be created. Fig. 13B illustrates a similar protocol for creating smooth boundaries. The case where the boundary is parallel to the plane defined by the pair of cell vectors is particularly simplified. For such boundaries, the only difference between the original case and the dual case is the displacement of the half unit vector in the vertical direction.

The effect of this measurement mode is to terminate the block, create a boundary, which can then be used as a feature to encode and manipulate the logical qubit. FIG. 13D shows an example of how these boundaries are macroscopically assembled to prepare state |0> (or |1 >) in bulk-coded logical qubits fault-tolerant.

Pauli frame tracking

In FBQC, the logic state has only a direct physical counterpart up to the Pauli correction, which is tracked in typical logic by so-called Pauli frames, e.g., within the typical computing system 307 of fig. 3A. The use of Pauli frames is essentially unavoidable due to the inherent randomness of the electrical introduction performed by the Bell measurements. For example, at the logic level, the same component that prepares for the |0> state in fig. 13 will also represent preparing for the |1> = x|0> state. In general, this means that for some tracked Pauli correction operators P, state |ψ > can be physically represented with different states P|ψ >.

When relying on Pauli frame tracking, the general n-qubit state can be tracked by any 4 ⁿ The possible physical quantum states are represented with 2n typical bits describing the frame. The stabilizer code used to protect the logic information is effectively half the number of bits needed to describe the frame. A key feature of this technique is that most of the computations that can be described by the Clifford operation can be performed independently of typical trace information. In cases such as magic state injection and distillation, typical Pauli frame data only affects the quantum operations performed at the logic level. This allows typical Pauli frame processing to occur at a logical clock rate rather than at a potentially faster physical fusion clock rate. In the following we explain how Pauli frame tracking naturally fits into fault tolerant FBQC, explaining why this technique only imposes minimal quantum and typical processing requirements.

C. Universal logic

To implement a universal gate set, the Clifford gate is supplemented with state injection, which in combination with the magic state distillation protocol can be used to implement a T gate or other small angle revolving gate. By performing a modified fusion operation, by performing a single qubitMagic state injection can be implemented in FBQC by measuring or by replacing the resource states with special "magic" resource states. These methods and the configuration of injection points provide various ways to optimize noise coding state preparation.

D. Decoding and other typical processing

In FBQC, as with other methods of fault tolerant quantum computing, a typical error correction protocol is responsible for extracting reliable logical measurement information from unreliable and noisy physical measurements. In FBQC, it is helpful to look at the decoding results as logical Pauli frame information. Keeping track of the logical Pauli frames is necessary to interpret future measurements.

When logic level feedforward is required, this logic Pauli frame produces time sensitive information. That is, when a logic measurement junction is usedIf future logic gates are to be determined, the relevant Pauli frame information must be made available. An example of this is the implementation of a T gate by magic state injection, where S orIs applied on the condition of the logical measurement result.

One widely discussed challenge of decoding is that it must be performed in real-time during quantum computing. However, this feed forward operation occurs on a logical time scale is a key feature and does not require decoding results on a fusion (or physical qubit) time scale. If decoding is slower than the logic clock rate, buffered or auxiliary logic qubits may be used to allow computation of "waiting" for the decoded result. However, it is worth emphasizing that these are tools used at the logic level and that no modification of any physical operations is necessary. Fusion can always be performed without decoding results. An important implication of this is that the slow decoder does not affect the threshold.

However, a fast decoder is needed to reduce unnecessary overhead.

E.FBQC architecture

There are many possible variations of physical architecture for any given converged network.

Fig. 14A-14E illustrate examples of a converged router providing routes that use photonic resource state generators, optical routes, and linear optical fusion to generate a 6-ring converged network. This example demonstrates several features of the FBQC scheme:

1. the resource state generator may be reused to create a large converged network. Converged networks contain many resource states, but they do not all need to coexist at the same time. A resource state generator (resource state generator, RSG) that generates states in each clock cycle may be reused. One natural way to time-order the creation of a large converged network is to divide it into "time slices". The 3D converged network is divided into 2D layers, wherein one layer is created at each time stage and converged with a previous layer. This time ordering allows the generation of a 3D network using a 2D array of resource state generators.

2. The converged routes may be fixed such that no rerouting between each clock cycle is required. A good design principle in converged routing is to minimize the need for switching between clock cycles. This reduces losses and errors in the switches and minimizes the need to input typical control signals. In this exemplary layout, each resource state generated at a given location is destined for the same fusion device. This means that the connection of the devices is fixed and no handover is required to generate the block.

3. Logic may be implemented by modifying the fusion measurements. To allow logic to be implemented (at least), a subset of the fusion devices should be reconfigurable, as indicated in fig. 14E. Boundaries or other topological features in the block are achieved by changing the basis of the fused measurements or switching to a single qubit measurement.

The example in fig. 14 represents a simplified physical architecture. Particularly for optical qubits, there is great flexibility in how such an architecture is constructed. Depending on how long the resource state can wait in memory before experiencing excessive decoherence or loss, more extreme time ordering approaches can be taken. With photon resource states, the interleaving policy described in WO2020257772A1 can be applied, i.e. a single RSG is used to create an entire block of the converged network by creating one resource state at a time. Any temporal ordering of state creation is possible as long as it is compatible with the advanced feed forward constraints of the logic level. This example includes only local connections, which is not a requirement of optical qubits. The remote connection may allow for creation of aperiodic boundary conditions, other topologies, or code embedded in non-euclidean space. Higher-dimensional fusion networks can also be created by combining time-ordered structures with appropriate optical connectivity.

i. Circuit symbol

Fig. 14A to 14D introduce a set of schematic circuit symbols used in subsequent figures for ease of understanding of the description. These circuit symbols represent photon/electron circuits operating on physical qubits, and each input or output line represents a (physical) qubit. According to drawing conventions, inputs are shown on the left and outputs are shown on the right, with the understanding that schematic circuit drawings need not correspond to a particular physical layout.

Fig. 14A shows a symbol representing a Resource State Generator (RSG) circuit 1400. As described above, the RSG circuit may be implemented using any photonic/electronic circuit that generates resource states. The output of the RSG circuit 1400 is qubits, with each qubit indicated by a line; the number of outputs depends on the particular resource state. In the embodiments described herein, it is assumed that one resource state is generated per clock cycle by the RSG circuit, and the length of a clock cycle may be defined based on the time required for one RSG circuit to generate one resource state. The time required depends on the particular RSG circuit; for example, some existing RSG circuits may generate resource states within about 1ns, and the clock period may be 1ns. In some embodiments, the clock period may be longer than the time required for the RSG circuit to generate one resource state; RSG is not required to run at maximum speed. For purposes of this description, it is assumed herein that RSG circuit 1400 outputs all qubits of a resource state in the same clock cycle; however, those skilled in the art having access to the present disclosure will appreciate that the timing may vary.

Fig. 14B shows a symbol representing the type II fusion circuit 1405. Type II fusion circuits may be implemented, for example, as described below with reference to fig. 23-36, and may be reconfigured as described above with reference to fig. 3D. The input is two qubits (indicated by the solid line with inwardly pointing arrows) that are consumed by a type II fusion operation, as described below. The type II fusion circuit 1405 may provide a typical output signal 1406 that indicates the measurement of the fusion and the success or failure of the fusion operation and/or a particular type of success or failure (e.g., a pattern of detected photons) where the pattern of detected photons indicates the number of photons detected at each of the detectors of the fusion circuit.

Fig. 14C shows a symbol representing the switching circuit 1410. The inputs and outputs to the switching circuit 1410 may include any number of qubits, and the number of inputs need not be equal to the number of outputs. Switching circuit 1400 may incorporate any combination of one or more active optical switches, mode couplers, phase shifters, and the like. The switching circuitry may be configured to perform active operations that reconstruct the input modes (e.g., implement a change in the radix of the qubit by coupling the modes of the qubit) and/or apply phases to one or more of the input modes (which may affect subsequent coupling between the modes). In some embodiments, the operation of switching circuit 1410 may be dynamically controlled in response to typical control signals 111, the state of which may be determined based on the results of previous operations, the particular calculations to be performed, configuration settings, timing counters (e.g., for periodic switching), or any other parameter or information.

Fig. 14D shows a symbol representing the delay circuit 1415. The delay circuit delays the qubit for a fixed length of time and may serve as a memory for the qubit information stored in the qubit. The length of time (in clock cycles) is indicated by a number, in this example +1 indicating a 1 clock cycle delay. In the case of an optical qubit, the delay circuit may be implemented, for example, by providing one or more suitable lengths of optical fiber or other waveguide material, such that photons of the delayed qubit travel a longer path than photons of non-delayed qubits.

Fig. 14E shows a schematic diagram of a system for FBQC that uses RSG circuits, referred to herein as networking, wherein these circuits may generate the above-described topological features to implement fault-tolerant quantum logic gates, in accordance with some embodiments. The circuit symbols are as described above with reference to fig. 14A to 14F, except that typical inputs and outputs are not shown for clarity of illustration. Fig. 14A shows a system employing four representative network elements 1400, 1400', 1400 "and 1400'". Fig. 14A also shows the coupling between adjacent instances of network element 1400 within the network, such that the coupling forms an example of a converged network router implementing quantum error correction codes as described above. Each network element 1400 may include RSG circuitry 1402 that generates a resource state (e.g., the 6-ring resource state 1410 shown here) having six peripheral qubits. RSG 1402 provides two qubits to adjacent network elements as shown by the qubit 5 output path, herein referred to as the "x-fusion direction", and the qubit 6 output path, herein referred to as the "y + fusion direction". Network element 1400 also receives qubits from two adjacent network elements. Specifically, the qubit 2 output path, referred to herein as the "x+ fusion direction," is coupled at the fusion circuit to the output qubit 5 'output path of the neighboring network element 1400'. Likewise, the output path of qubit 3 is coupled to the output qubit path of qubit 6 "of network element 1400".

The network units 1400, 1400', 1400", 1400 '" may also include reconfigurable fusion circuitry, e.g., reconfigurable fusion circuitry 1420', and as such may implement the quantum logic depicted in fig. 13A-13D. Any or all of the other fusion circuits may also be reconfigured according to the architecture implemented, as described in detail with reference to fig. 3D. Furthermore, to allow fusion between qubits within resource states generated in different clock cycles, the illustrated offset may be implemented. For example, in each RSG shown, qubit 1 from a resource state generated in a first clock cycle will be fused with qubit 4 from a (different) resource state generated in a subsequent clock cycle. Delays other than the +1 clock circuit may be implemented and/or delays may be applied to other qubits to achieve fusion between any layers and also to achieve interleaving policies (e.g., as described in international patent application publication number WO2020257772A1, which is incorporated herein by reference in its entirety for all purposes).

Fig. 15A and 15B illustrate examples of networked RSG circuits that may be used, for example, in a substance-based qubit architecture (e.g., trapped ions, superconducting qubits, etc.), in accordance with some embodiments. These embodiments also use 6-ring resource states as shown, including qubits as numbered circles. Solid lines show qubit coupling that is capable of performing entanglement operations between qubits. For example, such a gate may enable two qubit bell projection measurements by performing a CZ gate between two qubits and then performing two single qubit measurements on a computational basis. For example, in fig. 15A, to implement FBQC protocols similar to those described elsewhere herein, resource states are first generated by applying CZ gates between the following qubit pairs: (1, 6), (1, 2), (2, 3), (3, 4), (4, 5) and (5, 6). Next, two qubit projection bell measurements are applied between resource states, for example by performing two qubit measurements (fusion) on the qubit pairs (2, 5), (3, 6) and (4, 1'). Then, qubit 1 states are transferred to qubit 1'. In the next time step, a new resource state is generated and the process is repeated as described previously.

