WO2015042660A1 - Quantum simulation - Google Patents

Abstract

This disclosure relates quantum simulation, which enables the use of quantum systems as analog hardware simulators to simulate complex systems, such as high temperature superconductors, which have proven to be difficult to simulate with existing methods. In particular, this disclosure relates to simulating a first quantum system characterised by a first quantum interaction using a second, different quantum system. This is achieved by determining a set of stimulation pulse sequences to stimulate the second quantum system characterised by a second quantum interaction, such that the second quantum system evolves in substantially the same way as the first quantum system. The step of determining the set of stimulation pulse sequences comprises combining multiple basis pulse sequences according to a combination of basis functions. The combination of basis functions approximates a transformation of the second quantum interaction to the first quantum interaction.

Description

Quantum simulation

Cross-Reference to Related Applications

The present application claims priority from Australian Provisional Patent Application No 2013903715 filed on 26 September 2013, the content of which is incorporated herein by reference.

Technical Field

This disclosure relates to quantum simulation. In particular, but not limited to, this disclosure relates to a method, software and a system for quantum simulation.

Background Art

Many important physical effects can be modelled by the interaction of quantum patticies. For example, models for high temperature superconductivity are based on complex interacting quantum particles, For a small number of particles, such as four, it is possible to simulate the interactions using a conventional digital computer.

However, simulations need to consider the interaction of each particle with every other particle and as a result, with a rising number of particles the number of interactions increases dramatically which leads to a dramatic increase in computational complexity for the computer. As a result, problems of practical size, such as more than 40 particles, are computationally too complex to simulate even with the most powerful supercomputer, One approach for overcoming this problem is to create a quantum system, such as a trapped-ion system with hundreds of qubits, such that the qubits can be easily observed and the interactio between the qubits is identical to the interaction of particles in the physical problem. This approach is referred to as quantum simulation because the physical problem is simulated using the created quantum system. However, it is difficult to create a quantum system such that the interaction of qubits i the same as the interaction of particles in the physical system to be simulated,

Further, the level of funability of such a quantum system is generally low and the system is specific to a narrow range of physical problems. As a result, the quantum system needs to be created for each physical problem separately which means that the cost per physical problem for creating the q antum system is very high. Throughout this specification the word "comprise", or variations such as "comprises" or "comprising", will he understood to imply the inclusion of a stated element integer or step, or group of elements, integers or steps, but not the exclusion of any other element, integer or step, or group of elements, integers or steps.

Any discussion of documents, acts, materials, devices, articles or the like which has been included in. the present specification is not to be taken as an admission that any or all of these matters form part of the prior art base or were common general knowledge in the field relevant to the present disclosure as it existed before the priority date of each claim of this application.

Disclosure of Invention

A method for simulating a first quantum system characterised by a first quantum interaction comprises:

determining a set of stimulation pulse sequences to stimulate a second quantum system characterised by a second quantum interaction, such that the second quantum system evolves in substantially the same way as the first quantum system,

wherein the step of determining the set of stimulation pulse sequences comprises combining multiple basis pulse sequences according to a combination of basis functions, the combination of basis functions approximating a transformation of the second quantum interaction to the first quantum interaction.

Since stimulation pulse sequences are determined such that: the second quantum system evolves in the same way as the first quantum system the stimulation pulse sequences can be used to program the second quantum system to be usable as a simulator of the first, quantum system. This is an advantage over non-programmable quantum simulators because these are limited to the interactions defined by their native hardware. With the above method, the second quantum system can be used to simulate the first quantum system even if the required first quantum interaction is not native to the hardware of the second quantum system.

Since the transformation from the second to the first quantum interaction is approximated by multiple basis functions, a wide range of different interactions can be realised with the same set of basis functions. Further, combining the basis pulse sequences according to the combination of basis functions is computationally inexpensive and therefore efficient.

Since the basis pulse sequences are combined to determine stimulation pulse sequences, this is a modular approach, which means that the same basis pulse sequences can be used to realise oilier transformations and do not need to be determined separately for each different transformation.

The method provides a solution to quantum simulation which is easy to use, which means that it requires only a small amount of processing po wer. Therefore , the method enables the use of quantum systems as analog hardware simulators to simulate complex systems, such as high temperature superconductors, which have proven to be difficult to simulate with existing methods. Performing quantum simulation using the above method eliminates the exponential increase in required resources, which is one of the draw-backs of other methods. The above method is resource efficient which means that it only requires a small amount of energy and time to perform the simulation.

The approximation of the transformation may be a decomposition of the transformation into the combination of the basis functions which may comprise a weighted sum of basis functions.

