Classifications

• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
  • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
  • G06N10/20—Models of quantum computing, e.g. quantum circuits or universal quantum computers
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
  • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
  • G06N10/40—Physical realisations or architectures of quantum processors or components for manipulating qubits, e.g. qubit coupling or qubit control
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
  • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
  • G06N10/70—Quantum error correction, detection or prevention, e.g. surface codes or magic state distillation
• • B—PERFORMING OPERATIONS; TRANSPORTING
  • B82—NANOTECHNOLOGY
  • B82Y—SPECIFIC USES OR APPLICATIONS OF NANOSTRUCTURES; MEASUREMENT OR ANALYSIS OF NANOSTRUCTURES; MANUFACTURE OR TREATMENT OF NANOSTRUCTURES
  • B82Y10/00—Nanotechnology for information processing, storage or transmission, e.g. quantum computing or single electron logic

Definitions

• This application is generally directed to quantum computing (QC), and more particularly to quantum computer architectures employing lattice array structures of more than one dimension.
• Certain implementations disclosed herein provide a quantum computer architecture employing a lattice array structure of more than one dimension for implementing and interconnecting quantum gates in which more than two logical qubits (e.g., each logical qubit comprising one or more physical qubits) can be simultaneously entangled.
• Certain implementations disclosed herein provide quantum microprocessor configuration and gate array design platforms for quantum processing chips, analogous to field programmable gate arrays (FPGAs) which can advantageously provide a degree of reconfigurability.
• Certain implementations disclosed herein provide quantum microprocessor configuration and gate array design platforms for quantum processing chips or boards (e.g., electrical and/or optical circuits; integrated optical lattices), analogous to application specific integrated circuits (ASICs) which can advantageously be optimized for a specified application and can advantageously provide custom design flexibility.
• quantum processing chips or boards e.g., electrical and/or optical circuits; integrated optical lattices
• ASICs application specific integrated circuits
• Certain implementations disclosed herein provide a quantum computing (QC) system comprising a plurality of qubits arranged substantially in a plurality of substantially planar regions (e.g., planes; layers) that are substantially parallel to one another, at least some of the substantially planar regions comprising two or more qubits and one or more qubits of each substantially planar region configured to interact with one or more qubits of at least one neighboring substantially planar region.
• QC quantum computing
• the QC system can comprise a plurality of 2-D lattice layers (e.g., atomic; photonic), the lattice layers substantially parallel to each other, and the QC system can comprise a multiple-qubit gate array comprising the plurality of qubits arranged as a plurality of multiple-qubit gates positioned in a region spanning multiple lattice layers.
• 2-D lattice layers e.g., atomic; photonic
• the QC system can comprise a multiple-qubit gate array comprising the plurality of qubits arranged as a plurality of multiple-qubit gates positioned in a region spanning multiple lattice layers.
• the multiple-qubit gate array can comprise lattices configured to contain and/or selectively address neutral (e.g., uncharged) atoms, Rydberg atoms, and/or other qubits confined in multilayer lattice arrays, such as nitrogen- vacancy or NY centers (e.g., engineered in diamond or other crystal), arrays of trapped Bose-Einstein condensates (BECs), and others.
• neutral e.g., uncharged
• NY centers e.g., engineered in diamond or other crystal
• BECs Bose-Einstein condensates
• these substantially parallel lattice array layers can be arranged substantially in a plurality of substantially planar regions (e.g., planes; levels), with at least some of the qubits of at least one substantially planar region configured to interact (e.g., to be quantum-mechanically entangled) with at least some of the qubits of at least one other (e.g., neighboring) substantially planar region.
• substantially planar regions e.g., planes; levels
• the qubits of at least one substantially planar region configured to interact (e.g., to be quantum-mechanically entangled) with at least some of the qubits of at least one other (e.g., neighboring) substantially planar region.
• Certain implementations disclosed herein comprise lattice configurations comprising multiple fully-connected qubits arranged as arrays of three- dimensional (3-D) lattice structures (e.g., to form gates), with simultaneous multi-qubit gate operations enabled by qubits arrayed in geometric layouts.
• Certain implementations disclosed herein address a common set of key challenges to building an optimally efficient quantum computer. Principal among these challenges is how to engineer implementable geometric structures of qubits in which maximum numbers of nearest neighbor, next nearest neighbor, next-next nearest neighbor, etc. qubits can be simultaneously entangled to effect up to many-qubit quantum gate operations natively, or geometrically within a single gate operation (see, e.g., U.S. Provisional Appl. No.
• a quantum computing (QC) system comprises a first plurality of logical qubits in a first substantially planar region and a second plurality of logical qubits in a second substantially planar region that is substantially parallel to the first substantially planar region. At least some of the first plurality of logical qubits are configured to interact with one another, and at least some of the second plurality of logical qubits are configured to interact with one another and to interact with the at least some of the first plurality of logical qubits.
• the QC system comprises additional pluralities of logical qubits in additional substantially planar regions that are substantially parallel to the first and second substantially planar regions and at least some of the second plurality of logical qubits are configured to interact with one or more of the additional pluralities of logical qubits.
• a quantum computing (QC) system comprises a plurality of confinement regions configured to contain logical qubits forming quantum gates in a multilayer qubit lattice array comprising more than two dimensions.
• the quantum gates are configured to perform quantum logic operations involving three or more logical qubits natively without reliance on concatenations of one- and two-qubit gates.
• FIG. 1 schematically illustrates a perspective view of an example quantum computing (QC) system comprising logical qubits in substantially planar regions in accordance with certain implementations described herein.
• QC quantum computing
• FIG. 2A schematically illustrates a perspective view of an example system comprising 36 logical qubits in three substantially planar regions that are substantially parallel to one another in accordance with certain implementations described herein.
• FIG. 2B schematically illustrates a perspective view of another example system comprising 60 logical qubits in three substantially planar regions that are substantially parallel to one another in accordance with certain implementations described herein.
• FIG. 3A schematically illustrates a perspective view of a plurality of logical qubits arranged in three substantially planar regions that are substantially parallel to one another (e.g., compatible with the example systems of FIGs. 2A and 2B) in accordance with certain implementations described herein.
• FIGs. 3B and FIG. 3C each schematically illustrate a perspective view of a plurality of logical qubits arranged in five substantially planar regions that are substantially parallel to one another in accordance with certain implementations described herein.
• FIGs. 4A and 4B schematically illustrate a top view (e.g., along the z-axis) and a perspective view, respectively, of a portion of an example system comprising three planar regions each comprising a corresponding plurality of logical qubits in accordance with certain implementations described herein.
• FIGs. 5A and 5B schematically illustrate a top view (e.g., along the z-axis) and a perspective view, respectively, of a portion of another example system comprising three planar regions each comprising a corresponding plurality of logical qubits in accordance with certain implementations described herein.
• FIG. 5C schematically illustrates a perspective view similar to that of FIG. 5B in accordance with certain implementations described herein.
• FIGs. 6A and 6B schematically illustrate a top view (e.g., along the z-axis) and a perspective view, respectively, of a portion of still another example system comprising three planar regions each comprising a corresponding plurality of logical qubits in accordance with certain implementations described herein.
• FIG. 6C schematically illustrates a perspective view similar to that of FIG. 6B in accordance with certain implementations described herein.
• FIGs. 6D and 6E schematically illustrate a top view (e.g., along the z-axis) and a perspective view, respectively, of a portion of still another example system comprising three planar regions, each comprising a corresponding plurality of logical qubits in accordance with certain implementations described herein.
• FIG. 6F schematically illustrates a perspective view similar to that of FIG. 6E in accordance with certain implementations described herein.
• FIGs. 7A-7E schematically illustrate various views of an example multiple-qubit gate cell (e.g., a C 20 NOT gate cell or C 20 (p gate cell having 21 qubits in a Pyrochlore configuration with a target qubit in the center and control qubits elsewhere) formed among multiple 2-D planar optical trap regions and having a plurality of addressing laser light beams in accordance with certain implementations described herein.
• a multiple-qubit gate cell e.g., a C 20 NOT gate cell or C 20 (p gate cell having 21 qubits in a Pyrochlore configuration with a target qubit in the center and control qubits elsewhere
• FIGs. 8A-8E schematically illustrate various views of trapping laser light beams in accordance with certain implementations described herein.
• FIGs. 9A-9B schematically illustrate perspective views of a three-layer, 10 x 10 atom trap lattice created by intersecting arrays of trapping laser light beams in accordance with certain implementations described herein.
• FIG. 9C schematically illustrates a top view of the structure of FIGs. 9A-9B in accordance with certain implementations described herein.
• FIG. 10A schematically illustrates a perspective view of a partially assembled QC system comprising a three-layer, 10 x 10 atom lattice with three sets of addressing laser light beams and two intersecting arrays of trapping laser light beams in accordance with certain implementations described herein.
• FIGs. 10B-10D show other views from different directions of the three sets of addressing laser light beams and two intersecting arrays of trapping laser light beams of FIG. 10A in accordance with certain implementations described herein.
• FIG. 10E schematically illustrates a perspective view of another partially assembled QC system comprising a three-layer, 10 x 10 atom lattice with four sets of addressing laser light beams and two intersecting arrays of trapping laser light beams in accordance with certain implementations described herein.
• FIGs. 10F-10H show other views from different directions of the four sets of addressing laser light beams and two intersecting arrays of trapping laser light beams of FIG. 10E in accordance with certain other implementations described herein.