In the embodiment shown in fig. 15B, the FBQC protocol proceeds as follows. In step 1, the 6-ring resource states are prepared as described previously by applying a CZ gate between the following appropriate qubit pairs: (1, 6), (1, 2), (2, 3), (3, 4), (4, 5) and (5, 6). In step 2a, fusion is applied between resource states, e.g. between qubits (2, 5) and (3, 6) from adjacent (different) resource states and between (4, 1') within each resource state. Then in step 2b the state of qubit 1 is transferred to qubit 1' within each resource state.

fFBQC error

The thresholds presented herein are based on a simple error model. In a physical implementation, there are many factors that affect the performance of the system. The error channel may have more structure than a random i.i.d Pauli error, including error bias and correlation, and the chronological order of the operations may propagate the error. When logic gates are performed by creating topological features (e.g., boundaries or distortions), these require different physical operations, resulting in different error models at these locations.

However, there are a number of reasons why the results we provide here nonetheless remain significant in many types of physical hardware.

Resource state and fusion errors are local in nature. Construction of FBQC limits the distance over which errors can spread. We want dependencies to exist only in resource states and not between resource states. This desire is particularly strong for linear optics, where photons at different locations cannot "accidentally" intertwine. Furthermore, each qubit in the protocol has a short finite lifetime, limiting the possibility of error propagation in its neighborhood. Furthermore, the resource states and fusion in our model are the same, so a reasonable assumption is that they have the same or similar error rates in a physical implementation.

The correlation in fusion can only make the performance better. The possible location where the correlation error occurs is between the two measurements of the fusion operation. Our model treats these errors as uncorrelated. Since we decode the original and dual subgraphs here, respectively, if the fusion error is correlated, it will have no effect on our threshold. If there is a way to consider this information, it will only improve performance.

Block determination threshold. The topology features used to implement the logic are 2-dimensional or 1-dimensional objects. As a result, the error threshold is determined by the block (bulk).

Although evaluating the complexity of the overall system, many hardware-level error models can be well approximated by our model through error channel remapping. For example, if a resource state is to be constructed from a series of noisy two qubit gates that suffer from errors with probability p_physical, then under a standard gate error model, pauli-X and Pauli-Z accumulated by each qubit during state preparation can be interpreted and re-expressed as accumulated error rates. When a cluster is constructed from two qubit gates, errors cannot propagate farther than their nearest neighbors, so any correlation resulting from propagating errors can only be between original and dual.

Discussion of X

At the fault tolerant logic gate level, fusion-based quantum computing allows the same operations as circuit-based quantum computing (CBQC) or measurement-based quantum computing (MBQC). However, when observing the physical process for implementing a logic gate, a significant difference occurs: the dependencies of the resources required for the computation protocol, the connectivity required for the physical qubits, the handling of typical information, the occurrence, propagation and impact of errors.

One difference between FBQC and MBQC is the nature of the respective entanglement states required to achieve fault tolerance. MBQC requires a large entangled cluster state, the size of which is amplified as the calculations are performed. FBQC, on the other hand, requires a constant size of resource states, with the number of resource states required increasing for larger quantum calculations. The types of measurements used also differ significantly. MBQC uses a single qubit measurement to perform the computation, and the work previously proposed for the LOQC architecture to implement fault tolerant MBQC does so: large cluster state resources are first created from finite size operations and then computational (single qubit) measurements are made. There is no such separation in the FBQC protocol, where multiple qubit projection measurements (e.g., fusion gates) integrate the entanglement measurements required to create remote entanglement with measurements that enable fault tolerance and computation. Although not required, in some FBQC variants, the protocol may also contain a small number of single qubit measurements, for example to create topological features as depicted in fig. 13A-13D.

When fault tolerant computing of linear optics is involved, more differentiation occurs at the architecture level. The LOQC has a long history, the earliest proposal relies on extremely large gate stealth transmissions or strategies that repeat until successful to handle the probability gates, and requires quantum memory. Recent architectural suggestions eliminate the need for memory, and fault tolerance schemes are based on building large entangled states, and then making single qubit measurements of the states to achieve fault tolerance and computation by MBQC. These schemes have a low constant depth, which means that each photon sees a small fixed number of components during its lifetime, regardless of the calculated size. The highest performing scheme is based on the percolation method to handle probability fusion. Other schemes use branch resource states to add redundancy that can tolerate probabilistic fusion at the cost of lowering the threshold for loss and Pauli errors.

The FBQC schemes disclosed herein use constant size resources in architectures with constant depth, but provide significant threshold improvements compared to the best results in the literature. In addition to fault tolerance, FBQC offers a key advantage in terms of architecture feasibility compared to these previous schemes, where typical processing and feed forward needs to occur during the lifetime of the photons. This typical process is often complex and the need to perform such global computation places special demands on the loss of photon delay as photons wait on the delay line. FBQC eliminates this requirement and makes the fault tolerance threshold independent of the time scale of typical feedforward. According to some embodiments, feed forward is still needed, as is the case in any quantum computing architecture, but in FBQC, this requirement is only at the logic level, with a time scale completely separate from the physical operation.

FBQC is a modular architecture consisting of different small functional blocks: generating resource state, fusing network and fusing operation. The blocks are required to be compatible with each other, but the physical implementation of each block may be independent and in fact have multiple options. As an example of its application in a practical physical system, we give an example of an all-linear optical implementation of FBQC. However, it finds more general application to other physical platforms, and in particular, it critically supports applications in hybrid quantum systems. For example, photon fusion operations and fiber-based fusion networks can be integrated with substance-based resource state generators that produce photon entangled states. Modularity is also a key aspect in ensuring that quantum computing architectures are reliable and manufacturable.

XI decoder buffering

The challenges of typical processing and feedforward at the decoding and processing logic level coexist between all quantum computing models. The decoder is unlikely to run as fast as the physical clock speed of the quantum processor. Typical feed forward is required at the logic level (see fig. 16), so the decoder system should be designed to handle long delays and increase decoder throughput. Here, we describe decoder buffering and decoder parallelization as techniques to solve these problems.

Fig. 16 shows an example of a quantum circuit requiring logic feed forward. More specifically, FIG. 16 shows a circuit implementing a logical pi/8 rotation. The Pauli product is measured and,is performed on the target qubit and magic state,/-> Logic X measurements were performed on the magic states. These two steps are referred to as "state injection". Based on the results of the two measurements, a correction circuit is applied to produce pi/8 rotation on the input qubit.

A. Decoder system

We first introduce the basic functionality of the decoder system and how it interleaves with the quantum processing of logical qubits, as well as other typical processing. Fig. 17 shows a schematic diagram of typical information flowing into and out of a quantum system. The figure shows the evolution of the system over time when two logic gates are implemented. Here we consider an example of topological quantum computation, where a 2D measurement information layer is obtained in each time step. Physically this can be implemented in an FBQC setup, where fusion measurements are made on resource states in a 3D fusion network. Or it may be a 2D surface code in which parity measurements are made in each time step. The system comprises three subsystems:

a quantum processor 1701 containing quantum information. The quantum processor may receive and execute measurement instructions that allow fault tolerance and logic gates to be implemented.

A logic gate control system 1703 containing a program for executing a quantum algorithm. This will contain instructions for the logic gates, as well as feed forward instructions regarding which gates are implemented based on previous measurements. Which receives output from the decoder and sends logic gate instructions to the vector sub-processor.

A decoder system 1705 that receives the measurement information from the quantum processor and performs typical calculations to decode it. The decoder system sends its output to the logic gate control system.

B. Information flow

We can understand the information flow by the following steps, which are represented in the figure by numbered circles.

In step 0, logic gate control includes a quantum program. This results from user input and can be compiled offline before the runtime of the quantum processor. The quantum program may include a feed forward step, wherein future instructions depend on measurements made on the quantum system. After some steps of the program have been performed, the logic gate control has the current program state.

1. First, instructions are issued by the logic gate control system and sent to the quantum processor.

2. The instructions are executed on a quantum system. To implement a logic gate, this may include executing multiple instruction layers over an L time step. These instructions may be gate sequences, fusion measurement modes, single qubit measurements, or other physical quantum instructions depending on the nature of the quantum hardware.

3. After the logic gates have been implemented, we accumulate the measurement information of the L-layers that are bundled together and passed to the decoder.

4. The decoder receives the measurement information, which we call decoding problems and calculates corrected measurement results.

5. When decoding is complete, the result is passed back to the algorithm control where it is used to calculate which logic gate instructions should be issued next.

G. Time scale

There are several time scales associated with determining how the system should be set to ensure that logic gate instructions are available when needed:

layer clock time-which is the time between each layer of computation. We use t _c To represent. The time scale may vary significantly between different physical systems. In a photonic system without interleaving, this can be as fast as 1ns. By interleaving, it can be reduced to 1 μs or more.

Logic block time-the time to implement logic gates on L layers, which requires time t_log=l×t_c. The value of L required to achieve the target logical error rate is typically in the range of 30-50. Let us take the value of l=40 as an example. T_log may be as low as 40ns or greater than 40 mus, depending on the level of interleaving applied.

Decoder latency-time taken to run decoder, t _D . Here we consider this delay as from the final measurement in the quantum system to the point where the first logic finger can be executedThe time of the order, i.e. the time it takes for the signal to be transmitted to/from the output subsystem, and the calculation of the algorithm instructions. Latency cannot be reduced with parallelization. This time scale is indicated by the thick arrow in fig. 17. The decoder runtime depends on the system size, error rate, decoding algorithm. The run time also depends on the specific measurement configuration, so that in practice the time of each run will vary, following some distribution.

We can consider three different modes of these time scales, which would require different arrangements and methods of the decoding system.

1. And (3) decoding in real time: t is t _D <t _c In the simplest case, the decoder evaluation is completed within one layer clock cycle. In this case, the logic instructions for gate 2 are available in time so that gate 1 executes immediately after gate without hysteresis.

2. Fast decoding: t_c < t_d < t_log-in the second case, the decoder is slower than one clock cycle, but faster than the time it takes to execute a complete logic gate. In this case, there must be a hysteresis between the completion of logic gate 1 and logic gate 2. However, only one decoder processor is needed for this logic qubit, as it will complete execution of the decoding problem for logic gate 1 before the time it needs to start decoding logic gate 2. In other words, the decoder throughput is large enough to keep pace with the rate at which information is generated.

3. Slow decoding: t_c > t_log-finally we consider the case where the decoder run time is longer than the logic gate time. In this case, there must be a lag between the gates, but additional problems are created with the throughput of the decoder. For a decoder processor, the first decoding problem will still be running when the second arrives. This results in a "backlog problem" in which the delay increases substantially with the increase of each subsequent gate as the waiting decoding problem queue increases. Fortunately, this throughput problem can be solved using multiple decoder processors to increase throughput to match the quantum system.

In quantum computers, the "just-in-time decoding" case is unlikely to be implemented, and we almost certainly encounter the need to handle some decoder delay and parallelize the decoder. In the next section we will introduce the concept of decoder buffering, which can be used as a technique to handle these decoding timing scenarios.

H. Decoder system design to handle slow decoding

To address the problems that occur in the "fast decoding" and "slow decoding" schemes defined in the previous section, we can use a combination of modified logic circuitry to allow decoding delays and adding additional processors to increase throughput.

A. Decoder buffering

When the decoder delay time is longer than the layer clock time, we can use the decoder buffer to allow the target logic qubit to wait until the next logic gate instruction becomes available. The buffer area only implements feed forward of the identification area where the measurement is needed after each logic gate. This identification operation has no logical impact on the qubit. It is critical that the identification "buffer area" does not have to be decoded in order to determine the next logic gate measurement instruction. The buffer area needs to be decoded at a later point, but the result of this decoding provides an update to the Pauli frame that will be used to interpret future measurement results, but this cannot change the determination of what the logical measurement instruction will be.