Combining multiple basis pulse sequences may comprise determining a duration of one or more of the multiple basis pulse sequences based on the weights of the weighted sum of the basis functions.

The combination of the basis functions may comprise a multiplication of basis functions and then combining multiple basis pulse sequences may comprise detemiining a concatenation of one or more of the multiple basis pulse sequences. The method may further comprise detemiining the approximation of the desired behaviour using linear programming. It is an advantage that linear programming is used to determine the approximation because linear programming is a resource efficient way of determining an approximation that is optimised. The linear programming may comprise minimising the duration of stimulation pulse sequences. It is an advantage that the duration of stimulation pulse sequences is minimised. This ensures thai the stimulation pulse sequences are useful and not too long for a practical application..

The basis functions may comprise triangular functions.

The second quantum system may comprise multiple qubits and the stimulafion pulse sequences may comprise one or more stimulation pulse sequences for each qubit.

The transformation may be discretised according to a lattice structure of the second quantum system.

The method may further comprise determining the transformation of the second quantum interaction to the first quantum interaction, The method may further comprise determining the basis pulse sequences and associated basis functions based on the second quantum interaction of the second quantum system.

The first quantum interaction may be an interaction during adiabatic simulation. The method may further comprise performing a Suzuki -Trotter-type decomposition where a transverse magnetic field is present.

Software installed on a computer causes the computer to perform the above method. A computer system for simulating a first quantum system characterised by a first quantum interaction comprises:

a processor to generate timing data defining a set of stimulation pulse sequences to stimulate a second quantum system characterised by a second quantum interaction, such that the second quantum system evolves in substantially the same way as the first quantum system,

wherein the processor is to determine the set of stimulation pulse sequences by combining multiple basis pulse sequences according to a combination of basis functions,, the combination of basis functions approximating a transformation of the second quantum interaction to the first quantum interaction; and

aft output data port to send the timing data to the- second quantum system. Optional features described of any aspect of method, software or computer system, where appropriate, similarly apply to the other aspects also described here.

Brief Description of Drawings

An example will be described with reference to

Fig. 1 illustrates a schematic of a quantum simulation set-up.

Fig. 2 illustrates a computer system for quantum simulation.

Fig. 3 illustrates a method for simulating a quantum system.

Fig. 4a illustrates how a pulse is applied to one of two qubits halfway through the evolution period.

Fig. 4b illustrates how pulses are applied at the start and end of a sequence. Fig. 5 illustrates a translationally invariant dynamical filter generation for a linear chain of coupled qubits.

Fig. 6a illustrates accessible filters achievable using dynamical mapping pulse sequences.

Fig 6b is a representation of the set C. and the universal filter space contained within it.

Fig. 6c is a representation of the set Q and the universal filter space contained within it.

Fig. 7 illustrates an application of dynamic filtering to the adiabatic evolution of a quantum simulator.

Bes Mode for Carrying Out the In vention

Fig, 1 illustrates a schematic of a quantum simulation set-up 100. The operation of the quantum simulation will be first explained in a simplified maimer while the mathematical details are provided later in this description.

The set-up 100 comprises a physical problem 102 to be simulated, simulator hardware 104 and programming unit 106. The physical problem 102 comprises a problem quantum system 108, including quantum particles shown as dots, such as particle 1 10. The problem quantum system 108 is characterised by a problem quantum interaction 112. In this example, the problem quantum interaction 1 12 is translation invariant and quantified over the distance between the interacting particles, such that particles close to each other interact strongl while distant particles interact weakly.

Similarly, the simulator hardware 104 comprises a simulator quantum system 1 14 with qubits shown as dots, such as qubit 1 16. The simulator quantum system 114 is stimulated by pulse generators, such as pulse generator 1 17.

In one example, the simulator quantum system is a multi -qubit trapped ion quantum system. The simulator quantum system 114 is characterised by native simulator quantum interaction 118, which is also translation invariant in this example and quantified over the distance between the interacting qubits. The term 'native' means that this quantum interaction characterises the native behaviour of the simulator quantum system 1 14 without changing the behaviour externally.

It is noted that the tenn particle and qubit are used separately to clearly distinguish between the problem system to be simulated and the simulator system although from a physical point of view, these terms may have an identical meaning. In the following description the concept of a Hamiltonian is used extensively. The Hamiltonian Ή is an operator that can be applied to a quantum state |(//(t)J at time t.