• FIG. 11A schematically illustrates a view of the three-layer, 10 x 10 3- D atom lattice of FIGs. 9A-9C and 10A-10D viewed through the first optical detector port 242a in accordance with certain implementations described herein.
• FIG. 1 IB schematically illustrates a side cross-sectional view of the structure of FIG. 11A in accordance with certain implementations described herein.
• FIG. 12 schematically illustrates a perspective view of an example assembled QC system corresponding to FIGs. 8E, 9A, 10A, and 11A in accordance with certain implementations described herein.
• FIG. 13 shows three tables comparing the total number of qubits that can be entangled simultaneously at given gate fidelities for various multilayer qubit lattice arrays in accordance with certain implementations described herein.
• Certain implementations disclosed herein comprise lattice configurations comprising multiple fully-connected qubits arranged as arrays of three- dimensional (3-D) lattice structures (e.g., cells), with simultaneous multi-qubit gate operations enabled by qubits arrayed in geometric layouts.
• Certain implementations disclosed herein advantageously add to the number of lattice layers and increase the number of qubits that can be entangled simultaneously over the limits currently achievable (e.g., with acceptable gate fidelities) using current square-based photonic lattices for Rydberg atoms, and potentially over qubit densities achievable in the hexagonal multilayer surface- trap geometries for ions.
• Certain implementations disclosed herein address challenges to building an optimally efficient quantum computer.
• One such challenge is how to engineer implementable geometric structures of qubits in which maximum numbers of nearest neighbor, next-nearest neighbor, next-next-nearest neighbor, etc. qubits can be simultaneously entangled to provide many-qubit quantum gate operations natively (e.g., geometrically within a single gate operation).
• This engineering challenge comprises two coequal parts.
• One part is how to confine (e.g., trap; engineer crystal lattice site locations of) the qubits in optimal entangling geometries.
• confinement generally entails engineering complex intersections of several lasers along with precise periodic location of defect sites in crystals, electrodes, and/or magnetic fields.
• the other part of the engineering challenge is how to address (e.g., illuminate; excite; manipulate) and read out (e.g., detect; image) the qubits as selectively as possible (e.g., individually).
• This addressing can go beyond the creation of entanglement simultaneously between as many qubits as possible, which can often be done with global addressing. Rather, exploiting the full computational power of a number of entangled qubits can also include manipulating and reading out individual qubit states of a desired multiqubit gate operation, which utilizes the ability to address each qubit individually as needed.
• each quantum computing gate is exponential in (e.g., proportional to 2 to the power of) the number of qubits of simultaneously entangled qubits that can fully participate as logical qubits in each gate operation.
• overhead e.g., error correcting qubits; redundant qubits; ancilla qubits
• is to be factored in which grows dramatically (e.g., by orders of magnitude) as the size of quantum computing architectures increases to scales where surface codes are utilized to correct the errors of very large numbers of qubits.
• the total advantage of entangling more individually addressable (e.g., logical) qubits simultaneously in gate cells up front can greatly surpass even the initial exponential speedup indicated above by further greatly reducing or precluding the use of orders of magnitude of added overhead that surface-code error correction entails.
• Certain implementations disclosed herein can be configured as multiple two-dimensional (2-D) (e.g., planar) qubit arrays (e.g., layers) that are substantially parallel to one another and that form arrays of three-dimensional (3-D) cells.
• the arrays of cells can be analogized to or referred to as 3-D crystal structures.
• While various implementations are described herein as utilizing trapped neutral (e.g., uncharged) atom qubits or Rydberg atom qubits in optical lattice structures (see, e.g., I.
• NV centers nitrogen-vacancy centers
• fidelity coherence time
• full-connectivity between up to 10 qubits and multipartite entangled states with up to 7 qubits
• C. E. Bradley et al. “A 10-qubit solid-state spin register with quantum memory up to one minute,” Phys. Rev. X 9, 031045 (2019)
• Such properties along with the atom-like properties, that NV centers exhibit make them one more example of qubit technology that can be directly applied to certain implementations described herein, such as substantially equivalent solutions to the problem of forming 3-D lattices that enable both entanglement of maximum numbers of qubits simultaneously and optimal geometric accesses for addressing (e.g., initializing, performing gate operations on, and reading out the value of) qubits individually in multilayer optical lattices.
• Still other implementations can use other alternative qubit technologies, for example, trapped Bose-Einstein condensates (BECs); neutral molecules; phonons; photons; and others.
• BECs trapped Bose-Einstein condensates
• Certain implementations of the quantum computing (QC) system described herein advantageously provide a multilayer architecture to enable optimal numbers of qubits to be entangled simultaneously between nearest neighbors, next-nearest neighbors, and next-next-nearest neighbors, and potentially beyond.
• Certain implementations include geometries configured to enable optimal combinations of electrical, magnetic, and optical addressing, control, detection, and readout as are needed to engineer scalable quantum processors.
• Arrays of qubits that are fully connected provide more efficient, flexible choices for executing quantum algorithms in hardware than designs in which entangled gate operations are limited to specific pairs, due to type of qubit employed or their layout. This improved efficiency and flexibility grows rapidly with the number of qubits in an array.
• Adding a capability to perform gate operations involving more than two qubits at a time can significantly accelerate the efficiency gains over designs limited to one- and two-qubit gates, tens of which can be replaced with one four-qubit gate.
• Certain implementations described herein use multiple qubit arrays (e.g., multiple directly- connected planar lattices) to advantageously avoid problems of one- and two-dimensional geometry configurations (e.g., limits on connectivity; crowding of components needed to control each qubit in an increasingly dense gate array when limited to only two dimensions).
• multi-dimensional cells of qubits are formed that resemble crystals, such as pyrochlores.
• multi-dimensional cells of qubits are reconfigurable and can be formed into various crystal structures according to which nearest neighbor, next-nearest-neighbor qubits (e.g., within a cell; across cells; across layers) and beyond are simultaneously entangled for a selected gate operation. Utilizing many qubits to participate per gate can also reduce requirements of circuit depth, error correction and interference mitigation.
• Reconfigurable component cells can enable quantum FPGA (QFPGA) and ASIC (QASIC) layout (e.g., chips).
• Certain implementations of the QC system described herein comprise multi-layer configurations comprising multiple fully-connected qubits arranged as arrays of three-dimensional (3-D) lattice structures (e.g., cells) with simultaneous multi-qubit gate operations enabled by qubits arrayed in geometric layouts.
• 3-D lattice structures e.g., cells
• multiple planar qubit arrays in layers e.g., rows and columns; lattice
• the qubits can be localized (e.g., trapped; suspended; confined) within multiple qubit lattice containment zones (e.g., parallel neutral- atom trap arrays; NV centers) to enable optimal coherent connectivity (e.g., entanglement) directly between nearest neighbor qubits, next-nearest neighbor qubits, etc. across multiple substantially planar array regions (e.g., layers; levels; planes) without requiring inefficient photonic or other interconnects (e.g., quantum teleportation) entailing lossy conversions of in situ processing qubits to other species or data bits or significant time delays.
• multiple qubit lattice containment zones e.g., parallel neutral- atom trap arrays; NV centers
• optimal coherent connectivity e.g., entanglement
• substantially planar array regions e.g., layers; levels; planes
• quantum teleportation e.g., quantum teleportation
• Certain such implementations utilize geometrically symmetric cellular structures that provide the capability to perform gate operations involving more than two qubits at a time that can significantly accelerate the efficiency gains over designs limited to one- and two-qubit gates.
• Certain other implementations can be configured as multiple two-dimensional (2- D) qubit lattice arrays (e.g., planar lattices; grids; rows and columns) that are substantially parallel to one another to enable 3-D structures to be formed between multiple 2-D layers.
• Certain implementations of the QC system described herein comprise a plurality of multiple-qubit three dimensional (3-D) gate cells, each cell comprising at least three qubits that can be fully connected simultaneously across three dimensions, and a plurality of multiple-qubit cells configured for gate operations of two or more of the multiple-qubit gates.
• the QC system of certain such implementations can comprise multiple substantially parallel qubit containment lattices, such as parallel neutral atom trap arrays, that enable optimal coherent connectivity or entanglement directly between nearest neighbor qubits, next-nearest neighbor qubits, etc.
• the multiple-qubit cells can be configured using geometric symmetry to enable multiple-qubit gates to be effected natively, in one gate operation, without reliance on concatenations of multiple one- and two-qubit gates. Leveraging the symmetry of equilateral coupling distances between multiple qubits in a cell enables more than two entangled qubits at a time to perform gate operations that would otherwise require many more qubit gate operations comprising only one- and two-qubit gates.
• Certain other implementations disclosed herein provide a QC system comprising a plurality of qubits arranged substantially in a plurality of 2-D array layers (e.g., 2-D lattices; planes; grids) that are substantially parallel to one another, at least some of the 2-D array layers comprising two or more qubits and one or more qubits of each array layer configured to interact with one or more qubits of at least one neighboring 2-D array layer.
• 2-D array layers e.g., 2-D lattices; planes; grids
• the QC system can comprise a plurality of atom trap layers (e.g., 2-D optical lattice traps) substantially parallel to one another, and the QC system can further comprise multiple-qubit, 3-D gate arrays comprising the plurality of qubits arranged as a plurality of multiple-qubit gates positioned in regions between and/or within the trap layers.