The buffer time may be selected to be a fixed duration prior to each feed forward operation that is long enough to cover all decoding run times. Alternatively, the duration of the buffer region may be adaptively selected such that the logical qubit waits in memory until the next gate instruction becomes available.

B. Decoder parallelization

If the logic delay is slower than the logic clock speed, then in addition to the buffer, we can include an additional decoding processor to increase throughput so that decoding can be performed at a rate that can "keep up" with the generated information.

C. Exemplary decoding System

In fig. 18, we show an exemplary configuration of a decoding system including buffering and decoder parallelization. We consider the logic feed forward example required for a magic state injection circuit. In this case, the first logic gate is a state injection that involves coupling the targeted logic qubit with the distilled magic state and making a logic measurement. The second logic gate is a correction circuit that is executed according to the logic result of the first gate. The example in fig. 18 shows a case where the decoder run time is about twice the logic clock time and the total decoding delay is about three times the logic clock time. A buffer having a duration longer than the delay is added so that the correction gate instruction is ready at the end of the buffer area. To increase decoder throughput to match a quantum system, we use two decoder processors per logical qubit and then pass decoding problems to them in an alternating fashion.

XII physical stage assembly and operation

Fig. 19 illustrates photonic hardware components within a linear optical quantum computer according to some embodiments. In photonic chips for linear optical quantum computing, a single photon is emitted by a source, passing along a waveguide through a range of linear optical elements including retarders, directional couplers, and active phase shifters, before being detected by a photon counting detector.

In some embodiments, qubit state initialization involves a single photon source. When the source is successful, it produces only one and only one photon. The photons generated by each source are nearly identical, including frequency, pulse shape, and timing. In some embodiments, the source may generate photons at a very high repetition rate (e.g., about 1 GHz). Suitable photon generation techniques include spontaneous four-wave mixing to probabilistically generate photon pairs at mid-infrared frequencies near the optical communication band. Detection of one of the two photons produces an electrical signal indicative of the success of the light source.

Since spontaneous four-wave mixing sources do not operate with unity probability, it is desirable to multiplex them. By multiplexing multiple sources operating with low probability, a single source operating with high probability can be generated.

In multiplexed sources, it may be desirable to delay a single photon after generation, for example, by one or two nanoseconds. This delay provides a trigger time for the predictive detector and a time for the required execution logic to activate the optical switch. The switch routes photons from a successful source to the desired output waveguide.

According to some embodiments, the delay is created using an ultra low loss waveguide. Optical fibers may be used in some parts of the architecture, which can allow for longer delays. Note that the delay employed in the FBQC architecture is short and fixed (i.e., does not increase with increasing computational size).

According to some embodiments, the optical switch may be implemented using a generalized Mach-Zehnder interferometer (GMZI). These interferometers consist of an active phase shifter array sandwiched between two perfectly mixed interfering networks (e.g., hadamard networks). An active phase shifter is a device capable of achieving optical phase shifting when a voltage is applied. The fully hybrid interference network may be implemented with passive linear optical elements. The main feature of these interfering networks is that they transform any single photon input into a wave function that spreads equally across all modes. At this point, each mode enters an active phase shifter that implements one of the two phases as an optical mode (0 or pi). After this, the mode enters another fully mixed interfering network. This embodiment allows routing any input mode of the optical switch to any output mode. There is only one active phase shifter in the path of the photons, which minimizes losses in the switching network. The second interfering network can be significantly simplified if we only need to switch N input modes to a single output mode.

According to some embodiments, a photon number resolution detector may perform qubit measurements, as well as detect predicted photons emitted by a source, and may perform measurements needed to produce a resource state. Many such detectors can be used, but the technique chosen should have a very high quantum efficiency so that if a photon hits the detector it is detected with a very high probability. The detector should also have a low dark count so that the probability of the detector emitting without incident photons in each time frame is very low. The detector should also be quantitative such that when two photons hit the detector, the probability that two counts are reported is high. Finally, to operate with a 1GHz single photon source, the detector should have very low timing jitter and fast reset times.

According to some embodiments, a Superconducting Nanowire Single Photon Detector (SNSPD) may be used as a preferred single photon detection technique for near infrared photons. The combination of speed, timing accuracy and detection efficiency is superior to many alternatives, but any detector technology may be used without departing from the scope of the present disclosure. In general, SNSPD requires cryogenic operation at several Kelvin (Kelvin), which is much hotter than the Mi Like lin temperature of many substance-based qubits. Furthermore, a design involving a number-resolved detector of multiple SNSPDs may be employed. A simple way to achieve this conceptually is to place the fanout of the incident waveguide on the SNSPD set, but other designs are possible without departing from the scope of the invention. The number of SNSPDs should be such that the probability of two incident photons striking a single SNSPD is sufficiently low.

According to some embodiments, these hardware components-source, detector, delay and switch-are present in a multiplexed single photon source as shown in fig. 20.

A single photon source receives the pump laser input pulse and generates a pair of photons, a signal photon, and a prophetic photon. Indicating that a photon may be incident on a photon-number resolving detector set. The signal photons are delayed before being sent to the optical switching network. The schematic indicates GMZI with 6 input modes and a single output. The hadamard network is a directional coupler network that implements the hadamard transformation on the input modes. In fig. 20, only optical elements are shown for simplicity, and electronic components and interconnections for implementing logic and feedforward are not shown.

In each time frame, the electrical signal from each detector passes through some typical logic that determines which source produced the photon. The signal from the logic unit actuates the phase shifter in GMZI. These electrical components are not shown in the schematic. The optical delay should be long enough to allow detection, logic, and actuation of the phase shift to occur.

Substantially the same multiplexing method can be used at several stages in the architecture. In one example, a typical resource state generator takes as input multiplexed single photons and then generates bell or GHZ states using standard methods from the literature. These relatively small entangled states are referred to herein as seed states. The resource state generator needs to provide multiplexed seed states for building larger entangled resource states by fusion. According to some embodiments, the fusion step itself may be multiplexed.

The overhead associated with multiplexing depends on the probability of success required for a single photon source, seed state, or resource state. Advantageously, in the FBQC architecture, the probability of success of a single photon source can be chosen independent of the size of the computation. The same is true for both seed state generation and resource state generation. This is true, in part, because failed single photon generation, seed state generation, or resource state generation will ultimately result in the loss of qubits at known locations, i.e., these errors are predicted. Error correction codes can correct these lost qubits as long as the system remains below the error correction threshold.

XIII additional examples

Fig. 21 illustrates one possible example of a fusion site 6001 configured to operate with a fusion controller 319 to provide measurement results for fault tolerant quantum computing to a decoder, according to some embodiments. In this example, fusion site 6001 can be an element of fusion array 321 (shown in fig. 3), and although only one example is shown for purposes of illustration, fusion array 321 can include any number of instances of fusion site 6001.

As described above, the qubit fusion system 305 may receive two or more qubits (qubit 1 and qubit 2, shown here as double track encoding) to be fused. Qubit 1 is one qubit entangled with one or more other qubits (not shown) as part of a first resource state, and qubit 2 is another qubit entangled with one or more other qubits (not shown) as part of a second resource state. Advantageously, in contrast to MBQC, none of the qubits from a first resource state need to be entangled with any of the qubits from a second (or any other) resource state in order to facilitate fault tolerant quantum computing. Also advantageously, at the input of fusion site 6001, the sets of resource states do not intertwine to form cluster states in the form of quantum error correction codes, and thus there is no need to store and/or maintain large cluster states with remote entanglement across the entire cluster states. Also advantageously, the fusion operations that occur at these fusion sites may be a completely destructive joint measurement of qubit 1 and qubit 2, so that everything left after the measurement is typical information representing the measurement results on these detectors (e.g., detectors 6003, 6005, 6007, 6009). At this time, typical information is all information required for the decoder 333 to perform quantum error correction, and no additional quantum information propagates through the system. This can be contrasted with MBQC systems, which can employ fusion sites to fuse resource states into cluster states, which themselves serve as topological codes, and only then generate the required typical information via single-event measurements of individual qubits in a large cluster state. In such MBQC systems, not only is a large cluster state required to be stored and maintained in the system prior to taking the single-event measurements, but an additional single-event measurement step (in addition to the fusion used to generate the cluster state) is required to be applied to each qubit of the cluster state in order to generate the typical information needed to calculate the syndrome data required by the decoder to perform quantum error correction.

FIG. 21 shows an illustrative example of one way to implement fusion sites as part of a photonic computer architecture. In this example, qubits 1 and 2 may be double-track encoded optical qubits. A brief description of dual track encoding of optical qubits is provided in the following XIV section with reference to fig. 26A-29. Thus, qubit 1 and qubit 2 can be input on waveguide pairs 6021, 6023 and waveguide pairs 6025, 6027, respectively. Interferometers 6024 and 6028 may be placed in line with the respective qubits and within one arm of the respective interferometers 6024, 6028 programmable phase shifters 6030, 6032 may optionally be applied to affect the basis of applying the fusion operation, e.g., by implementing the particular mode coupling shown in fig. 21 to implement what is referred to herein as XX, XY, YY, or ZZ fusion. The programmable phase shifters 6030, 6032 may be coupled to the fusion controller 319 via control lines 6029 and 6031 such that signals from the fusion controller 319 may be used to set the basis for which the fusion operation is applied to the qubits. In some embodiments, the base may be hard coded within the fusion controller 319, or in some embodiments, the base may be selected based on an external input (e.g., instructions provided by the fusion pattern generator 313). Additional mode couplers (e.g., mode couplers 6033 and 6032) may be applied after the interferometer, followed by single photon detectors 6003, 6005, 6007, 6009 to provide a readout mechanism for performing joint measurements.

In some embodiments, the fusion may be a probabilistic operation, i.e., it implements probabilistic bell measurements, where measurements sometimes succeed and sometimes fail, as described below in fig. 35. In some embodiments, the probability of success of such an operation may be increased by using additional quantum systems in addition to the quantum system on which the operation acts. Embodiments using additional quantum systems are commonly referred to as "enhanced" fusion. Those of ordinary skill in the art will appreciate that any type of fusion operation (and may or may not be enhanced) may be applied without departing from the scope of the present invention. Additional examples of type II fusion circuits for polarization encoding and dual-rail path encoding are shown and described in section XIV below. In some embodiments, the fusion controller 319 may also provide control signals to the detectors 6003, 6005, 6007, 6009. The control signals may be used, for example, to gate the detector or to otherwise control the operation of the detector. Each of the detectors 6003, 6005, 6007, 6009 provides a photon detection signal (representing the number of photons detected by the detector, e.g., 0 photons detected, 1 photon detected, two photons detected, etc.), and the photon detection signals may be pre-processed at the fusion site 6001 to determine the measurement (e.g., fusion success or failure) or passed directly to the decoder 333 for further processing.

FBQC example employing GHZ resource state

Fig. 22A-22B illustrate FBQC schemes for fault tolerant quantum computing in accordance with one or more embodiments. In this example, a topological code called the Lawson-Duff lattice (also called a leaf-like surface code) is used, but any other error correcting code may be used without departing from the scope of the present invention. For example, FBQC may be implemented for various volume codes (e.g., diamond codes, mitsubishi codes, etc.), various color codes, or other topology codes may be used without departing from the scope of the invention.