For example,

describes the quantum state after time t when the quantum state at t =^■ 0 is known. The Hamiltonian includes terms relating to the kinetic energy, such as the speed, of the particles and terms relating to the potential energie of the particles. The potential energies are influenced by other particles, such as an electric field of a nearby electron. Therefore, the Hamiltonian as an operator reflects the structure, that is, the interaction, of the particles, such as the qubits in simulator quantum system 114. The Hamiltonian H can be described using the interaction Ω of the particles in the simulator quantum system 1 14. The aim is then to realise a different Ω_εβ· which is the problem interaction 112 by applying pulses to stimulate the simulator quantum system.

The transformation performed by the stimulation pulses is captured by w such that 0.,_ff = w_j w_? is the transformed interaction (assuming two particles). It is noted that in this description the term "filter' is used interchangably wit 'transformation' unless noted otherwise,

As can be seen in Fig. 1, the problem quantum interaction 112 is different to the simulator quantum interaction 3 18 and as a result, without the use of the programming unit 106 the problem quantum system 108 would evolve different to the simulator quantum system 1 14.

A transformation 120 describes an operation that transforms the simulator quantum interaction 1 18 to the problem quantum interaction 112. The following description explains the physical way of applyin the transformation 120 to the simulator quantum system 114, such that the simulator quantum system 114 evolves in substantially the same way as the problem quantum system 108.

'Substantially the same' in this context means that the evolution of the two systems 1 OS^¬ and 114 is sufficiently similar such that the observed state of the simulator quantum system 114 after a certain simulation time resembles the state of the problem quantum system 108. For longer time periods than the simulation time, the evolution of the simulator quantum system 114 and the evolution of the problem quantum system may^¬ be quite different.

Transforming the simulator quantum interactio 118 is generally achieved by applyin pulse sequences generated by the programming unit 106 to the qubits. In the example of Fig. 1, a pulse sequence 119 is applied to qubit 116. The task of the programming unit 106 is to determine a set of stimulation pulse sequences to stimulate the simulator quantum system 114 such that it evolve in substantially the same way as the problem quantum system 108.

It is noted here that the quantum simulation system 100 can be implemented several different ways while the functionalities are assigned to hardware' components in different ways. For example, the programming unit 106 may be an individual computer system that performs the necessary calculations and sends the resulting data to the simulator hardware 104, which may be located remotely from th programming unit 106 and from where the actual radiation pulses are generated and physically applied to the qubits. In another example, the programming unit 106 is implemented as an integral part of the simulator hardware,, -such as a microcontroller or FPGA.

In one example, the programming unit 106 receive the problem quantum interaction 112 and the simulator quantum interaction 1 18 and determines the transformation 120. In a different example, the transformation i determined elsewhere and reveived by the programming unit 106 via a LAN network interface or a storage medium, such as a USB flash memory. In one example, the programming unit 106 has stored on a data store multiple basis functions 1.24, such as triangular functions, and determines a combination of the basis functions 106 that approximates the transformation 120. This decomposition of the transformation into triangular functions is conceptually similar to a Fourier decomposition and results in a set of coefficients, such that each coefficient represents the weight of one basis function. For example, the combination of three basis functions 124 could be [0,1 , ], which means that the transformation has the exact same form as the second basis function.

Each of the basis functions 124 is associated with one or more pulse sequences. A pulse sequnce may comprise one or more individual pulses. Physically, the pulse sequence may be a list of timin values that define the times at. which inverting pulses are to be applied to the simulator quantum system 114. A pulse sequence may be written to an ASCII text, csv or XML file or may be streamed directl to the simulator quantum system 1 14 or pulse generato 117, The pulse generator 1 17 generates the actual pulse 11 , such as a pulse of a laser or other elecromagnetic radiation, to stimulate the simulator quantum system 114. In the example of four qubit in the simulator quantum system 114, eac basis function is associated with four stimulation pulse sequences - one for each qubit. The programming unit 106 combines the basis pulse sequences 122 according to the combination of ba i functions 124 to determine a stimulation pulse sequence for each qubit. This means that the combination of basi functions 124, such as [0,1.0], defines how the basis pulse sequences 123 are to be combined. In one example, the combination of basis functions 124 is a linear- combination, that is a weighted sum, or a weighted product of basis functions 124, In that case, combining the basis pulse sequences 122 according to the combination of basis functions 124 means that the basis pulse sequences 122 are transformed based on the weights of each basis function 124 in the combination.