• the qubits of the multiple-qubit gate array can comprise 2-D trap arrays configured to contain atomic qubits comprising neutral atoms (e.g., uncharged atoms, Rydberg states), NV centers, or other qubit species.
• qubits are confined in lattices (e.g., trapped in potential wells; in atomic vacancy centers), which can be created by optical-beam (e.g., laser) lattices, and can be arranged substantially in a plurality of substantially parallel trap array layers (e.g., lattices; levels; planes), with at least some of the quhits of at least one 2- D trap array layer configured to interact (e.g., to be quantum-mechanically entangled) directly with at least some of the qubits of at least one other (e.g., neighboring) substantially parallel trap array layer.
• lattices e.g., trapped in potential wells; in atomic vacancy centers
• optical-beam lattices e.g., laser lattices
• substantially parallel trap array layers e.g., lattices; levels; planes
• the symmetric or equilateral coupling geometries of the multiple qubits per cell can enable more complex quantum gates to be performed in a single gate operation and can additionally reduce requirements of circuit depth, error correction and interference mitigation.
• the interchangeable component cells can enable quantum FPGA (QFPGA) and ASIC (QASIC) layout (e.g., chips) and can be highly reconfigurable.
• Certain implementations of the QC system described herein advantageously provide a three-dimensional (3-D) layout of qubits and/or qubit gates that facilitate many more qubits and/or qubit gates being used for computations than are provided using one-dimensional (1-D) layouts or two-dimensional (2-D) layouts (e.g., J.I. Cirac and P. Zoller, “A scalable quantum computer with ions in an array of microtraps,” Nature, Vol. 404, p. 579 (2000); J. Chiaverini et al., Quant. Inf. Comp. Vol. 5, 419 (2005)).
• certain implementations described herein provide a 3-D layout of qubit gates, each comprising multiple atomic qubits (e.g., three or more simultaneously entangled neutral atoms, charged atoms, NV center qubits, etc.), while providing sufficient spacing to facilitate electrical connections and optical pathways for addressing, manipulation, control, readout, and potential sideband cooling of each qubit, and providing line of sight access angles.
• atomic qubits e.g., three or more simultaneously entangled neutral atoms, charged atoms, NV center qubits, etc.
• the qubits can be described as being “fully connected.”
• ions e.g., charged atoms
• 1-D linear atom trap can demonstrate measurably greater potential processing power than can the same number of qubits that are only pair-wise connected (see, e.g., N.M. Linke et al., “Experimental comparison of two quantum computing architectures”, PNAS, Vol. 114, no.13 (2017)).
• Quantum computing (QC) designs demonstrated over the past two decades indicate that the parameters which most affect how quickly a quantum computer can outperform its classical computer counterpart are not based simply on how many qubits are wired together in some fashion. Demonstrated performance of such systems has come down to qubit fidelities (e.g., how precisely the system can perform gate operations), how the qubits are interconnected, and how much overhead is used to allow the qubits to work together to compute solutions to hard problems.
• qubit fidelities e.g., how precisely the system can perform gate operations
• One-qubit gates simply entail flipping a qubit by itself from “0” to “1” or to a special quantum superposition of “0” and “1”.
• Two-qubit gates connect two qubits using superposition combined with quantum entanglement such that anything done to one of the qubits affects the other.
• a target qubit may start out in state “0” or state “1” and can be in any superposition of “0” and “1” (e.g., halfway between “0” and “1”).
• the function of a quantum controlled-NOT (CNOT) gate is to flip the target qubit if the control qubit is in state “1”; otherwise it does nothing.
• CNOT quantum controlled-NOT
• One- or two- qubit gates can be implemented directly in many different quantum gate-based architectures. For more complex gate operations, implementations that can entangle more than two qubits at once can have a significant impact on the total number of qubits and steps performed to effect the operation and the algorithms that incorporate them (see, e.g., C. Figgatt et al., “Parallel entangling operations on a universal ion-trap quantum computer,” Nature, Vol. 571 (2019); Y. Lu et al., “Global entangling gates on arbitrary qubits,” Nature, Vol. 571 (2019)).
• Certain implementations described herein use multiple fully connected, high-fidelity qubits.
• the advantages of such certain implementations e.g., how much more efficient a particular quantum gate operation can be, as opposed to using combinations of one- and two-qubit gates
• C 3 NOT quantum triply-controlled-NOT
• the C 3 NOT gate is also called a super-Toffoli gate.
• three control qubits must all be in a specified state (e.g., “1,1,1”) in order to flip a fourth target qubit from “1” to 0”.
• Multiply-controlled NOT gates are described in reference sources generally as comprising extended series of one- and two- qubit gate operations (see, e.g., M.A. Nielsen and I.L. Chuang, “Quantum Computing and Quantum Information,” 1st ed. (Cambridge Univ. Press, 2000)).
• the extent to which these one- and two-qubit gate series increase even more in physical implementations depends on type of qubits used and on how many qubits can be fully connected and entangled with one another.
• a C 3 NOT gate implemented using four fully-connected, multiply-entangled ions at the same time, in an appropriate physical layout, can be configured with a small fraction of the number of the quantum gate operations used in a C 3 NOT gate implemented only with one- and two-qubit gates.
• Methods for implementing the simpler C 2 NOT Toffoli gate were first described using trapped ions (see, e.g., J.I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. Vol. 74 (20) (1995)) and later demonstrated (see, e.g., T. Monz et al., “Realization of the Quantum Toffoli gate with Trapped Ions,” Phys.
• the improved efficiency example referenced above can be greatly multiplied according to the number of controls in each C n NOT gate through concomitant reductions in quantum gates needed to implement them.
• Other multiply-controlled gate operations, including phase rotations, can exhibit similar improvements in efficiency using a physical configuration that involves more than two entangled qubits simultaneously. These improvements up front can greatly reduce error correction.
• the overhead of error-correction qubits plus ancillas can represent an increase in the number of qubits of one order of magnitude or roughly a factor of ten, with a proportionate decrease in efficiency.
• the overhead can increase further to multiple orders of magnitude.
• a quantum computer that benefits from aggregate efficiencies of fully-connected, high-quality qubits and employs multi-qubit gate operations (e.g., performed natively by exploiting multi-dimensional geometry) can use significantly fewer steps and significantly fewer total numbers of qubits.
• native gate operations indicates that more than two qubits can participate at the same time, by virtue of a geometric layout.
• Certain native multi-qubit gate implementations described herein can advantageously perform algorithms without using extensive error correction overhead.
• significant improvements in overall design efficiency, due to reduced overhead can be achieved, using multiple orders of magnitude fewer quantum resources, both to perform basic quantum computing algorithms or subroutines and to show practical utility with increased speed as compared to classical computing systems.
• Optical interconnects can be employed by transferring a qubit state from an atomic qubit to a photon, then sending the photon to another linear trap where the quantum state is transferred to another atom.
• Quantum teleportation One type of protocol that is commonly used for such a process is termed “quantum teleportation.” Such interconnections impart time delays and potential inefficiencies in the conversions (e.g., from an atomic qubit to a photon, and to a second atom). Certain implementations described herein advantageously provide an alternate configuration to optimize more direct qubit-to-qubit interactions simultaneously than can be done efficiently using linear or 2-D elements with optical interconnects. When scaling up to larger numbers of qubits, certain such implementations can advantageously reduce or stave off the number of optical interconnections between nodes with their associated penalties (e.g., time delays; ion- photon conversion losses).
• Rectangular two-dimensional (2-D) grid configurations have been previously adopted for some trapped ion approaches, and superconducting qubit (SCQ) schemes.
• SCQ superconducting qubit
• the interactions between the qubits generally have been limited to one- and two-qubit operations that occur within the trapped ion grid lanes (e.g., by shuttling qubits in and out of lanes through intersections), and usually rely on significant redundancy to add the degree of fault tolerance for raising the probability of success in running an algorithm to usable levels.
• some approaches use global addressing of ensembles of qubits, which are shuttled in and out of aligned intersections in a grid in order to effect a single one- or two-qubit operation redundantly among the many qubits, and then average to reduce errors.
• the overhead in such an approach in terms of numbers of redundant qubits for effecting a single logic operation with sufficient fidelity, grows rapidly with the scale of the logic operations to be performed by the quantum computer.
• entanglement between more than two qubits is enabled by having the qubits arranged in two or more dimensions, to be involved simultaneously and to effect multi-qubit gate operations directly or natively.
• 2-D periodic crystal structures such as regular geometric lattice traps
• triangular ion traps e.g., Penning traps
• traps can be used to form energy topologies that mimic those of the quantum system to be modeled (e.g., to find the lowest electron energy configuration of a given molecule).
• tetrahedral, hexagonal Kitaev models see, e.g., A. Kitaev, “Anyons in an exactly solved model and beyond,” Ann. Phys. Vol. 321, 2 (2006); R. Schmied etal. “Quantum simulation ofthe hexagonal Kitaev model with trapped ions,” NewJ. Phys. 13 115011 (2011)); hybrid lattices; kagome optical lattices (see, e.g., R.
• a quantum simulator having a 2-D hexagonal lattice of ions (e.g., following the Kitaev model) or neutral atoms (e.g., in an optical kagome lattice) in accordance with certain implementations described herein generally can use much simpler qubit control, addressing, and readout schemes with fewer lasers (e.g., global addressing instead of individually addressing lasers), simpler detection schemes, fewer electrode structures, etc. than the complexity and number of such structures of a “full-up” quantum computer capable of quantum gate operations.