Fig. 22A illustrates one unit cell 2202 of the lawsonde lattice. For the case of measurement-based quantum computation, to determine at the center of the unit cell, this is referred to herein as P _cell Is measured on the x-base, resulting in a quantum bit on six faces of the unit cell for six M _x Each of the measurements determines a set of 0 or 1 eigenvalues. These eigenvalues are then combined as follows:

where S1, S2, S6 correspond to six sites on the face of a unit cell, mx (Si) corresponds to a measurement result (0 or 1) obtained by measuring the corresponding face qubit in the x base. ( S1, S2 and S3 are marked in fig. 22; s4, S5 and S6 are located on the hidden surface of the cell 2202. )

In FBQC, the goal is to generate a typical set of data corresponding to the error syndromes of some quantum error correction codes by a series of joint measurements (e.g., positive operator value measurements, also known as POVMs) of two or more qubits. For example, using Lawson-Duff unit cell of FIG. 22A as an illustrative example, the set of measurements that may be used to generate the syndrome values in the FBQC method is shown in FIG. 22B. In this example, the GHZ state is used as the resource state, but persons of ordinary skill in the art having benefit of the present disclosure will appreciate that any suitable resource state may be used without departing from the scope of the invention. To change from the MBQC scheme shown in fig. 22A to the FBQC scheme shown in fig. 22B, each face qubit of fig. 22A is replaced with a separate qubit from a distinctly separate (i.e., non-entangled) resource state. For example, four resource states R1, R2, and R3 (surrounded by a dashed oval) are labeled in FIG. 22B, each contributing at least one qubit to the content of the face qubit S2 that would be a Lawson Duff cell. For example, the surface qubit S2 in fig. 22A is replaced by 4 qubits from three different resource states: the resource state R1 contributes two qubits; the resource state R2 contributes a third qubit; and resource state R3 contributes a fourth qubit. In operation, the system will perform a fusion twice on each facet (e.g., circles 2221, 2222 in FIG. 22B represent the fusion between contributing qubits of resource states R2 and R1 and R3 and R1, respectively). In the example where the fusion is a type II fusion, all four face qubits are measured, generating four measurements. In the example where the fusion is a type II fusion, all four face qubits are measured, generating four measurements. The corrector-chart value of the unit cell is obtained by equation (2) above, but now:

M _x (S _i )＝[F _1,XX (S _i )+F _2,XX (S _i )]mod 2 (3)

Wherein, for the ith face, F _1,XX (S _i ) Is a measurement result obtained by performing joint measurement on qubits associated with fusion 1 (e.g., as indicated by circle 721), wherein fusion 1 is a type II fusion performed in XX base, and wherein F _2,XX (S _i ) Is a measurement obtained by performing a joint measurement on the qubits associated with fusion 2 (e.g., as indicated by circle 722), where fusion 2 is also a type II fusion performed in XX base. As with the measurement associated with X observables described above with reference to equation (2), the fused measurement of observables XX (and ZZ) takes the value of zero or 1, respectively, corresponding to the positive or negative eigenvectors of the measured operators (XX and ZZ in this example). In view of equation (3), in order to obtain the opposite plane M _x (S _i ) Is expected to be for fusion measurement F _1,XX (S _i ) And F _2,XX (S _i ) Proper fusion of the twoAs a result. However, if the fusion fails such that the operator's value cannot be recovered due to some error, in some embodiments, the measurement of the face is deemed to fail and results in at least one erased edge in the syndrome data. Persons of ordinary skill in the art having benefit of the present invention will appreciate that the decoder may process errors in a manner similar to that described above with reference to fig. 1A-1C. Those of ordinary skill in the art will also recognize that while our description of equation (3) focuses on XX observables, fusion may also produce measures of ZZ observables, and that these results may also be combined according to equation (3) to produce an independent set of syndrome data. In some embodiments, the two sets of syndrome data are referred to as original and dual syndrome graphs.

Fig. 22C shows an example of a cluster state consisting of several unit cells of the lawsonde lattice. In the MBQC method, it would be necessary to generate such an entire cluster state, thereby forming an entangled state of many qubits, where the entanglement of the state extends across the lattice from one surface boundary to another. In the MBQC method, it is this large entangled cluster state that is used as a quantum error correction code, so that logical qubits can be encoded. The calculation is performed by performing a single qubit measurement on each qubit of the entangled state to generate a measurement result that is used to generate a syndrome fed to the decoder, as described above with reference to fig. 1A-1C. It follows that increasing the fault tolerance of the calculation requires increasing the size of the lattice and therefore the size of the entangled state. In one or more embodiments of the FBQC methods disclosed herein, such large entangled cluster states are not necessary, but rather smaller resource states are generated, where the size of the resource states is independent of the required fault tolerance. As described in detail above with reference to fig. 22, the FBQC method may be constructed from any fault-tolerant lattice by replacing individual nodes of the lattice with a set of fusions between two or more adjacent resource states. This construction of individual nodes of a replacement lattice with resource states/fusion is only one example of obtaining an FBQC scheme, and one of ordinary skill in the art, having the benefit of the present disclosure, will recognize that many different ways of constructing an FBQC scheme from an error-tolerant lattice may be employed without departing from the scope of the present invention.

Furthermore, as described in more detail below, the process may proceed by generating resource state layers in a given clock cycle and performing fusion within the various layers, as described below in fig. 23-24. For example, in fig. 22C, the horizontal direction represents the time in the sense that all or a subset of the qubits in any given layer in the x-y plane may be generated/initialized at the same clock cycle, e.g., the qubits in layer 1 may be generated at clock cycle 1, the qubits in layer 2 may be generated at clock cycle 2, the qubits in layer 3 may be generated at clock cycle 3, and so on. As will be described in more detail below, some subset of the qubits in each layer may be stored/delayed so that they can be used to fuse with qubits from resource states in subsequent layers, if necessary, to achieve fault tolerance.

In some embodiments, to generate the desired error syndrome, a Lattice Preparation Protocol (LPP) may be designed that generates the appropriate syndrome from a fusion of multiple smaller entangled resource states. Fig. 23-24 illustrate examples of lattice preparation protocols according to some embodiments. For purposes of illustration, a resource state is a state such as resource state 2300 shown in FIG. 23A; however, other resource states may be used without departing from the scope of the invention. Resource state 2300 is equivalent to the GHZ state until hadamard is applied to a single qubit. For example, the states used in the examples disclosed herein are equivalent to the GHZ state until H-Hadamard is applied to the two termination end qubits 2300a-3 and 2300a-4 in FIG. 23A. More specifically, the 4-GHZ state may be identified as a stable sub-state with the following stable sub-states: < XXXX, ZZII, ZIZI, ZIIZ >. The resource state 2300 shown in FIG. 23A is closely related to this GHZ state, but the stabilizer of state 2300 is < XXZZ, ZZII, ZIXI, ZIIX > (where the ordering of the operators corresponds to qubits 2300a-1, 2300a-2, 2300a-3, and 2300a-4, respectively). Those of ordinary skill in the art will appreciate that the 4-GHZ state and the resource state 2300 are equivalent with application of hadamard gates on the qubits 2300a-3 and 2300a-4.

The time direction in fig. 23-24 is perpendicular to the page such that a resource state having a shape such as resource state 2310 represents a set of qubits: qubits 1, 2 and 3 entangled with each other in the same clock period and qubit 4 entangled with, for example, qubits 2 and 3 in the time dimension. Such a resource state may be created by, for example, generating a complete 4-qubit resource state in a single clock cycle and then storing the qubit 4 in memory for a fixed period of time (e.g., one clock cycle). As used herein, the term "memory" includes any type of memory, such as, for example, quantum memory, quantum bit delay lines, shift registers for the quantum bits, the quantum bits themselves, and the like. In the case of photon resource states, qubit memories such as these are equivalent to qubit delays and can therefore be implemented using optical fibers. In the example shown in fig. 23C, the delay to qubit 4 is represented schematically by a loop of extra optical path length (e.g., provided by an optical fiber) that is placed in-line with the existing optical path of the qubit, but is not present in the optical paths of qubits 1-3. In this example, the length of the fiber is such that it achieves a single clock cycle delay of duration T, but other delays are possible, such as 2T, 3T, etc. Such delays may be in the range of 500ps-500ns in terms of physical delay time, but any delay is possible without departing from the scope of the invention.

Turning back to the FBQC procedure disclosed herein, fig. 2-24 show examples of how the lattice preparation and measurement protocols of FBQC may proceed according to layers. Fig. 23A shows a portion of a base layer of the lawsonde lattice, shown as layer 2310 (corresponding to a portion of layer 1 shown in fig. 22C). In the example illustrated herein, to process a layer similar to that shown in fig. 23A, a plurality of resource states 2300 (e.g., in the qubit entanglement system 303 of fig. 3) are first generated. In this example, resource state 2300 is an entangled state comprising 4 physical qubits (also referred to herein as a quantum subsystem): qubits 2300a-1, 2300a-2, 2300a-3, 2300a-4. In some embodiments, resource state 2300 may take the form of a 4-GHZ state in which two terminating qubits 2300a-4 and 2300a-3 have undergone hadamard operations (e.g., by applying a 50:50 splitter between the two tracks forming the qubits in the case of a double track encoded qubit). In some embodiments, not all of the qubits in a layer are subject to fusion in this clock cycle, but rather some of the qubits generated from certain resource states during this clock cycle may be delayed, e.g., the measurement of the qubit 2320, the redundantly encoded qubit 2305, or any other qubit may be delayed so that the qubit will be available in the next clock cycle. Such delayed qubits may then be used to fuse with one or more qubits from a resource state that would only be available for fusing at the next clock cycle.

In examples employing a photonic implementation, the qubits from the resource states may then be appropriately routed (via integrated waveguides, optical fibers, or any other suitable photonic routing technique) to a qubit fusion system (e.g., the qubit fusion system 305 of fig. 3) to enable a set of fusion measurements that enable quantum error correction, i.e., that would result in collection of measurements corresponding to the selected error syndrome. Although this example explicitly uses a topology code based on the Lawson doffer lattice, any code may be used without departing from the scope of the invention.

Fig. 23B shows an example of a set of GHZ resource states that are arranged (i.e., they have been pre-routed) such that qubits to be sent to a given fusion gate are positioned graphically adjacent to each other. For qubits adjacent to each other in this illustration, a respective fusion may be performed between pairs of qubits (also referred to herein as respective quantum subsystems, where respective qubit inputs from the pairs of qubits are at fusion sites belonging to different respective resource states). For example, at site 2302, two type II fusion measurements may be applied, one between qubits 2322 and 2324 and one between qubits 2326 and 2328. It should be noted that prior to performing fusion, qubits 2322 and 2324 (or qubits 2326 and 2328) do not become entangled with each other, but are each part of a different resource state. It follows that there is no large entangled cluster state called the lawsondorf lattice before fusion measurements are performed.

Referring to fig. 24A, a portion of a second layer of the base code structure is shown as layer 2410 (corresponding to layer 2 shown in fig. 22C). In the FBQC system, in order to process a single layer as shown in fig. 24B, the FBQC method is performed along the same line as described above with reference to fig. 23A to 23B, and thus details will not be repeated here.

Fig. 25A-25E illustrate a method for performing FBQC in further detail in accordance with one or more embodiments. More specifically, the methods described herein include steps for performing joint measurements for particular quantum error correction codes, according to some embodiments, where different layers of code may be generated at different time steps (clock cycles) as described above with reference to fig. 23-24, and entangled together in a manner that provides fused measurements to extract the necessary syndrome information for performing quantum error correction. As with the other examples provided herein, the lawsonde lattice is used for illustration, but other codes may be used without departing from the scope of the invention.