Later in this description, if will be explained how operations on the basis functions, such as addition and multiplication of basis functions, correspond to operations on the basis pulse sequences, such as addition and concatenation of basis pulse sequences. It will also be explained how the weights in the combination of basis functions 124 are reflected when combining the basis pulse sequences 122, Fig. 2 illustrates a computer system 200 for quantum simulation. The computer system comprises a processor 202 connected to a program memory 204,. a data memory 206, an input data port 208 and an output data port 210, The program memor 104 is a non- transitory computer readable medium, such as a hard drive, a solid state disk or CD- ROM,

Software, that is, an executable program, stored on program memory 204 causes the processor 202 to perform the method in Fig. 3, that is. the processor 202 receives via input data port 208 data 212 based on the quantum interaction 112 of the problem quantum system 108. The data 212 may be the quantum interaction 112 itself, such as in form of an analytical expression. In one example, the analytical expression is encoded in Matlab syntax and received by processor 202 in form of an ASCII text file or text stream. In another example, the data 212 is the transformation 120 that was explained with reference to Fig, 1.

Similarly, the transformation 120 may be represented by an analytical expression in Matlab syntax contained in a text file, The analytical expressions may also be replaced by sampled data or other forms of data. In one example, a processor 202 is .connected to a displa (not shown) presenting a user interface to a user and the user is allowed to enter the data 212 manually, such as. by using a keyboard. In. this case, the processor 202 receives the data 212 from the display.

The processor 202 then generates timing data 214 that defines a set of stimulation pulse sequences to stimulate the simulator quantum system 1 14 in Fig. 1, such that the simulator quantum system 114 evolves substantiall the same way as the problem quantum system 108. The timing data 214 in Fig. 2 comprises four stimulation pulse sequences for the four qubits of the simulator quantum system 114. The processor 202 then sends the timing data 214 to the simulator quantum system 1 14 via output port 210, The timing data 214 defines the set of stimulation pulse sequences as a combination of basis pulse sequences and may be a digital trigger signal, such as rising edge, a set of time values or an expression describing the sequence (eg. 5*3ns, 2* Ins). In one example, sending the timing data 214 to the simulator quantum system 1 14 means that the timing data is provided to the simulator hardware 104 and the generators, such as generator 1.17, then generate the actual stimulation pulses for the individual qu its of the simulator quantum system 114 according to the timing data 214.

Of course, the first data port 208 and the second data port 210 may be combined into a single data port, such as a LAN or USB connection. The processor 202 may receive data, such as data 212 based on the quantum interaction 112 of the problem quantum system 108, from dat memory 206 as well a from the input port 208.

Althoug input port 208 and output port 210 are shown as distinct entities, it is to be understood that any kind of data port may be used to receive data, such as a. network connection, a memory interface, a pin of the chip package of processor 202, or logical ports, such as IP sockets or parameters, of functions stored on program memory 204 and executed by processor 202. These parameters may be stored on data memory 206 and may be handled by-value or by -reference, that is, as a pointer, in the source code.

The processor 202 may receive data through all these interfaces, hich includes memory access of volatile memory, such as cache or RAM, or non-volatile memory, such as an optical disk drive, hard disk drive, storage server o cloud storage. The computer system 200 may further be implemented as a web service, such as within a cloud computing environment, such as a managed group of interconnected servers hosting a dynamic number of virtual machines.

It is to be understood that any receiving step may be preceded by the processor 102 determining or computing the data that is later received. For example, the processor 102 determines the transformation 120 and stores the transformation 120 in data memory 206, such as RAM or processor register. The processor 202 then requests the data from the data memory 206, such as by providing a read signal together with a memory address. The data memory 206 provides the data a a voltage signal on a physical bit. line and the processor 202 receives the transformation via a memory interface as the inpu port. Fig. 3 illustrates a method 300 as performed by processor 202 for simulating a problem quantum system 108 characterised by a problem, quantum interaction 12. Method 300 may serve as a blueprint or pseudo-code for software implemented, compiled and stored on program memory 204 in Fig. 2,

The method 300 comprises the step of determining 302 a set of stimulation pulse sequences as described in detail below. In essence, this set is determined to stimulate the simulator quantum system 11.4, such that the simulator quantum system 114 evolves in■substantially the same wa as the problem quantum system 108.

The step of determining 302 the set of stimulation pulse sequences comprises combining multiple basi pulse sequences 122 according to a combination of basi functions, such as triangular basis functions according to the examples below. The combination of basis functions approximates the transformation 120 of the simulator quantum interaction 118 to the problem quantum interaction 112.

The followin description demonstrates how a broad class of interacting spin-lattice models characterised by quantum interaction 112, may be generated through a combination of an arbitrary "native"' inter-particle interaction, such as simulator quantum interaction 118, and time-domain Hamiltonian engineering by applying stimulating pulse sequences to realise single-quhit unitary operations.