• Certain implementations described herein advantageously facilitate solutions to hardware challenges that can grow rapidly and appear daunting or infeasible when designing gate-model QC structures that are scaled up from 2-D trapped atom lattices.
• certain implementations described herein integrate optical elements (e.g., lasers; optical ports; fibers; detectors) into the QC structure for addressing, manipulation, readout, and potential sideband trapping and/or cooling of each qubit, and provide line of sight access angles.
• Certain implementations described herein advantageously provide scalable hardware configurations on which it is possible to directly “write” and run complex quantum algorithms by enabling simultaneous entanglement between optimal numbers of neighboring qubits.
• a multidimensional quantum gate implementation analogous to conventional firmware such as a field programmable gate array (FPGA), can be written directly in the form of multi-qubit gates and reprogrammed flexibly.
• FPGA field programmable gate array
• quantum gate operations advantageously enable multiply controlled quantum gate operations to be run natively by exploiting multidimensional geometry.
• the quantum gate operations are run on a quantum firmware platform in the least number of steps (e.g., without resorting to concatenations of one- and two-qubit gate operations to effect a multiply controlled NOT operation).
• Certain implementations described herein advantageously provide a realizable engineering configuration that enables the quantum firmware platform to be scaled up as needed by integrating electrical and optical accesses for full control and readout of each qubit in a circuit-model architecture to enable universal quantum computing.
• Certain implementations described herein advantageously provide a multi-layer quantum computing structure configured to enable optimal numbers of qubits to be entangled simultaneously between nearest neighbors, next-nearest neighbors, next- next-nearest neighbors, and beyond.
• Certain such implementations include electrical channels and optical accesses for addressing, control, and readout of qubits in scalable quantum processors.
• arrays of qubits that are fully connected advantageously provide more efficient, flexible choices for executing quantum algorithms in hardware than do other designs in which entangled gate operations are limited to specific pairs (e.g., due to the type of qubit employed or their layout). This improved efficiency and flexibility can grow rapidly with the number of qubits in an array.
• Certain implementations described herein are advantageously able to perform gate operations involving more than two qubits at a time, thereby providing significant improvements of efficiency over previous designs that are limited to one- and two-qubit gates (e.g., by replacing tens of such gates with one four-qubit gate).
• Certain implementations described herein advantageously overcome connectivity limitations found in one- and two-dimensional qubit geometries (e.g., linearly trapped ions; ions trapped in 2-D grids, NV centers in 2-D diamond lattices; other 1-D and 2-D qubit lattices).
• arraying qubits in multiple qubit arrays e.g., multiple planar and/or linear qubit arrays
• direct connectivity e.g., entanglement
• Certain implementations described herein advantageously provide multi-dimensional entangling geometries that resemble complex 3-D crystal structures (e.g., pyrochlores).
• specific, unobstructed lines of sight designed into certain implementations advantageously provides optical accesses from multiple angles for addressing each atomic qubit individually and uniquely as well as ensembles of the qubits globally.
• Certain implementations advantageously provide multiple optical accesses for readout of individual atomic qubit states by detectors.
• Certain implementations described herein advantageously utilize neutral atom qubits (e.g., uncharged Rydberg atoms in which an outer electron is elevated to a highly excited state) to form multilayer qubit lattice arrays, which both demonstrates the applicability of this system and methods to alternative qubit types and helps further illustrate the processing potential.
• neutral atom qubits e.g., uncharged Rydberg atoms in which an outer electron is elevated to a highly excited state
• Recently improved techniques for entangling Rydberg atoms make them increasingly viable for quantum computing (see e.g., M. Khazali and K. Molmer (2020); D. Petrosyan et al. (2017)).
• Rydberg atoms are also well suited to serve as another example qubit for the system and methods described herein.
• Techniques for forming Rydberg qubit lattices can include crossing coherent light beams (e.g., lasers) and creating electromagnetic field interactions with the optical sidebands to form energy potential wells, either within intersecting optical beams (e.g., in potential wells formed by red sidebands) or in spaces (e.g., cells) between the light beams (e.g., in potential wells formed by blue sidebands) to trap individual neutral atom qubits, analogous to using electro-potential to trap ion qubits, with many of the same benefits of trapped ion lattice arrays and some specific advantages.
• crossing coherent light beams e.g., lasers
• electromagnetic field interactions with the optical sidebands to form energy potential wells, either within intersecting optical beams (e.g., in potential wells formed by red sidebands) or in spaces (e.g., cells) between the light beams (e.g., in potential wells formed by blue sidebands) to trap individual neutral atom qubits
• Rydberg qubits have yet to demonstrate the highest qubit gate fidelities and longest lifetimes associated with trapped ions
• Rydberg atoms can be configured into hexagonal lattice layers to create crystalline geometries (e.g., pyrochlore structures) equivalent to and/or continuations of those described in U.S. Pat. Appl. Publ. No. 2021/0142204 Al using ion qubits as primary examples.
• the photonic lattice structures utilized to trap the Rydberg atoms can enable very close qubit spacing and additional array layers in accordance with certain implementations described herein.
• Certain implementations described herein advantageously utilize Rydberg atoms in qubit arrays of multilayer lattice structures in geometries equivalent to those described in U.S. Pat. Appl. Publ. No. 2021/0142204 Al (e.g., using primarily charged-atom qubit examples), with the potential to add to the number of lattice layers and to increase the number of qubits that can be entangled simultaneously over the limits currently achievable (e.g., with acceptable gate fidelities) using current square-based photonic lattices for Rydberg atoms, and potentially over qubit densities achievable in the hexagonal multilayer surface-trap geometries described herein for ions.
• certain implementations described herein advantageously provide multi-dimensional entangling geometries equivalent to those of charged atomic qubits (e.g., ions) and resembling 3-D crystal structures (e.g., apyrochlore) of equal, and in some cases increased, complexity and processing power.
• charged atomic qubits e.g., ions
• 3-D crystal structures e.g., apyrochlore
• certain implementations described herein utilize Rydberg atoms in photonic multilayer qubit lattice arrays that advantageously enable minimized inter-qubit spacing (e.g., to less than 10 microns in a photonic trap lattice).
• Certain implementations described herein incorporate hexagonal multilayer lattices arrays equivalent to those described herein for ions to enable full connectivity of more than 12 simultaneously entangled qubits while enabling the addition of lattice layers (e.g., to 3, 4, 5 or more layers). These example implementations can yield the ability to entangle more qubits simultaneously.
• Certain implementations described herein utilize multi-dimensional configurations (e.g., cells; nodes) of qubits, which can resemble 3-D crystal structures (e.g., pyrochlores). Certain implementations utilize many qubits per gate, advantageously reducing (e.g., minimizing) circuit depth, error correction, and interference mitigation. Certain implementations utilize interchangeable component cells which can advantageously enable quantum FPGA (QFPGA) and quantum ASIC (QASIC) chips.
• QFPGA quantum FPGA
• QASIC quantum ASIC
• qubits compatible with certain implementations described herein include but are not limited to: subatomic particles; neutral atoms; ions; neutral molecules; charged molecules; Bose-Einstein condensates (BECs); electrons; electron holes; excitons; magnetic qubits; nitrogen-vacancy (NV) centers (e.g., in diamond); phonons; photons; quantum dots; Rydberg atoms; spins in silicon; and possibly superconducting qubits.
• the qubits are suitable to effect gate operations directly (e.g., natively), between more than two qubits in the specified configuration.
• the physical architecture of certain implementations can advantageously provide complex gate operations directly, such as a multiply-controlled NOT or phase rotation, without resorting to serial concatenations of one- and two-qubit gates.
• the trapped atom qubits utilized in certain implementations described herein illustrate the nature of quantum interactions to be optimized.
• Germane figures of merit that trapped atoms exhibit include but are not limited to: (i) the fact that they are identical within a given species and therefore extensive calibration or tuning can advantageously be avoided, (ii) the ability to form qubits that have good stability and coherence times relative to their potential gate cycle times, and (iii) continued, demonstrated high fidelity gate operations as compared to competing qubit technologies.
• simultaneous multi-qubit gate operations can be enabled by atoms arrayed in 3-D geometric layouts of multiple identical, fully-connected qubits.
• 3-D geometric layouts of fully-connected qubits can advantageously integrate more than one qubit type (e.g., employ atoms of one species for the multiple control qubits and a second atomic species as a target; employ different atomic species for neighboring cells).
• Certain implementations described herein comprise multilayer qubit arrays utilizing various qubit technologies, including but not limited to: naturally occurring atoms; neutral atoms; charged atoms; ions; molecules; artificially formed atoms; Rydberg atoms; nitrogen-vacancy (NV) centers in diamond; Bose-Einstein condensates (BECs); electrons; photons; quantum particles; quantum dots; phonons; transmons; quantum states that behave as quantum particles. While the figures herein depict various example multilayer qubit arrays comprising uncharged atoms (e.g., neutral atoms; Rydberg states; neutral Rydberg atoms), it is understood that these example multilayer qubit arrays are compatible with other qubit technologies as well.
• uncharged atoms e.g., neutral atoms; Rydberg states; neutral Rydberg atoms
• FIG. 1 schematically illustrates a perspective view of an example quantum computing (QC) system 100 comprising logical qubits 110 in substantially planar regions 120 in accordance with certain implementations described herein.
• the system 100 comprises a first plurality of logical qubits 110a in a first substantially planar region 120a, at least some of the first plurality of logical qubits 110a configured to interact with one another.