For example, fig. 25A and 25B show portions of layers 1 and 3 and layers 2 and 4, respectively, from the lawsonde lattice of fig. 22C (referred to herein as Quantum Error Correction (QEC) codes). FIGS. 25C and 25D illustrate methods for processing these layers in an FBQC system, including example resource states that may be used. For the sake of example, the description is limited to vertices 1, 2, 3 and 4 of the QEC code, and this example focuses on how resource state generation and measurement is performed in the FBQC system.

Returning to FIG. 25A, in step 2501, a first set of resource states is provided during a first clock cycle. Fig. 25D shows an example in which instead of providing single qubits at vertices 1, 2, 3, 4, 5, etc., where the single qubits are entangled with each other across the lattice (as is the case with MBQC systems), two or more qubits are provided, each of which originates from a different, non-entangled resource state (e.g., respective resource states A, B, C, D, E, F and G). As used herein, the symbol Aij is used to represent the jth qubit from the ith resource state of the ith layer. For example, the A-th resource state of layer 1 in FIG. 25D is the GHZ state, which includes 4 qubits, labeled A11, A12, A13, A14, as shown. Likewise, qubits comprising resource state B provided as part of layer 1 may be labeled B11, B12, B13, B14 (although labels not explicitly shown in the figures are used this time to avoid cluttering the figure). In fig. 25D, the qubits to be fused to generate syndrome information associated with vertices 1, 2, 3, 4, 5 are also shown as enclosed by solid ellipses 1, 2, 3, and 4. As used herein, these vertices are each associated with hardware for performing type II fusion at a fusion site, as described above.

In some embodiments, the resource states of any given layer may be generated/provided by a qubit entanglement system (e.g., the qubit entanglement system described above with reference to fig. 3). However, persons of ordinary skill in the art having benefit of the present disclosure will appreciate that any qubit entanglement system may be employed, and that a given qubit entanglement system may employ many different types of resource state generators, even different types of resource states. In this sense, the FBQC system is completely unaware of the selection of resource states and the selection of the architecture of the qubit entanglement system or even the architecture of the qubit itself, allowing great flexibility for the system designer to implement a system that produces the highest threshold for a given dominant error/noise source.

In step 2503, fusion instructions (also referred to herein as fusion patterns) in the form of typical data are provided to the fusion site. Referring back to fig. 3, for example, fusion pattern data frame 317 is one example of a set of fusion instructions (e.g., type II fusion measurements in XX base) that can be applied between qubit pairs from different entangled resource states at fusion sites during a certain clock cycle when quantum applications are executed on the FBQC system. Also as described above, in some embodiments, several fused mode data frames may be stored as typical data in memory. In some embodiments, the fusion mode data frame may specify whether XX II type fusion will be applied (or whether any other type of fusion will be applied) for a particular fusion gate within the fusion site. Furthermore, the fusion pattern data frame may indicate that type II fusion is to be performed in a different base, such as XX, XY, ZZ, etc.

Returning to fig. 25D, the fusion instructions for layer 1 may include fusion parameters (qubit positions and bases) to fuse two or more qubits from different resource states (also referred to herein as respective quantum subsystems because the qubits reside in or are part of respective individual resource states). For example, for fusion site 1, the fusion instruction may specify a fusion parameter to indicate that XX II type fusion is to be performed between qubits A1, B1 and C1 from the resource state (similarly, for site 3, between E1, F1 and G1). More specifically, two type II fusions to be made at fusion site 1 can be designated as between a14 and B12 and between C11 and B13. Similar instructions are provided for other fusion sites in the layer. For example, for fusion site 2, the fusion instruction may specify a fusion parameter to indicate that XX II type fusion is to be performed between qubits B1, D1 and F1 from the resource state. More specifically, two type II fusions to be made at fusion site 2 can be designated as between B14 and D12 and between D13 and F14. However, unlike the case of fusion site 1, which measures all qubits, fusion site 2 includes qubits that remain unmeasured until the second clock cycle. This is because the infrastructure of the QEC lattice requires that the quantum state of this qubit be preserved until it merges into the qubit from a different layer at a different clock cycle, i.e. if this is an MBQC scheme, the qubit associated with this vertex will be one entangled with the qubit in another layer, e.g. qubits 2 and 6 shown in fig. 10B and 10C, respectively.

Returning to the explicit example shown in fig. 25D, the fuse instruction may specify that D14 is not measured until the next clock cycle, where it will fuse from the qubits in a later layer (e.g., layer 2 shown in fig. 25E). In a photonic implementation, the fiber can implement a qubit delay for the above functions, acting as a reliable quantum memory to store the qubits until they are needed for future clock cycles. As used herein, these unmeasured (delayed) qubits are referred to as unmeasured quantum subsystems.

Moving to fusion site 4, which is an example of a fusion between layers that includes a resource state generated in that clock cycle, and a quantum bit from a resource state generated in a previous clock cycle but not measured at that time but rather delayed or equivalently stored until the next clock cycle. For fusion site 4, the fusion instruction may specify a fusion parameter to indicate that XX II type fusion is to be performed between qubits C1, B0 and B2 from resource states in three different layers. The fused instruction may also include an instruction to delay (not measure) the qubits C12 and C13 until the next clock cycle. For example, in this case, the fuse instruction may indicate that in the next time step, C12 will be fused with B04 and C13 will be fused with B21.

In step 2503, a fusion operation specified by the fusion instruction is performed, thereby generating typical data in the form of fusion measurements. As described above with reference to fig. 3 and equation (2), this typical data is then passed to a decoder and used to construct a syndrome to be used for quantum error correction.

These examples are illustrative. The selection of the error correction code determines a set of qubit pairs that are fused from certain resource states such that the output of the qubit fusion system is typical data from which the syndrome pattern can be constructed directly. In some embodiments, typical error syndrome data is generated directly from the qubit fusion system without requiring additional single-event measurements of any remaining qubits. In some embodiments, the joint measurements performed at the qubit fusion system are destructive to the qubits on which the joint measurements are performed.

Introduction to qubit and Path coding

The kinetics of quantum objects (e.g., photons, electrons, atoms, ions, molecules, nanostructures, etc.) follow the rules of quantum theory. More specifically, in quantum theory, the quantum state of a quantum object (e.g., a photon) is described by a set of physical properties, the complete set of which is called a mode. In some embodiments, the pattern is defined by specifying values (or a distribution of values) of one or more characteristics of the quantum object. For example, again for a photon, a mode may be defined by the frequency of the photon, the position of the photon in space (e.g., which waveguide or superposition of waveguides the photon propagates within), the associated propagation direction (e.g., the k-vector of the photon in free space), the polarization state of the photon (e.g., the direction (horizontal or vertical) of the electric and/or magnetic fields of the photon), and so forth.

For the case of photons propagating in a waveguide, it is convenient to express the state of the photon as one of a set of discrete spatiotemporal modes. For example, the spatial mode k of photons _i Is determined based on which of a finite set of discrete waveguides the photons can propagate in. Furthermore, time pattern t _j Is determined by which of a set of discrete time periods (referred to herein as "bins") a photon may be present in. In some embodiments, time discretization of the system may be provided by the timing of the pulsed laser responsible for generating the photons. In the following examples, spatial modes will be mainly used to avoid complications of the description. However, one of ordinary skill in the art will appreciate that the systems and methods may be applied to any type of mode, such as temporal mode, polarization mode, and any other mode or set of modes for specifying quantum states. Furthermore, in the following description, an embodiment will be described in which a photonic waveguide is employed to define a spatial mode of photons. However, persons of ordinary skill in the art having benefit of the present disclosure will appreciate that any type of mode (e.g., polarization mode, temporal mode, etc.) may be used without departing from the scope of the invention.

For quantum systems of a plurality of non-distinguishable particles, it is useful to describe the quantum states of the entire multi-body system using the form of the Fock (sometimes referred to as the occupancy representation), rather than the quantum states of the individual particles in the system. In the Fock state description, the multi-bulk quantum state is specified by how many particles are present in the various modes of the system. Because the pattern is a complete set of characteristics, the present description is sufficient. For example, multimode, two-particle Fock state |1001> _1,2,3,4 Two-particle quantum states are specified, one in mode 1, zero in mode 2, zero photonsIn mode 3, 1 photon is in mode 4. Again, as described above, a mode may be any set of characteristics of a quantum object (and may depend on the single-particle ground state used to define the quantum state). For the case of photons, any two modes of electromagnetic field may be used, for example, the system may be designed to use modes related to degrees of freedom that may be passively steered with linear optics. For example, polarization, spatial degrees of freedom, or angular momentum may be used. For example, from two particles Fock state |1001> _1,2,3,4 The represented four-mode system may be physically implemented as four different waveguides, where two of the four waveguides (representing mode 1 and mode 4, respectively) have one photon traveling therein. Other examples of states of such a multi-body quantum system are four-photon Fock state 1111, which represents individual waveguides containing one photon > _1,2,3,4 And four-photon Fock state |2200 representing waveguides one and two respectively accommodating two photons and waveguides three and four accommodating zero photons> _1,2,3,4 . For modes where zero photons are present, the term "vacuum mode" is used. For example, for four photon Fock state |2200> _1,2,3,4 Mode 3 and mode 4 are referred to herein as "vacuum modes" (also referred to as "assist modes").

As used herein, a "qubit" is a physical quantum system having associated quantum states that can be used to encode information. In contrast to typical bits, qubits may have a state that is a superposition of logical values, e.g., 0 and 1. In some embodiments, the qubit is "double-track encoded" such that the logical value of the qubit is encoded by one of the two modes being occupied by exactly one photon (a single photon). For example, consider two spatial modes of a photonic system associated with two different waveguides. In some embodiments, the logical 0 and 1 values may be encoded as follows:

|0> _L ＝|10> _1,2 (3)

|1> _L ＝|01> _1,2 (4)

wherein the subscript "L" indicates that the right arrow represents a logical value (e.g., a qubit value), and as beforeThe right-hand side symbols |ij > of the above equations (3) - (4) _1,2 Indicating that i photons are present in the first waveguide and j photons are present in the second waveguide, respectively (e.g., where i and j are integers). In this symbol, it has a logical value of |01> _L The two-qubit state (representing the state of two qubits, the first being in the "0" logic state and the second being in the "1" logic state) of (i) 1001 may be passed through> _1,2,3,4 Represented using photon occupancy across four different waveguides (i.e., one photon in the first waveguide, zero photon in the second waveguide, zero photon in the third waveguide, and one photon in the fourth waveguide). In some cases, in the present invention, various subscripts are omitted to avoid unnecessary mathematical confusion.

Introduction to XIV.LOQC

A. Double track optical qubit

Qubits (and operations on qubits) may be implemented using a variety of physical systems. In some examples described herein, qubits are provided in an integrated photonic system employing a waveguide, beam splitter (or directional coupler), photonic switch, and single photon detector, and the modes that can be occupied by photons are spatiotemporal modes corresponding to the presence of photons in the waveguide. A mode coupler (e.g., an optical beam splitter) may be used to couple the modes to achieve the transformation operation, and the measurement operation may be achieved by coupling a single photon detector to a particular waveguide. Those of ordinary skill in the art having access to the present invention will appreciate that modes defined by any suitable set of degrees of freedom, such as polarization mode, temporal mode, etc., may be used without departing from the scope of the present invention. For example, for modes that differ only in polarization (e.g., horizontal (H) and vertical (V)), the mode coupler may be any optical element that coherently rotates the polarization, such as a birefringent material, e.g., a waveplate. For other systems, such as ion trap systems or neutral atom systems, the mode coupler may be any physical mechanism that can couple two modes, e.g., a pulsed electromagnetic field tuned to couple two internal states of atoms/ions.

In some embodiments of optical quantum computing systems using dual-rail encoding, the qubits may be implemented using a pair of waveguides. Fig. 26A shows two representations (2600, 2600') of a portion of a pair of waveguides 2602, 2604 that can be used to provide a double-track encoded optical qubit. At 2600, photons 1106 are in waveguide 1102 and no photons are in waveguide 2604 (also referred to as vacuum modes); in some embodiments, this corresponds to the |0> state of the optical qubit. At 2600', photon 2608 is in waveguide 2604 and no photon is in waveguide 2602; in some embodiments, this corresponds to the |1> state of the optical qubit. To prepare the optical qubit in a known state, a photon source (not shown) may be coupled to one end of one waveguide. The photon source may be operated to emit single photons into a waveguide to which it is coupled, thereby preparing the optical qubit in a known state. Photons travel through the waveguides and by periodically operating the photon sources, a quantum system with qubits whose logical states map to different temporal modes of the photon system can be created in the same pair of waveguides. In addition, by providing pairs of waveguides, a quantum system with qubits whose logic states correspond to different spatiotemporal modes can be created. It should be appreciated that the waveguides in such a system need not have any particular spatial relationship to each other. For example, they may be, but need not be, arranged in parallel.

An occupied mode may be created by generating photons using a photon source, which then propagate in a desired waveguide. The photon source may be, for example, a source based on a resonator emitting photon pair, also known as an announced single photon source. In one example of such a source, the source is driven by a pump (e.g., an optical pulse) coupled into an optical resonator system that can generate a pair of photons through a nonlinear optical process (e.g., spontaneous four-wave mixing (SFWM), spontaneous parametric down-conversion (SPDC), second harmonic generation, etc.). Many different types of photon sources may be employed. Examples of photon pair sources may include micro-ring based spontaneous four wave mixing (SPFW) declarative photon sources (HPS). However, the exact type of photon source used is not critical, and any type of source may be used, employing any process, such as SPFW, SPDC, or any other process. Other kinds of sources may also be used which do not necessarily require non-linear materials, such as sources employing atomic and/or artificial atomic systems, such as quantum dot sources, color centers in crystals, etc. In some cases, the source may or may not be a photonic cavity or may not be coupled to a photonic cavity, as is the case, for example, for an artificial atomic system such as quantum dots coupled to a cavity. Other types of photon sources exist for SPWM and SPDC, such as opto-mechanical systems and the like.

In this case, the operation of the photon source may be deterministic or non-deterministic (also sometimes referred to as "random") such that a given pump pulse may or may not produce a photon pair. In some embodiments, coherent spatial and/or temporal multiplexing of several non-deterministic sources (referred to herein as "active" multiplexing) may be used to allow the probability that one mode is occupied during a given period to be close to 1. Those of ordinary skill in the art will appreciate that many different active multiplexing architectures are possible that incorporate spatial and/or temporal multiplexing. For example, active multiplexing schemes employing logarithmic trees, generalized Mach-Zehnder interferometers, multimode interferometers, linked sources with dump pumping schemes, asymmetric polycrystalline single photon sources, or any other type of active multiplexing architecture may be used. In some embodiments, the photon source may employ an active multiplexing scheme with quantum feedback control, or the like.

The measurement operation may be accomplished by coupling the waveguide to a single photon detector that generates a typical signal (e.g., a digital logic signal) that indicates that a photon has been detected by the detector. Any type of photodetector that has sensitivity to single photons may be used. In some embodiments, detection of a photon (e.g., at the output end of the waveguide) indicates an occupied mode, while absence of a detected photon may indicate an unoccupied mode. In some embodiments, the measurement operation is performed in a particular basis (e.g., a basis defined by a Pauli (Pauli) matrix and referred to as X, Y or Z), and mode coupling as described below may be applied to transform qubits to the particular basis.

Some embodiments described below relate to a physical implementation of what forward transform operation of a mode of a coupled quantum system, which may be understood as transforming a quantum state of the system. For example, if the initial state of the quantum system (prior to mode coupling) is a state in which one mode is occupied by probability 1 and the other mode is unoccupied by probability 1 (e.g., state |10 in the Fock symbol>Where the numbers indicate the occupancy of the respective states), the mode coupling may result in a state where both modes have a non-zero probability of being occupied, e.g., state a ₁ |10>+a ₂ |01>Wherein, |a ₁ | ² +|a ₂ | ² =1. In some embodiments, this may be accomplished by coupling the modes together using a beam splitter and applying a phase shift to one or more of the modes using a variable phase shifter. The amplitudes a1 and a2 depend on the reflectivity (or transmissivity) of the beam splitter and any phase shift introduced.

Fig. 26B shows a schematic diagram 2610 (also referred to as a circuit diagram or circuit symbol) for coupling two modes. The modes are drawn as horizontal lines 2612, 2614, and the mode coupler 2616 is indicated by a vertical line ending with a node (solid dot) to identify the coupled mode. In a more specific language of linear quantum optics, the mode coupler 2616 shown in fig. 26B represents a 50/50 beam splitter implementing a transfer matrix:

Where T defines the linear mapping of the photon-generating operator over the two modes. (in some contexts, the transfer matrix T may be understood as implementing a first order imaginary hadamard transform.) conventionally, if the system includes more than two modes, the first column of the transfer matrix corresponds to a generation operator on the top mode (referred to herein as mode 1, labeled horizontal line 1112) and the second column corresponds to a generation operator on the second mode (referred to herein as mode 2, labeled horizontal line 1114), and so on. More specifically, the map may be written as:

wherein the subscript on the generating operator indicates the mode being operated on, the subscript input and output identify the form of the generating operator before and after the beam splitter, respectively, and wherein:

for example, the application of the mode coupler shown in fig. 26B results in the following mapping:

therefore, the mode coupler described in equation (4) functions to change the input states |10>, |01>, and |11> to:

fig. 26C illustrates a physical implementation of mode coupling according to some embodiments that implements the transfer matrix T of equation (4) for two photon modes. In this example, mode coupling is achieved using a waveguide splitter 2620 (sometimes also referred to as a directional coupler or a mode coupler). The waveguide splitter 2620 may be implemented by bringing two waveguides 2622, 2624 close enough so that the evanescent field of one waveguide may be coupled to the other waveguide. By adjusting the spacing d between the waveguides 2622, 2624 and/or the length l of the coupling region, different couplings between the modes can be obtained. As such, waveguide beam splitter 2620 may be configured to have a desired transmissivity. For example, the beam splitter may be designed to have a transmissivity equal to 0.5 (i.e., a 50/50 beam splitter for implementing the particular form of transfer matrix T described above). If other transfer matrices are desired, the reflectivity (or transmissivity) may be designed to be greater than 0.6, greater than 0.7, greater than 0.8, or greater than 0.9 without departing from the scope of the present invention.

In addition to mode coupling, some positive transformations may involve phase shifting applied to one or more modes. In some photon implementations, the variable phase shifter may be implemented in an integrated circuit, providing control of the relative phase of photon states spread over multiple modes. An example of a transfer matrix defining such a phase shift is given by the following formula (for applying +i and-i phase shifts, respectively, to the second mode):

for silicon-based silicon dioxide materials, some embodiments implement a variable phase shifter using thermo-optic switches. The thermo-optical switch uses a resistive element fabricated on the chip surface that can provide a change in refractive index n by raising the temperature of the waveguide by an amount on the order of 10-5K via the thermo-optical effect. Those skilled in the art having access to the present disclosure will appreciate that any effect of changing the refractive index of a portion of a waveguide may be used to generate a variable, electrically tunable phase shift. For example, some embodiments use beam splitters based on any material that supports the electro-optic effect, which is a so-called χ2 and χ3 material (e.g., lithium niobate, BBO, KTP, BTO, PZT, etc.), even doped semiconductors (e.g., silicon, germanium, etc.).

B. Photon mode coupler: beam splitter

By combining a directional coupler and a variable phase shifter in a mach-zehnder interferometer (MZI) configuration 2630, a beam splitter having variable transmissivity and an arbitrary phase relationship between output modes can also be implemented, for example, as shown in fig. 26D. By varying the phase imparted by the phase shifters 2636a, 2636b, and 2636c and the length and proximity of the coupling regions 2634a and 2634b, complete control of the relative phase and amplitude of the two modes 2632a, 2632b in dual track encoding can be achieved. Fig. 26E shows a somewhat simplified example of an MZI 2640 that allows for variable transmissivity between modes 2632a, 2632b by varying the phase imparted by phase shifter 2637. Fig. 26D and 26E are examples of how a mode coupler may be implemented in a physical device, but any type of mode coupler/splitter may be used without departing from the scope of the invention.

In some embodiments, beam splitters and phase shifters may be employed in combination to implement various transfer matrices. For example, fig. 26A shows, in schematic form similar to fig. 26A, a mode coupler 2600 implementing the following transfer matrix:

thus, mode coupler 2700 applies the following mapping:

transfer matrix T of equation (10) _r The shift in phase through the second mode is related to the transfer matrix T of equation (4). This is schematically illustrated in fig. 27A by a closed node 2707 coupled to a first mode (line 2712) by a mode coupler 2716 and an open node 2708 coupled to a second mode (line 2714) by mode coupler 2716. More specifically, T _r = sTs, and as shown on the right hand side of fig. 27A, mode coupler 2716 can be implemented using mode coupler 2716 (described above) with forward and backward phase shifts (represented by open squares 2718a, 2718 b). Thus, the transfer matrix T _r This can be achieved by a physical beam splitter as shown in fig. 27B, where the open triangles represent +i phase shifters.

C. Exemplary photon diffusion Circuit

A network of mode couplers and phase shifters may be used to achieve coupling between more than two modes. For example, fig. 28 shows a four-mode coupling scheme that implements a "diffuser" or "mode information erasure" transformation for four modes, i.e., that acquires photons in any one of the input modes and delocalizes the photons between each of the four output modes such that the probability of the photons being detected in any one of the four output modes is equal. (the well-known hadamard transform is one example of a diffuser transform.) as in fig. 26A, horizontal lines 2812-2815 correspond to modes, and mode coupling is indicated by vertical lines 2816, which have nodes (points) identifying the coupled modes. In this case, four modes are coupled. Circuit symbol 2802 is an equivalent representation of circuit diagram 2804, which is a network of first order mode couplings. More generally, in the case of a network where higher order mode coupling can be implemented as first order mode coupling, a circuit symbol similar to symbol 2802 (with an appropriate number of modes) can be used.

FIG. 29 illustrates an example optical device 2900 that can implement the four-mode diffusion transformation schematically illustrated in FIG. 28, in accordance with some embodiments. The optical device 2900 includes a first set of optical waveguides 2901, 2903 formed in a first material layer (represented by solid lines in fig. 29) and a second set of optical waveguides 2905, 2907 formed in a second material layer (represented by dashed lines in fig. 29) that is different and separate from the first material layer. The second material layer and the first material layer are located at different heights on the substrate. Those of ordinary skill in the art will appreciate that an interferometer such as that shown in fig. 29 may be implemented in a single layer if appropriate low loss waveguide crossover is employed.

At least one optical waveguide 2901, 2903 of the first set of optical waveguides is coupled to an optical waveguide 2905, 2907 of the second set of optical waveguides using any type of suitable optical coupler. For example, the optical device shown in fig. 29 includes four optical couplers 2918, 2920, 2922, and 2924. Each optical coupler may have a coupling region in which two waveguides propagate in parallel. Although two waveguides are illustrated in fig. 29 as being offset from each other in the coupling region, the two waveguides may be positioned directly above and below each other in the coupling region without offset. In some embodiments, one or more of the optical couplers 2918, 2920, 2922, and 2924 are configured to have a coupling efficiency between the two waveguides of approximately 50% (e.g., a coupling efficiency between 49% and 51%, a coupling efficiency between 49.9% and 50.1%, a coupling efficiency between 49.99% and 50.01%, and a coupling efficiency of 50%, etc.). For example, the length of the two waveguides, the refractive index of the two waveguides, the width and height of the two waveguides, the refractive index of the material between the two waveguides, and the distance between the two waveguides are selected to provide a 50% coupling efficiency between the two waveguides. This allows the optocoupler to operate like a 50/50 splitter.