• the system 100 further comprises a second plurality of logical qubits 110b in a second substantially planar region 120b that is substantially parallel to the first substantially planar region 120a. At least some of the second plurality of logical qubits 110b are configured to interact with one another and to interact with the at least some of the first plurality of logical qubits 110a.
• the logical qubits 110 of FIG. 1 form multiple-qubit gates across multiple 2-D planar optical trap layers in accordance with certain implementations described herein.
• the example system 100 further comprising a third plurality of logical qubits 110c in a third substantially planar region 120c that is substantially parallel to the second substantially planar region 120b. At least some of the third plurality of logical qubits 110c are configured to interact with one another and to interact with the at least some of the second plurality of logical qubits 110b.
• example systems 100 compatible with certain implementations described herein can comprise at least one additional plurality of logical qubits 110 in at least one additional substantially planar region 120 that is substantially parallel to the first, second, and third substantially planar regions 120a,b,c, with the at least one additional plurality of logical qubits 110 configured to interact with one another and to interact with the at least some of the first, second, and/or third pluralities of logical qubits 110a, b,c.
• the system 100 can comprise four, five, six, or more pluralities of logical qubits 110 arranged in four, five, six, or more substantially planar regions 120 that are substantially parallel to one another and the logical qubits 110 of the various planar regions 120 configured to interact with one another (e.g., intra-planar interactions and inter-planar interactions).
• the pluralities of logical qubits 110 comprise at least one physical qubit selected from the group consisting of: naturally occurring atoms; neutral atoms; charged atoms; ions; molecules; artificially formed atoms; Rydberg atoms; nitrogen-vacancy (NV) centers in diamond; Bose-Einstein condensates (BECs); electrons; photons; quantum particles; quantum dots; phonons; transmons; quantum states that behave as quantum particles.
• the logical qubits 110 of the pluralities of logical qubits 110 are individually addressable.
• At least some of the logical qubits 110 are configured to interact directly with one another to form at least one three- dimensional (3-D) gate cell array (e.g., comprising a plurality of multiple-qubit 3-D gate cells 130) configured to undergo multiple-qubit gate operations in which more than two logical qubits 110 participate simultaneously.
• 3-D gate cell array e.g., comprising a plurality of multiple-qubit 3-D gate cells 130
• each logical qubit 110 of a 3- D gate cell 130 can be configured to be quantum-mechanically entangled with at least one other logical qubit 110 of the 3-D gate cell 130.
• each first logical qubit 110a in the first planar region 120a can be configured to interact directly with some (e.g., nearest neighboring and next-nearest neighboring) other first logical qubits 110a in the first planar region 120a.
• each second logical qubit 110b in the second planar region 120b can be configured to interact directly with some (e.g., nearest neighboring and next-nearest neighboring) other second logical qubits 110b in the second planar region 120b
• each third logical qubit 110c in the third planar region 120c can be configured to interact directly with some (e.g., nearest neighboring and next-nearest neighboring) other third logical qubits 110c in the third planar region 120c.
• each first logical qubit 110a in the first planar region 120a can be configured to interact directly with some (e.g., nearest neighboring and next-nearest neighboring) second logical qubits 110b within the second planar region 120b
• each second logical qubit 110b in the second planar region 120b can be configured to interact directly with some (e.g., nearest neighboring and next-nearest neighboring) third logical qubits 110c within the third planar region 120c
• each third logical qubit 110c in the third planar region 120c can be configured to interact directly with some (e.g., nearest neighboring and next-nearest neighboring) second logical qubits 110b within the second planar region 120b.
• At least some of the logical qubits 110 can also directly interact with even more distant logical qubits 110 (e.g., next-next-nearest neighboring logical qubits 110; next-next-next-nearest neighboring logical qubits 110) in the same planar region 120 and/or in an adjacent planar region 120.
• logical qubits 110 e.g., next-next-nearest neighboring logical qubits 110; next-next-next-nearest neighboring logical qubits 110
• FIG. 2A schematically illustrates a perspective view of an example system 100 comprising 36 logical qubits 110 in three substantially planar regions 120a, b,c that are substantially parallel to one another in accordance with certain implementations described herein.
• FIG. 2B schematically illustrates a perspective view of another example system 100 comprising 60 logical qubits 110 in three substantially planar regions 120a,b,c that are substantially parallel to one another in accordance with certain implementations described herein.
• each comprise a 3-D lattice qubit array comprising 2-D planar optical lattice layers forming multiple-qubit gates (e.g., in a 3-D hexagonal regular or “pyrochlore” lattice qubit array configuration with the logical qubits 110 suspended or trapped in multiple planar layers with orthogonal connections between logical qubits 110 in adjacent substantially planar regions 120).
• 3-D lattice qubit array comprising 2-D planar optical lattice layers forming multiple-qubit gates (e.g., in a 3-D hexagonal regular or “pyrochlore” lattice qubit array configuration with the logical qubits 110 suspended or trapped in multiple planar layers with orthogonal connections between logical qubits 110 in adjacent substantially planar regions 120).
• the logical qubits 110 shown in FIGs. 2A or FIG. 2B are all the logical qubits 110 of the example system 100, while in certain other implementations, the logical qubits shown in FIG. 2A or FIG. 2B are only a subset of the logical qubits 110 of the example system 100.
• the logical qubits 110 form at least one multiple-qubit 3-D gate cell 130. For example, each pair of adjacent 3-D gate cells 130 share two first logical qubits 110a in the first planar region 120a, two second logical qubits 110b in the second planar region 120b, and two third logical qubits 110c in the third planar region 120c.
• Each of the planar regions 120a,b,c of FIG. 2A comprises 12 logical qubits 110 arranged in a substantially hexagonal 2-D pattern and each of the planar regions 120a,b,c of FIG. 2B comprises 20 logical qubits 110 arranged in a substantially hexagonal 2-D pattern.
• the logical qubits 110 of FIGs. 2A and 2B have an intraplanar nearest- neighbor (nn) distance d 1 between each logical qubit 110 and its nearest neighboring logical qubits 110 within each planar region 120a,b,c and for FIGs.
• the intraplanar nearest-neighbor (nn) distance between each logical qubit 110 and its nearest neighboring logical qubits 110 in the other planar regions 120 equals the interplanar distance d 2.
• the intraplanar nn distance d 1 of each of the planar regions 120a,b,c is substantially equal to one another, while in certain other implementations, the intraplanar nn distance d 1 of at least two of the planar regions 120a,b,c differ from one another.
• the interplanar distance between planar regions 120a, b and between planar regions 120b, c are substantially equal to one another, while in certain other implementations, the interplanar distance d between planar regions 120a,b and between planar regions 120b,c differ from one another.
• the intraplanar nn distance d 1 and the interplanar distance d 1 are substantially equal to one another (e.g., less than 15 microns; in a range of 3 microns to 15 microns; in a range of 8 microns to 11 microns), while in certain other implementations, the intraplanar nn distance d 1 and the interplanar distance ifc differ from one another (e.g., by a difference in a range of 1 micron to 3 microns), with either d 1 > d or d 1 ⁇ d .
• the 2A and 2B can have the intraplanar nn distance d 1 equal to the interplanar distance d , the interplanar next-nearest-neighbor (nnn) distance is equal to , the intraplanar nnn distance is equal to d 1 ⁇ 3 , the intraplanar next-next- nearest-neighbor (nnnn) distance is equal to 2 d 1, the interplanar nnnn distance is equal to 2d 1, the interplanar next-next-next-nearest-neighbor (nnnn) distance is equal to and the intraplanar nnnnn distance is equal to
• Other configurations of the 2-D hexagonal patterns of qubits 110 in the planar regions 120 can have different lateral offsets between the planar regions 120 (e.g., offsets parallel to the planar regions 120 such that the qubits 110 in one planar region 120 are not directly above the qubits 110 of an adjacent planar region 120; see, e.g., FIGs. 5A- 5C, 6A-6F).
• such configurations can have other values of the intraplanar nn distance d;, the interplanar distance d 1 , and the various intraplanar and interplanar nnn, nnnn, and nnnnn distances.
• 21 of the logical qubits 110 form a first multiple-qubit 3-D gate cell 130a (e.g., a C 20 NOT gate cell or a C 20 (p gate cell) and 21 of the logical qubits 110 form a second multiple-qubit 3-D gate cell 130b (e.g., a C 20 NOT gate cell or a C 20 ⁇ gate cell).
• a logical qubit 110 in the planar region 120b can be the target qubit of the first gate cell 130a, and can be simultaneously quantum-mechanically entangled with its 20 control qubits (shown in FIG.
• FIG. 2A shows dark grey
• its six intraplanar nearest neighboring logical qubits 110 its two interplanar nearest neighboring logical qubits 110 (e.g., one in each of the planar regions 120a,c), and its twelve interplanar next-nearest neighboring logical qubits 110 (e.g., six in each of the planar regions 120a, c).
• another logical qubit 110 in the planar region 120b can be the target qubit of the second gate cell 130b and can be simultaneously quantum-mechanically entangled with its 20 control qubits 110 (e.g., 14 of which are shown in FIG.
• 21 of the logical qubits 110 can form a third multiple-qubit 3-D gate cell 130c and 21 of the logical qubits 110 can form a fourth multiple-qubit 3-D gate cell 130d.