In addition, the optical device shown in fig. 29 may include two interlayer optical couplers 2914 and 2916. The optical coupler 2914 allows light propagating in the waveguide on the first material layer to pass to the waveguide on the second material layer, and the optical coupler 2916 allows light propagating in the waveguide on the second material layer to pass to the waveguide on the first material layer. The optical couplers 2914 and 2916 allow optical waveguides located in at least two different layers to be used in a multi-channel optical coupler, which in turn enables a compact multi-channel optical coupler.

In addition, the optical device shown in fig. 29 includes a non-coupling waveguide crossover region 2926. In some implementations, the two waveguides (2903 and 2905 in this example) cross each other, while there is no parallel coupling region at the intersection in the uncoupled waveguide intersection region 2926 (e.g., the waveguides may be two straight waveguides that cross each other at an angle of approximately 90 degrees).

Those skilled in the art will appreciate that the foregoing examples are illustrative and that photonic circuits using beam splitters and/or phase shifters may be used to implement many different transfer matrices, including transfer matrices for real and imaginary hadamard transforms, discrete fourier transforms, etc. of any order. One type of photonic circuit (referred to herein as a "diffuser" or "Mode Information Erase (MIE)" circuit) has the following characteristics: if the input is a single photon positioned in one input mode, the circuit delocalizes the photon between each of the plurality of output modes such that the photon has equal probability of being detected in any one of the output modes. Examples of diffusers or MIE circuits include circuits that implement a hadamard transfer matrix. (it should be appreciated that a diffuser or MIE circuit may receive inputs for single photons that are not localized in one input mode, and in which case the behavior of the circuit depends on the particular transfer matrix implemented.) in other cases, the photonic circuit may implement other transfer matrices, including one that provides unequal probabilities of detecting photons in different output modes for single photons in one input mode.

D. Exemplary Photonic Bell state Generator Circuit

A bell pair is a pair of qubits in any type of maximally entangled state (called bell's state). For a double-track encoded qubit, examples of bell states (also referred to as bell ground states) include:

in a computation base (e.g., a logical base) having two states, the Greenberger-Huo En-Cai Linge (Greenberger-Horne-Zeilinger) state is a quantum superposition of all the qubits in a first of the two states with all the qubits in a second state. Using the above logical basis, the general M-qubit GHZ state can be written as:

in some embodiments, the entangled state of the plurality of optical qubits may be generated by coupling two (or more) modes of qubits and measuring the other modes. By way of example, fig. 30 shows a circuit diagram of a bell state generator 3000 that may be used in some dual-track encoded photon embodiments. In this example, modes 3032 (1) -3032 (4) are each initially occupied by photons (represented by wavy lines); modes 3032 (5) -3032 (8) are initially vacuum modes. (those skilled in the art will appreciate that other combinations of occupied and unoccupied modes may be used).

First order mode coupling (e.g., implementing the transfer matrix T of equation (4)) is performed on occupied and unoccupied mode pairs as shown by mode couplers 3031 (1) -3031 (4). Thereafter, mode information erasure coupling (e.g., implementing a four-mode diffusion transformation as shown in fig. 13) is performed for the four modes (modes 3032 (5) -3032 (8)) as shown by mode coupler 3037. The modes 3032 (5) -3032 (8) act as "announce" modes that are measured and used to determine whether the bell state was successfully generated on the other four modes 3032 (1) -3032 (4). For example, detectors 3038 (1) -3038 (4) may be coupled to modes 3032 (5) -3032 (8) after second order mode coupler 3037. Each detector 3038 (1) -3038 (4) may output a typical data signal (e.g., a voltage level on a conductor) that indicates whether it detects a photon (or the number of photons detected). These outputs may be coupled to a representative decision logic 3040 that determines whether a bell state exists on the other four modes 3032 (1) -3032 (4) based on representative output data. For example, the decision logic 3040 may be configured such that the bell state (also referred to as the "success" of the bell state generator) is confirmed if and only if a single photon is detected by each of the exact two of the detectors 3038 (1) -3038 (4). Modes 3032 (1) -3032 (4) may map to the logic states of two qubits (qubit 1 and qubit 2), as indicated in fig. 30. Specifically, in this example, the logic state of qubit 1 is based on the occupancy of modes 3032 (1) and 3032 (2), and the logic state of qubit 2 is based on the occupancy of modes 3032 (3) and 3032 (4). It should be noted that the operation of bell state generator 3000 may be non-deterministic; that is, inputting four photons as shown does not guarantee the creation of bell states on modes 3032 (1) -3032 (4). In one implementation, the success probability is 4/32.

In some embodiments, it is desirable to form a resource state of a plurality of entangled qubits (typically 3 or more qubits, but a bell state can be understood as a resource state of two qubits). One technique for forming larger entanglement systems is through the use of "fusion" gates. The fusion gate receives two input qubits, each of which is typically part of an entanglement system. The fusion gate performs a "fusion" operation on the input qubits that produces one ("type I fusion") or zero ("type II fusion") output qubits in a manner such that the original two entanglement systems are fused into a single entanglement system. Fusion gates are a specific example of a general class of two-particle projection measurements that can be used to create entanglement between qubits and are particularly suited for photonic architectures. Examples of type I and type II fusion gates will now be described.

E. Examples of fusion gate photonics circuits

Fig. 31-36 illustrate some embodiments of photonic circuit implementations for a fusion gate or circuit of photonic qubits that may be used in accordance with some embodiments using type II fusion. It should be understood that these example embodiments are illustrative and not limiting. More generally, as used herein, the term "fusion gate" refers to a device that can implement two-particle projection measurements (e.g., bell projections) that can measure two operators, such as operators XX, ZZ, operator XX, ZY, etc., according to the bell base selected. In polarization encoding, a type II fusion circuit (or gate) employs two input modes, mixes them at a Polarizing Beam Splitter (PBS), and then rotates each of them 45 degrees before measuring them in a computational base. Fig. 31 shows an example. In path encoding, the type II fusion circuit uses four modes, exchanges the second mode and the fourth mode, applies a 50:50 beam splitter between two pairs of adjacent modes, and then detects all modes. Fig. 32 shows an example.

By using so-called "redundant coding" of qubits, fusion gates can be used to build larger entangled states. This is in that a single qubit is represented by multiple photons, i.e.,

such that logical qubits are encoded in n individual qubits. This is achieved by measuring adjacent qubits in the X-base.

This coding (represented graphically as n qubits with no edges between them (as in fig. 33 (b))) has the following advantages: the brix measurement on redundant qubits does not split clusters, but removes photons measured from redundant encoding, and combines adjacent qubits into one single qubit that inherits the combination of input qubits, possibly adding phase. In addition, another advantage of this type of fusion is that it is resistant to loss. Two modes are measured and thus a detection mode that declares successful when one photon is lost cannot be obtained. Finally, type II fusion does not need to distinguish between different photon numbers, as both detectors need to click to declare a successful fusion, and this only occurs when the photon count at each detector is 1.

Fusion was successful with 50% probability when single photons were detected at each detector in polarization encoding. In this case, the gate effectively makes a bell state measurement of the qubit sent through it, projecting the logical qubit pair to the maximum entangled state. When the gate fails (as declared by zero or two photons at one detector), it performs measurements on the individual photons in the computation base, removing them from the redundant code, but without destroying the logical qubits. The effect of fusing in generating clusters is depicted in fig. 33, where (a) shows the measurement of a qubit in a linear cluster in the X base to combine that qubit with its neighbors into a single logical qubit, and (c) and (c') show the effect of the success and failure of gates on the structure of the cluster. It can be seen that successful fusion allows two-dimensional clusters to be constructed.

The correspondence between the detection pattern and the claus (Kraus) operator implemented by the gate on that state can be retrieved. In this case, since both qubits are detected, these are projection operators (projectors):

where the first two rows correspond to "success" results, which project two qubits into bell states, and the second two rows correspond to "failure" results, in which case the two qubits are projected into a direct product state.

In some embodiments, the probability of success of type II fusion may be increased by using an auxiliary bell pair or a single photon pair. The use of a single auxiliary bell pair or two single photon pairs allows the success probability to be increased to 75%.

One technique for enhancing fusion gates comes from the following recognition: when it succeeds, it is equivalent to a bell state measurement of the input qubit. Thus, increasing the probability of success of a fusion gate corresponds to increasing the probability of success of the bell-state measurement it performs. Two different techniques have been developed by Grice (Grice) (using Bell pairs) and by Ewert (Ewert) and Fan Luke (van Loock) (https:// arxiv. Org/pdf/1403.4841. Pdf) (using single photons) to increase the probability of a district decibel state.

The former shows that the auxiliary bell pair allows a 75% probability of success to be achieved and that in theory the process can be iterated using increasingly complex interferometers and large entanglement states to achieve any probability of success. However, the complexity of the circuit and the size of the necessary entanglement states may make this impractical.

The second technique uses four single photons (input in pairs in two modes with opposite polarizations) to increase the probability of success to 75%. It has also been shown numerically that this process can be iterated a second time to obtain a probability of 78.125%, but not to be able to increase the success rate arbitrarily as other schemes.

Fig. 34 shows a type II fusion gate that is enhanced once using both techniques in polarization and path coding. The probability of success for both circuits is 75%.

The following describes a detection mode for two types of circuits that declares fusion successful.

When bell states are used to enhance fusion, the logic behind the "successful" detection mode is best understood by considering two pairs of detectors: the group corresponds to the input photon modes (polarization modes 1 and 2, top 4 modes of path encoding) and to the bell pair input modes (polarization modes 3 and 4, bottom 4 modes of path encoding). These pairs are referred to as "primary" and "secondary" pairs, respectively. Then, a successful fusion is declared whenever: (a) a total of 4 photons detected; and (b) detecting less than 4 photons in each set of detectors.

When 4 single photons are used as auxiliary resources, gate success is declared whenever: (a) detecting 6 photons in total; and (b) detecting less than 4 photons at each detector.

When the gate is successful, two input qubits are projected onto one of the four bell pairs, since these qubits can be distinguished from each other due to the use of auxiliary resources. The specific projection depends on the detection pattern obtained as described above.

If no assist is present or only some of it (in the case of four single photon assist), two enhanced type II fusion circuits, each designed to employ one bell pair and four single photons as assist, can be used to perform type II fusion with variable power probability. This is particularly useful as it allows the fusion to be performed in a flexible way using the same circuitry depending on the available resources. If aids are present, they can be entered into the gate to increase the probability of fusion success. However, if an adjunct is not present, a gate may still be used to attempt fusion with a low but non-zero probability of success.

In the case of using a bell pair reinforced fusion gate, the only consideration is the absence of aids. In this case, the logic of declaring a successful detection mode can be understood by considering the detectors in the pair again. Fusion was still successful in the following cases: (a) detecting 2 photons at different detectors; and (b) detecting 1 photon in the "primary" pair and 1 photon in the "secondary" detector pair.

In the case of using four single photon enhanced circuits, multiple modifications are possible, removing all or part of the auxiliary. This is similar to an enhanced bell state generator based on the same principle.

Consider first the case where no auxiliary is present at all. As expected, fusion was 50% successful with a probability, which is the success rate of the unenhanced fusion. In this case, fusion was successful as long as 2 photons were detected at any two different detectors.

For an increased BSG, the presence of an odd number of adjuncts proves to be detrimental to the success probability of the gate: if there are 1 photon, the gate is successful only 32.5% of the time, while if there are 3 photons, the probability of success is 50%, similar to the unenhanced case.

If only two of the four aids are present, two effects are possible.

If they are input in different modes in polarization encoding, i.e. in different adjacent auxiliary mode pairs in path encoding, the success probability is reduced to 25%.

However, if two aids are input in the same polarization mode, i.e. in the same adjacent pair of modes in the path coding, the success probability increases to 62.5%. In this case, by grouping the detectors into two pairs (pair (group 1) in the circuit branch of the input aid and pair (group 2) in the other branch), the mode of declaring success can be understood again. This distinction is particularly clear in polarization encoding diagrams. Considering these groups, the fusion was successful when: (a) detecting 4 photons in total; (b) Less than 4 photons are detected at each detector in group 1; and (c) detecting less than 2 photons at each detector in group 2.

In these examples, the fusion gate works by projecting the input qubit to the maximum entanglement upon success. The basis for encoding such states can be changed by introducing a local rotation of the input qubits before they enter the gate (i.e. before they are mixed at the PBS in polarization encoding). Changing the polarization rotation of the photons before they interfere at the PBS creates different subspaces into which the states of the photons are projected, resulting in different fusion operations on the cluster states. In path coding, this corresponds to the application of a local beam splitter or a combination of beam splitters and phase shifts, which corresponds to the desired rotation between the pairs of modes (adjacent pairs in the above figures) that make up the qubit.

This can be useful for implementing different types of cluster operations in case of success and failure, which is useful for optimizing the construction of large cluster states from small entangled states.

Fig. 35 shows a table with the effect of some rotational variants of type II fusion gates for fusing two small entangled states. The diagram of gates in polarization encoding, the effective projection performed and the resulting impact on cluster states are shown.

Rotation to different ground states is further illustrated in fig. 36, which shows an example of a photonic circuit using a path-encoded type II fusion gate implementation. Fusion gates for ZX fusion, XX fusion, ZZ fusion and XZ fusion are shown. In various cases, a combination of beam splitters and phase shifters (e.g., as described above) may be used.

Those skilled in the art having access to the present invention will appreciate that the embodiments described herein are illustrative rather than limiting and that many modifications and variations are possible. The measurements performed and the states they act on may be selected such that the measurement results have redundancy, which causes fault tolerance. For example, the code may be entered directly with the measurement, or correlations may be generated in the measurement that directly handle the destructive nature of the measurement and entanglement-destructive nature of the measurement in a fault-tolerant manner. This can be handled as part of a typical decoding; for example, a failed fusion operation may be treated as an erasure of the code.

Referring to the figures, components that may include memory may include a non-transitory machine-readable medium. The terms "machine-readable medium" and "computer-readable medium" as used herein refer to any storage medium that participates in providing data that causes a machine to operation in a specific fashion. In the embodiments provided above, various machine-readable media may be involved in providing instructions/code to a processor and/or one or more other devices for execution. Additionally or alternatively, a machine-readable medium may be used to store and/or carry such instructions/code. In many implementations, the computer-readable medium is a physical and/or tangible storage medium. Such a medium may take many forms, including but not limited to, non-volatile media, and transmission media. Common forms of computer-readable media include, for example, magnetic and/or optical media, punch cards, paper tape, any other physical medium with patterns of holes, RAM, programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), FLASH-EPROM, any other memory chip or cartridge, a carrier wave as described hereinafter, or any other medium from which a computer can read instructions and/or code.

The methods, systems, and devices discussed herein are examples. Various embodiments may omit, replace, or add various procedures or components as appropriate. For example, features described with respect to certain embodiments may be combined in various other embodiments. The different aspects and elements of the embodiments may be combined in a similar manner. The various components of the figures provided herein may be embodied in hardware and/or in software. Moreover, technology is evolving, so many elements are examples that do not limit the scope of the invention to those specific examples.

It has proven convenient at times, principally for reasons of common usage, to refer to such signals as bits, information, values, elements, symbols, characters, variables, terms, numbers, or the like. It should be understood, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels. Unless specifically stated otherwise, as apparent from the above discussion, it is appreciated that throughout the present disclosure, discussions utilizing terms such as "processing," "computing," "calculating," "determining," "ascertaining," "identifying," "associating," "measuring," "performing," or the like, refer to the action or processes of a particular apparatus (e.g., a special purpose computer or similar special purpose electronic computing device). Thus, in the context of the present invention, a special purpose computer or similar special purpose electronic computing device is capable of manipulating or transforming signals, which are typically represented as physical electronic, electrical, or magnetic quantities within the memories, registers, or other information storage devices, transmission devices, or display devices of the special purpose computer or similar special purpose electronic computing device.

Those of skill in the art would understand that information and signals used to communicate the messages described herein may be represented using any of a variety of different technologies and techniques. For example, data, instructions, commands, information, signals, bits, symbols, and chips that may be referenced throughout the above description may be represented by voltages, currents, electromagnetic waves, magnetic fields or particles, optical fields or particles, or any combination thereof.

The terms "and," "or," and/or "as used herein may include a variety of meanings that are also contemplated to depend at least in part on the context in which the terms are used. Generally, "or" when used in connection with a list of associations (e.g., A, B or C) is intended to mean A, B and C (used herein in an inclusive sense) and A, B or C (used herein in an exclusive sense). Furthermore, the term "one or more" as used herein may be used to describe any feature, structure, or characteristic in the singular or may be used to describe some combination of features, structures, or characteristics. It should be noted, however, that this is merely an illustrative example and claimed subject matter is not limited to this example. Furthermore, the term "at least one of …" when used in association with a list (e.g., A, B or C) can be construed to mean any combination of A, B and/or C, such as A, B, C, AB, AC, BC, AA, AAB, ABC, AABBCCC, etc.

Reference throughout this specification to "one example," "an example," "certain examples," or "example implementations" means that a particular feature, structure, or characteristic described in connection with the feature and/or example may be included in at least one feature and/or example of claimed subject matter. Thus, the appearances of the phrases "in one example," "an example," "in some examples," "in some implementations," or other similar phrases in various places throughout this specification are not necessarily all referring to the same feature, example, and/or limitation. Furthermore, the particular features, structures, or characteristics may be combined in one or more examples and/or features.

In some implementations, the operations or processes may involve physical manipulations of physical quantities. Usually, though not necessarily, such quantities take the form of electrical or magnetic signals capable of being stored, transferred, combined, compared, and otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to such signals as bits, data, values, elements, symbols, characters, terms, numbers, or the like. It should be understood, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels. Unless specifically stated otherwise as apparent from the discussion herein, it is appreciated that throughout the present invention, discussions utilizing terms such as "processing," "computing," "calculating," "determining," or the like, refer to the action or processes of a specific apparatus (e.g., a special purpose computer, special purpose computing device, or similar special purpose electronic computing device). Thus, in the context of the present invention, a special purpose computer or similar special purpose electronic computing device is capable of manipulating or transforming signals, which are typically represented as physical electronic or magnetic quantities within memories, registers or other information storage devices, transmission devices, or display devices of the special purpose computer or similar special purpose electronic computing device.

In the previous detailed description, numerous specific details have been set forth in order to provide a thorough understanding of the claimed subject matter. However, it will be understood by those skilled in the art that the claimed subject matter may be practiced without these specific details. In other instances, methods and apparatuses that would be apparent to one of ordinary skill have not been described in detail so as not to obscure the claimed subject matter. It is intended, therefore, that the claimed subject matter not be limited to the particular examples disclosed, but that such claimed subject matter may also include all aspects falling within the scope of the appended claims, and equivalents thereof.

For implementations involving firmware and/or software, the methods may be implemented with modules (e.g., procedures, functions, and so on) that perform the functions described herein. Any machine-readable medium tangibly embodying instructions may be used in implementing the methods described herein. For example, the software codes may be stored in a memory and executed by a processor unit. The memory may be implemented within the processor unit or external to the processor unit. As used herein, the term "memory" refers to any type of long-term, short-term, volatile, nonvolatile, or other memory and is not to be limited to any particular type of memory or number of memories, or type of media upon which memory is stored.

If implemented in firmware and/or software, the functions may be stored as one or more instructions or code on a computer-readable storage medium. Examples include computer readable media encoded with a data structure and computer readable media encoded with a computer program. Computer-readable media include physical computer storage media. A storage media may be any available media that can be accessed by a computer. By way of example, and not limitation, such computer-readable media can comprise RAM, ROM, EEPROM, compact disk read-only memory (CD-ROM) or other optical disk storage, magnetic disk storage, semiconductor storage, or other storage devices, or any other medium that can be used to store desired program code in the form of instructions or data structures and that can be accessed by a computer; disk and disc, as used herein, includes Compact Disc (CD), laser disc, optical disc, digital Versatile Disc (DVD), floppy disk and blu-ray disc where disks usually reproduce data magnetically, while discs reproduce data optically with lasers. Combinations of the above should also be included within the scope of computer-readable media.

In addition to being stored on a computer-readable storage medium, instructions and/or data may be provided as signals on a transmission medium included in a communication device. For example, the communication device may include a transceiver with signals indicative of instructions and data. The instructions and data are configured to cause one or more processors to implement the functions outlined in the claims. That is, the communication device includes a transmission medium having signals indicative of information to perform the disclosed functions. At a first time, a transmission medium included in the communication device may include a first portion of information that performs the disclosed function, and at a second time, a transmission medium included in the communication device may include a second portion of information that performs the disclosed function.

Claims (13)

1. A system, comprising:

a first input coupled to a first qubit and a first switch, wherein the first switch comprises a first output, a second output, and a third output;

a first single qubit measurement device coupled to the first output of the first switch;

a second single qubit measurement device coupled to the first output of the second switch;

a first two-qubit measurement device coupled to the second output of the first switch and a second output of the second switch; and

a second two-qubit measurement device coupled to the third output of the first switch and a third output of the second switch.

2. The system of claim 1, further comprising a converged network controller circuit coupled to the first switch and the second switch.

3. The system of claim 1, further comprising a decoder coupled to an output of the first single-qubit measurement device, an output of the second single-qubit measurement device, an output of the first two-qubit measurement device, and an output of the second two-qubit measurement device.

4. The system of claim 1, wherein the first qubit is entangled with one or more other qubits as part of a first resource state and the second qubit is entangled with one or more other qubits as part of a second resource state, wherein none of the qubits in the qubits from the first resource state are entangled with any of the qubits in the qubits from the second resource state.

5. The system of claim 1, wherein the first and second two-qubit measurement devices are configured to perform destructive joint measurements on the first and second qubits and output representative information representative of the joint measurement results.

6. The system of claim 1, wherein the first and second qubits are optical qubits.

7. The system of claim 6, wherein the coupling between the first and second qubits and the first and second switches comprises a plurality of photonic waveguides.

8. The system of claim 1, wherein the first single-qubit measurement device is configured to measure the first qubit on a Z-basis.

9. The system of claim 1, wherein the second single-qubit measurement device is configured to measure a second qubit on a Z-basis.

10. The system of claim 1, wherein the first two-qubit measurement device is configured to perform a projected bell measurement between the first and second qubits.

11. The system of claim 1, wherein the second two-qubit measurement device is configured to perform a projected bell measurement between the first and second qubits.

12. The system of claim 10, wherein the projected bell measurement is a linear optical type II fusion measurement.

13. The system of claim 11, wherein the projected bell measurement is a linear optical type II fusion measurement.