• Each C 20 NOT/C 20 (p gate cell 130 of FIGs. 2A and 2B can be referred to as a “Pyrochlore cell” due to the similarity of its arrangement to that of pyrochlore crystal structures.
• more logical qubits 110 can be simultaneously quantum-mechanically entangled with one another to form larger multiple-qubit gate cells 130 (e.g., C 34 NOT gate cells; C 34 (p gate cells; C 94 NOT gate cells; C 94 (p gate cells, etc.).
• These larger gate cells 130 can each be referred to as a “Super Pyrochlore cell.”
• FIGs. 2A and 2B indicate some of the entanglements between the various logical qubits 110 with dashed and solid lines of various weights
• all of the logical qubits 110 of a gate cell 130 e.g., first gate cell 130a
• all of the logical qubits 110 of a gate cell 130 can interact directly (e.g., can be simultaneously quantum-mechanically entangled; fully entangled; completely entangled) with one another
• all of the logical qubits 110 of another gate cell 130 e.g., second gate cell 130b
• can interact directly e.g., can be simultaneously quantum-mechanically entangled
• each logical qubit 110 is configured to interact directly with its intraplanar nearest neighboring (nn) logical qubits 110, its interplanar nearest neighboring (nn) logical qubits 110, and its interplanar next- nearest neighboring (nnn) logical qubits 110.
• each logical qubit 110 is further configured to interact directly with its intraplanar next-nearest neighboring (nnn) logical qubits 110, its interplanar next-next-nearest neighboring (nnnn) logical qubits 110, its intraplanar next-next-nearest neighboring (nnnn) logical qubits 110, and/or its interplanar next-next-next-nearest neighboring (nnnn) logical qubits 110.
• each logical qubit 110 is further configured to interact directly with its intraplanar next-next-next-nearest neighboring (nnnn) qubits 110.
• each multiqubit gate cell 130 is formed by 21 entangled qubits 110 (e.g., as shown in FIGs. 2A and 2B) that can be operated as a “native” C 20 NOT gate (e.g., a20-controlledNOT gate) and/or as a “native” C 20 (p gate (e.g., a 20-controlled phase gate).
• a “native” C 20 NOT gate e.g., a20-controlledNOT gate
• p gate e.g., a 20-controlled phase gate
• Such native C 20 NOT/C 20 p gates can be achieved with far fewer gate operations than would be used by one- or two-qubit gates alone.
• the net fidelity of the native C 20 NOT/C 20 p gate provided by the example 21 -qubit gate cell 130 is significantly greater than that of a C 20 NOT/C 20 (p gate comprising many two-qubit gates that may have much higher individual gate fidelities, since the 21 -qubit gate cell 130 does not aggregate the errors of many successive operations to achieve its result.
• the C 20 NOT/C 20 (p gate provided by the example 21-qubit gate cell 130 uses a very small fraction of the steps of a C 20 NOT/C 20 (p gate comprising many two-qubit gates, so the C 20 NOT/C 20 (p gate provided by the example 21 -qubit gate cell 130 is faster and less error-prone (e.g., thereby utilizing significantly less error correction at the outset), as can be seen in a comparison of the probabilities of successful gate operation using both structures and fidelity estimates.
• the number of logical qubits 110 per gate cell 130 can be increased by increasing the number of substantially planar regions 120 and/or by extending the distance at which the logical qubits 110 can directly interact with one another.
• FIG. 3A schematically illustrates a perspective view of a plurality of logical qubits 110 arranged in three substantially planar regions 120 that are substantially parallel to one another (e.g., compatible with the example systems 100 of FIGs. 2A and 2B) in accordance with certain implementations described herein.
• the perspective view of FIG. 3 A is along a direction in which all of the logical qubits 110 are viewable (e.g., none of the logical qubits 110 are obscured by another logical qubit 110).
• FIG. 3A schematically illustrates a perspective view of a plurality of logical qubits 110 arranged in three substantially planar regions 120 that are substantially parallel to one another (e.g., compatible with the example systems 100 of FIGs. 2A and 2B) in accordance with certain implementations described herein
• FIG. 3 A includes lines between some of the logical qubits 110, the lines denoting some of the direct interactions between the logical qubits 110 of a gate cell 130.
• Each logical qubit 110 is configured to interact directly with its intraplanar nearest neighboring logical qubits 110, its interplanar nearest neighboring logical qubits 110, and its interplanar next-nearest neighboring logical qubits 110 and the gate cell 130 of FIG. 3A is a C 20 NOT gate cell or a gate cell (e.g., with an example target qubit shown as black and its control qubits shown as dark grey).
• FIG. 3C schematically illustrates a perspective view of a plurality of logical qubits 110 arranged in five substantially planar regions 120 that are substantially parallel to one another in accordance with certain implementations described herein.
• the perspective views of FIGs. 3B and 3C are along a direction in which all of the logical qubits 110 are viewable (e.g., none of the logical qubits 110 are obscured by another logical qubit 110) and includes lines between some of the logical qubits 110, the lines denoting some of the direct interactions between the logical qubits 110 of a gate cell 130.
• each logical qubit 110 is configured to interact directly with its intraplanar nearest neighboring logical qubits 110, its interplanar nearest neighboring logical qubits 110, and its interplanar next-nearest neighboring logical qubits 110 and the gate cell 130 of FIG. 3B is a C 34 NOT gate cell or a C 34 (p gate cell (e.g., with an example target qubit shown as black and its control qubits shown as dark grey).
• p gate cell e.g., with an example target qubit shown as black and its control qubits shown as dark grey.
• each logical qubit 110 is configured to interact directly with its intraplanar nearest neighboring logical qubits 110, its interplanar nearest neighboring logical qubits 110, its intraplanar next-nearest neighboring logical qubits 110, and its intraplanar next- next-nearest neighboring logical qubits 110.
• the gate cell 130 of FIG. 3C is a C 94 NOT gate cell or a C 94 ⁇ gate cell (e.g., with an example target qubit shown as black and its control qubits shown as dark grey).
• 3A-3C illustrate the advantageously increased degree of simultaneous connectivity (e.g., entanglement) possible by using neutral (e.g., Rydberg) atom traps formed by hexagonal cell, N-layer optical lattices to form “Super Pyrochlore” cells (e.g., a 21-qubit C 20 NOT/ C 20 (p gate 130 in a 3-layer regular hexagonal atom lattice; a 35-qubit C 34 NOT/ C 34 (p gate 130 in a 5-layer regular hexagonal atom lattice; a 95-qubit C 94 NOT/ C 94 (p gate 130 in a 5-layer regular hexagonal atom lattice) in accordance with certain implementations described herein.
• neutral e.g., Rydberg
• FIGs. 4A-4B, 5A-5C, and 6A-6D schematically illustrate portions of example systems 110 with different relative displacements amongst the logical qubits 110 of the multiple substantially planar regions 120 in accordance with certain implementations described herein.
• FIGs. 4A and 4B schematically illustrate a top view (e.g., along the z-axis) and a perspective view, respectively, of a portion of an example system 100 comprising three planar regions 120a,b,c each comprising a corresponding plurality of logical qubits 110a,b,c in accordance with certain implementations described herein.
• the logical qubits 110 within each of the planar regions 120 are arranged in a substantially hexagonal 2-D pattern and the planar regions 120 are substantially parallel to one another.
• the 3-D configurations includes orthogonal connections between qubits in adjacent planar regions 120.
• the logical qubits 110 of a substantially planar region 120 are not substantially displaced along a direction substantially parallel to the planar region 120 from the logical qubits 110 of an adjacent substantially planar region 120. For example, as shown in FIG.
• the logical qubits 110a of the top-most planar region 120a are aligned with corresponding logical qubits 110b,c in the other two planar regions 120b,c such that the logical qubits 110a are viewable from above (e.g., along the z-axis) while the logical qubits 110b, c are obscured by the logical qubits 110a.
• This example system 100 of FIGs. 4A and 4B corresponds to those of FIGs.
• each logical qubit 110 is configured to interact directly with its intraplanar nearest neighboring logical qubits 110, its interplanar nearest neighboring logical qubits 110, and its interplanar next-nearest neighboring logical qubits 110).
• FIGs. 5A and 5B schematically illustrate a top view (e.g., along the z-axis) and a perspective view, respectively, of a portion of another example system 110 comprising three planar regions 120a,b,c each comprising a corresponding plurality of logical qubits 110a,b,c in accordance with certain implementations described herein.
• the logical qubits 110 within each of the planar regions 120 are arranged in a substantially hexagonal 2-D pattern and the planar regions 120 are substantially parallel to one another.
• the 3-D configurations includes nonorthogonal connections between qubits in adjacent planar regions 120, and such an alternative configuration can expand the engineering trade space of optimal access angle choices for the optical lattice.
• the logical qubits 110 of a substantially planar region 120 are displaced (e.g., offset) along a direction substantially parallel to the planar region 120 from the logical qubits 110 of an adjacent substantially planar region 120.
• FIG. 5C schematically illustrates a perspective view similar to that of FIG. 5B, in which the intraplanar nn distance d 1 and the distance between the interplanar nearest neighbor qubits are substantially equal to one another such that the interplanar distance d 1 is substantially equal to d 1 ⁇ 3/2, and in which each logical qubit 110 is configured to interact directly with its intraplanar nearest neighboring logical qubits 110, its interplanar nearest neighboring logical qubits 110, its intraplanar next-nearest neighboring logical qubits 110, its interplanar next-nearest neighboring logical qubits 110, and its interplanar next-next-nearest neighboring logical qubits 110 in accordance with certain implementations described herein.
• an example 23-qubit C 22 NOT/ C 22 ⁇ gates 130 has a target qubit (black) and 22 control qubits (dark grey).
• FIGs. 6A and 6B schematically illustrate a top view (e.g., along the z-axis) and a perspective view, respectively, of a portion of still another example system 110 comprising three planar regions 120a, b,c each comprising a corresponding plurality of logical qubits 110a,b,c in accordance with certain implementations described herein.
• the logical qubits 110 within each of the planar regions 120 are arranged in a substantially hexagonal 2-D pattern and the planar regions 120 are substantially parallel to one another.
• the logical qubits 110 of a substantially planar region 120 are displaced along a direction substantially parallel to the planar region 120 from the logical qubits 110 of an adjacent substantially planar region 120.
• the logical qubits 110a are displaced relative to the logical qubits 110b in a direction along the x-axis (e.g., perpendicular to the y- and z-axes; shown by a dashed double-headed arrow in FIG.
• FIG. 6C schematically illustrates a perspective view similar to that of FIG.
• each logical qubit 110 is configured to interact directly with its intraplanar nearest neighboring logical qubits 110, its interplanar nearest neighboring logical qubits 110, its intraplanar next-nearest neighboring logical qubits 110, its interplanar next-nearest neighboring logical qubits 110, and its interplanar next-next-nearest neighboring logical qubits 110 in accordance with certain implementations described herein.
• an example 25-qubit C 24 NOT/ C 24 cp gate 130 has a target qubit (black) and 24 control qubits (dark grey).
• FIGs. 6D and 6E schematically illustrate a top view (e.g., along the z-axis) and a perspective view, respectively, of a portion of still another example system 110 comprising three planar regions 120a,b,c, each comprising a corresponding plurality of logical qubits 110a,b,c in accordance with certain implementations described herein.
• the logical qubits 110 within each of the planar regions 120 are arranged in a substantially hexagonal 2-D pattern and the planar regions 120 are substantially parallel to one another.
• the logical qubits 110 of a substantially planar region 120 are displaced along a direction substantially parallel to the planar region 120 from the logical qubits 110 of an adjacent substantially planar region 120.
• the logical qubits 110a are aligned with the logical qubits 110b,c, such that the logical qubits 110b,c are obscured by the logical qubits 110a in FIG. 6D.
• each logical qubit 110 is configured to interact directly with its intraplanar nearest neighboring logical qubits 110, its interplanar nearest neighboring logical qubits 110, its intraplanar next-nearest neighboring logical qubits, its interplan
• an example 21-quhit C 20 NOT/ C 20 ⁇ gate 130 has a target qubit (shown as black) and 20 control qubits.
• target qubit shown as black
• FIGs. 5A-5C and 6A-6F schematically illustrate example displacements between the logical qubits 110 of the planar regions 120, the displacements having example directions and magnitudes.
• Certain other implementations have displacements in other directions (e.g., having magnitudes both along the x-axis and the y-axis) and/or having other magnitudes (e.g., 1 ⁇ 2 cell width; 1/3 cell width).
• Certain other implementations have differing interplanar distances ⁇ &.
• the separation between planar regions 120 can be reduced according to the angle from vertical between nearest logical qubits 110 across the adjacent planar regions 120 (e.g., ifc equal to dil2 ⁇ 2-di/3; 3-dil4; 0.85 - ⁇ 7j).
• FIGs. 5A-5C and 6A-6F schematically illustrate example systems 100 having 2-D substantially hexagonal patterns, other 2-D patterns are also compatible with certain implementations described herein.
• the first plurality of logical qubits 110a are individually addressable and arranged in a substantially planar first optical lattice
• the second plurality of logical qubits 110b are individually addressable and arranged in a substantially planar second optical lattice substantially parallel to the first optical lattice
• the third plurality of logical qubits 110c are individually addressable and arranged in a substantially planar third optical lattice substantially parallel to the second optical lattice.
• the system 100 comprises at least one additional plurality of logical qubits 110 in at least one additional substantially planar region 120 substantially parallel to the third substantially planar region 120c, the at least one additional plurality of logical qubits 110 configured to interact with one another and/or to interact with at least some of the third plurality of logical qubits 110c, wherein the at least one additional plurality of logical qubits 110 are individually addressable.
• the number of substantially parallel optical lattices of pluralities of qubits can further comprise a fourth optical lattice, a fifth optical lattice, a sixth optical lattice, a seventh optical lattice, and so on.
• the plurality of confinement regions can further comprise at least one additional confinement region arranged in at least one additional optical lattice substantially parallel to the third optical lattice, the at least one additional optical lattice comprising the at least one additional plurality of logical qubits.
• FIGs. 7A-7E schematically illustrate various views of an example multiple-qubit gate cell 130 (e.g., a C 20 NOT gate cell or C 20 ⁇ gate cell having 21 qubits 110 in a Pyrochlore configuration with a target qubit in the center and control qubits elsewhere) formed among multiple 2-D planar optical trap regions and having a plurality of addressing laser light beams 210 in accordance with certain implementations described herein.
• FIG. 7A depicts the atomic qubits of the multilayer lattice structure of the multiple- qubit gate cell 130. The lines between the qubits 110 in FIG.
• FIG. 7A only denote some of the entanglements between the qubits 110 and lines denoting the other entanglements between the qubits 110 are omitted for clarity.
• FIG. 7B depicts this multiple-quhit gate cell 130 with only some of the addressing laser light beams 210 (e.g., emitted from optical ports 212) that are configured to illuminate the atoms individually, and arrays of trapping laser light beams 220 that create the optical potential wells containing the atoms are omitted for clarity.
• each of the addressing laser light beams 210 illuminates a single atom of the multiple-qubit gate cell 130.
• a limitation of 2-D views is that one illustration by itself can make it appear that a single laser light beam is intersecting more than one atom.
• the laser light beams 210 are arranged in alternating rows and are illustrated as transparent.
• an atom that is between or behind multiple laser light beams 210 may be seen “through” beams that do not address it in a way that can make one atom appear to be in multiple laser light beams, or make one laser light beam appear to intersect multiple atoms when in fact each atom is intersected by at most one addressing laser light beam.
• Certain implementations described herein are configured to entangle a maximum number of qubits simultaneously between nearest neighbors, next- nearest neighbors, next-next-nearest neighbors, etc.
• FIGs. 8A-8E schematically illustrate various views of trapping laser light beams 220 in accordance with certain implementations described herein.
• These example implementations of FIGs. 8A-8D utilizes neutral (e.g., uncharged) Rydberg atoms trapped in 3-D optical lattices (see, e.g., I. Bloch, Nature Physics, 1, 23 (2005)) of potential wells formed in or between intersecting trapping optical beams 220 (e.g., trapping laser light beams emitted from optical ports 222).
• atoms are trapped in minimum potential energy (e.g., null) regions that are formed in lattice spaces between coaligned and crossing optical beams (see, e.g., FIGs. 8A and 8B), whereas in certain other implementations atoms can be trapped within intersecting optical beams (see, e.g., FIGs. 8C and 8D).
• minimum potential energy e.g., null
• FIGs. 8C and 8D atoms are trapped in minimum potential energy (e.g., null) regions that are formed in lattice spaces between coaligned and crossing optical beams
• atoms can be trapped within intersecting optical beams (see, e.g., FIGs. 8C and 8D).
• Such trapping within intersecting trapping optical beams 220 is described by S.E. Anderson, et al. Phys. Rev. Lett. 107, 263001 (2011); T.M. Graham et al., Phys. Rev. Lett. 123,
• FIGs. 8A-8E employ lattice spacings in a range of 8 microns to 11 microns (e.g., less than 12 microns) between nearest-neighbor qubits 110.
• a nominal 8 micron lattice cell size is within the range of optimal values for certain multiply-controlled quantum gate operations based on the referenced data curves.
• atomic qubits arranged in alternating columns can be in alternating planes with staggered offsets (e.g., 1/2 cell offset spacing along x or y, 1/3 cell offset spacing along x or y, 1 ⁇ 4 x 1 ⁇ 4 cell offset cell spacing along x and y) between adjacent planes, which can enable the formation of regular hexagonal prism configurations of fully-connected entangled atomic qubits and can enable slantwise hexagonal prism configurations as seen from certain views.
• staggered offsets e.g., 1/2 cell offset spacing along x or y, 1/3 cell offset spacing along x or y, 1 ⁇ 4 x 1 ⁇ 4 cell offset cell spacing along x and y
• a coordinate system can be defined as the length x and width y of a planar base (e.g., horizontal dimensions) along with height z (e.g., vertical dimension) of the example quantum computing system
• the potential wells formed between trapping laser light beams 220 can be substantially symmetric in a plane (e.g., with cross-sections that are square or diamond-shaped) when viewed at certain angles, which can create substantially uniform potential wells (e.g., saddle-shaped wells; saddle points).
• the first and second arrays of trapping laser light beams 220 can appear rectangular or trapezoidal from certain views.
• trapping laser light beams 220 appearing to be angled from lower left to upper right in FIGs. 8B and 8D, can also be angled downward in the 3-D structure, such that the intersections of the first and second arrays of trapping laser light beams depicted as horizontal in FIGs.
• the atom confinement structure can form regular hexagonal prisms across planes (e.g., as shown in FIGs 1A, IB, ID), and in certain other implementations, the atom-to-atom geometric structure can appear slanted or angular to form slantwise hexagonal prism configurations.
• the optical beam intersections can appear nonorthogonal from various views and angles, however, the angular 3-D lattice structure of certain implementations is designed to allow maximum spatial separation between atoms when viewed from those angles for optimal addressing and readout (e.g., detection) of qubits.
• FIGs. 9A-9B schematically illustrate perspective views of a three-layer, 10 x 10 atom trap lattice created by intersecting arrays of trapping laser light beams 220 in accordance with certain implementations described herein.
• FIG. 9C schematically illustrates a top view of the structure of FIGs. 9A-9B. Also shown are three interlaced arrays of addressing laser light beams 210 in accordance with certain implementations described herein.
• FIG. 10A schematically illustrates a perspective view of a partially assembled QC system 100 comprising a three-layer, 10 x 10 atom lattice with three sets of addressing laser light beams 210a,b,c and two intersecting arrays of trapping laser light beams 220a, b in accordance with certain implementations described herein.
• FIG. 10A schematically illustrates a perspective view of a partially assembled QC system 100 comprising a three-layer, 10 x 10 atom lattice with three sets of addressing laser light beams 210a,b,c and two intersecting arrays of trapping laser light beams 220a, b in accordance with certain implementations described herein.
• FIG. 10A schematically illustrates a perspective view of a partially assembled QC system 100 comprising a three-layer, 10 x 10 atom lattice with three sets of addressing laser light beams 210a,b,c and two intersecting arrays of trapping laser light beams 220a, b in accordance
• FIG. 10A also shows optical ports 214a,b,c through which the addressing laser light beams 210a,b,c extend, optical fiber ports 222a, b emitting the trapping laser light beams 220a, b, a magnetic field source 230, configured to generate a magnetic field as part of the optical trap configuration (electrodes 234 configured to generate an electric field as part of the optical trap configuration are shown in FIG. 10A), and first and second optical detectors 240a, b (e.g., CCD detector array) configured to receive and detect fluorescent light from the qubits via corresponding optical detector ports 242a, b.
• FIGs. 10B-10D show other views from different directions of the three sets of addressing laser light beams 210a,b,c and two intersecting arrays of trapping laser light beams 220a, b of FIG. 10A in accordance with certain implementations described herein.
• FIGs. 10A-10D schematically illustrate an example QC system 100 having three sets of addressing laser light beams 210a,b,c
• the QC system 100 can comprise four or more sets of addressing laser light beams 210.
• FIG. 10E schematically illustrates a perspective view of another partially assembled QC system 100 comprising a three-layer, 10 x 10 atom lattice with four sets of addressing laser light beams 210a,b,c,d and two intersecting arrays of trapping laser light beams 220a, b in accordance with certain implementations described herein.
• 10F-10H show other views from different directions of the four sets of addressing laser light beams 210a,b,c,d and two intersecting arrays of trapping laser light beams 220a, b of FIG. 10E in accordance with certain other implementations described herein.
• all directions lying in the x-y plane are defined as 0 degrees in elevation (e.g., horizontal) and the direction orthogonal to x and y (e.g., vertical) is defined as z.
• the direction along y is defined as 0 degrees in azimuth (e.g., rotation about z), and the direction 0 degrees in azimuth, 0 degrees in elevation is expressed as (0°, 0°).
• laser arrays can be oriented to intersect as shown in FIGs. 10A-10C (e.g., trapping laser light beams 220a at (0°, 0°), trapping laser light beams 220b at (90°, + 20°), and the rows of lasers 220a can be arranged with a cross-sectional tilt (e.g., elevation) of +20 degrees such that the intersection of trapping laser beams 220a and 220b forms a 2-D lattice laying in a plane angled at +20 degrees from the x-y plane.
• the trapping cells formed by the substantially orthogonal intersecting lasers can be substantially square-shaped.
• the intersection of trapping laser beams can be arranged advantageously at other angles to form trapping cells that are diamond shaped or rhombus shaped to enable the addition of more planar qubit lattice arrays while continuing to enable multiple addressing laser angles for individually addressing each qubit in the 3-D lattice.
• trapping laser beams 220a can be arranged at (30°, 0°) and trapping laser beams 220b can be arranged at (90°, +20°) with addressing lasers at (0°, 60°), (120°, 60°), (240°, 60°).
• this particular combination of angles as depicted in FIG. 10B is chosen for visual clarity; ease of discrimination of two compact arrays (e.g., clusters; banks) of trapping laser light beams 220 that appear laterally oriented versus the three compact arrays of addressing laser light beams 210 that appear at angles that appear closer to vertical.
• Other examples can utilize alternative combinations of composite angles that form substantially orthogonal intersections of lattice lasers within a 3-D structure while similarly optimizing opportunities for lines of sight for addressing and detecting individual qubits in accordance with certain other implementations described herein.
• FIG. 11A schematically illustrates a view of the three-layer, 10 x 10 3- D atom lattice of FIGs. 9A-9C and 10A-10D viewed through the first optical detector port 242a in accordance with certain implementations described herein. From this viewpoint, each atom of the multi-dimensional, multilayer qubit lattice arrays is separately visible to the first optical detector 240a, such that when addressed by one dedicated optical addressing beam, each atom’s state can be individually detected and discriminated.
• FIG. 1 IB schematically illustrates a side cross-sectional view of the structure of FIG. 11 A. While FIG.
• FIG. 11A shows a three-layer, 10 x 103-D atom lattice
• other atom lattices can also be configured to allow each atom in the lattice to be separately and individually addressed and seen (e.g., detected by the first optical detector 242a) from optimal access angles above and below the lattice structure.
• CAD computed aided design
• Certain such implementations allow for entangling a maximum number of qubits simultaneously between nearest neighbors, next-nearest neighbors, next-next-nearest neighbors, etc. such that many more qubits can participate in a single native gate operation.
• 11A also schematically illustrates the relative placement of electrode regions 234, magnetic field (e.g., B-field) source 230, detector array 240 (e.g., CCD camera, etc.) and optical beam ports 242a, 214a,b,c for addressing and manipulating qubits to effect gate operations, and readout in accordance with certain implementations described herein.
• magnetic field e.g., B-field
• detector array 240 e.g., CCD camera, etc.
• optical beam ports 242a, 214a,b,c for addressing and manipulating qubits to effect gate operations, and readout in accordance with certain implementations described herein.
• FIG. 12 schematically illustrates a perspective view of the example assembled QC system of FIG. 10A comprising a three-layer, 10 x 10 atom lattice with three sets of addressing laser light beams 210a,b,c and two intersecting arrays of trapping laser light beams 220a, b in accordance with certain implementations described herein.
• FIG. 12 also schematically illustrates the view through the first optical detector port 242a (e.g., as shown in FIG. 11 A) and two different views of the intersecting arrays of trapping laser light beams 210a, b,c (e.g., as shown in FIGs. 8E and 9A).
• FIG. 13 shows three tables comparing the total number of qubits that can be entangled simultaneously at given gate fidelities for various multilayer qubit lattice arrays in accordance with certain implementations described herein.
• any reference to “one implementation” or “some implementations” or “an implementation” means that a particular element, feature, structure, or characteristic described in connection with the implementation is included in at least one implementation.
• the appearances of the phrase “in one implementation” in various places in the specification are not necessarily all referring to the same implementation.
• Conditional language used herein, such as, among others, “can,” “could,” “might,” “may,” “e.g.,” and the like, unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain implementations include, while other implementations do not include, certain features, elements and/or steps.
• the articles “a” or “an” or “the” as used in this application and the appended claims are to be construed to mean “one or more” or “at least one” unless specified otherwise.
• the terms “generally parallel” and “substantially parallel” refer to a value, amount, or characteristic that departs from exactly parallel by ⁇ 10 degrees, by ⁇ 5 degrees, by ⁇ 2 degrees, by ⁇ 1 degree, or by ⁇ 0.1 degree
• the terms “generally perpendicular” and “substantially perpendicular” refer to a value, amount, or characteristic that departs from exactly perpendicular by ⁇ 10 degrees, by ⁇ 5 degrees, by ⁇ 2 degrees, by ⁇ 1 degree, or by ⁇ 0.1 degree.
• the ranges disclosed herein also encompass any and all overlap, sub-ranges, and combinations thereof. Language such as “up to,” “at least,” “greater than,” less than,” “between,” and the like includes the number recited.
• the terms “comprises,” “comprising,” “includes,” “including,” “has,” “having” or any other variation thereof, are open-ended terms and intended to cover a non-exclusive inclusion.
• a process, method, article, or apparatus that comprises a list of elements is not necessarily limited to only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.
• “or” refers to an inclusive or and not to an exclusive or.
• a condition A or B is satisfied by any one of the following: A is true (or present) and B is false (or not present), A is false (or not present) and B is true (or present), or both A and B are true (or present).
• a phrase referring to “at least one of’ a list of items refers to any combination of those items, including single members.
• “at least one of: A, B, or C” is intended to cover: A, B, C, A and B, A and C, B and C, and A, B, and C.
• Conjunctive language such as the phrase “at least one of X, Y and Z,” unless specifically stated otherwise, is otherwise understood with the context as used in general to convey that an item, term, etc. may be at least one of X, Y or Z. Thus, such conjunctive language is not generally intended to imply that certain implementations require at least one of X, at least one of Y, and at least one of Z to each be present.

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