Basis pulse sequence that permit the challenge of realising an arbitrary spin Hamiltonian can be mapped to a linear program in polynomial time. Thus, using these basis pulse sequences we provide an algorithmic framework for "programming" spin interactions into a generic and constrained hardware platform 114, and prove that this permits any hardware platform to realize a universal class of effective interactions 112 that are not native to the system 114 via linear programming. Our analyse extend from circuit model quantum information to adiahatic quantum evolutions, where our approach allow for the creation of non-native ground state solutions to a computation.

Analog quantum simulation provides a near-term path to providing solutions to currently intractable computations at the scale of just a few dozen interacting quantum systems. These systems leverage a well-controlled quantum coherent device 1 14 at a small scale in order to study the properties of much larger, poorly understood many- body system 108, e.g. interacting spins and quantum magnetism. Producing a general, efficient and extensible framework for programming, complex, interactions and broadening the range of accessible simulations with limited hardware capabilities is an advantage when expanding the utility of mesoscale quantum simulation.

Here we describe a technology-independent technique to develop a hybrid programmable quantum spin simulator 104 with constrained resources 1 14, capable of efficiently realising a universal class of interacting spin Hamiitonians. Using a simple set of hardware capabilities we show how to encode transformations 120 of the native hardware behaviour 118 through chains of discrete unitary operations realised by stimulation pulses.

In one example, processor 202 uses the family of one-dimensional Ising-type Hamiitonians with translation-invariant two-body couplings 118 of arbitrary form and homogeneous transverse on-site terms, Through application of time-domain sequences of single-axis Paul! operators 119 processor 202 can stroboscopically realise any other effective Hamiltonian in the family by filtering the time-averaged relative weight of different pairwise interactions.

The provided sequences give transformations 120 with wavelet-like properties that may be easily augmented through linear combinations or products, spanning a provably universal space of interactions. Thus, the challenge of determining the appropriate hardware operations required to map the native interaction form to any other may be reduced to an efficiently soluble linear program in the space of available sequences.

In addition to providing specific examples of useful filtering tasks, we show how this technique may be extended to adiabatie quantum simulation, permitting simulators to reach states that are otherwise inaccessible to the natural interactions.

A quantum system may be effectively isolated from its environment, not by eliminating the physical interaction, but by inducing a dynamical response which acts to average out the coupling to the environment. The result is an effective average Hamiltonian that appears as if the environmental interaction were not present, leading the problem to be recast as filtering of the system-bath interactions [Biereuk2011] , PCT/AU2013/000649, which is incorporated herein by reference, describes a method for preserving a quantum state of a quantum system in a quantum memory. A similar method can be ap lied here to dynamically decouple the quantum system from noise of the environment.

In order to see how these insights lead to the development of control protocols appropriate for producing arbitrary spin Hamiltonians in quantum simulators we consider a system of two qubits interacting via th Hamiltonian H = ·ΠΧ,Χ₂, where X . is the Pauli x matrix for the j th particle. The natural evolution of this system can be modified by the application of state-inverting operations to the qubits at regular time intervals to producing time-dependent operators X. (t) = w_j(t)X _j in the Heisenberg picture. The function w^t ) captures the modulation performed on the j th qubit, switching between ±1 at the time of an applied pulse and is referred to as the control propagator. It is noted that the terms 'modulation' and 'transformation' are used interchangeably unless noted otherwise.

In this circumstance the time-evolution operator is given by

with Q_siS ~ · w, , where the use of vector notation for the control propagator indicates that the propagators may take different values fo each discrete time step.

The modulated time~e volution operator is U (t) = J^"!^©^"'1*1"¹ '^,¾^ι·^,ι>^1¾ίι^χ2^Α _ j_{n one} example, the method is limited to evenly spaced 7C pulses and we assume discrete time and represent the functions Wj (t) as vectors w₍ = ±1 over n = t/At entries, In this case w' = _j(t) for (1 -l)At <t < lAt .■ This notation allows the propagator to be written as U (t) ~ e ¹ - ¹ - .

For example, a pulse sequence for qubit 1 with pulses at times ti, ti and t$ as determined by processor 202, can then be formall writted as

. Alternatively, if the time values are

for example, the stimulation pulse sequence is [+1, 0, ±1, 0, 0, +1], which means there are pulses applied to one of the qubits of the simulator quantum system 114 at. times 1, 3 and 6.

The original- strength of the interaction characterized by Ω has now been altered by the factor \¾ * whose magnitude and sign is controlled by processor 202- by the timing of the applied pulses. We thus see that it is the inner product of the discretized control propagator w that determines the resulting effective pairwise qubit coupling.

Such time-domain modulation actually provides the processor 202 wit an exceptional level of control over the effective qubit interaction. For instance, the two spins may be dynamically decoupled if w,^■ w₂ = 0, achievable via application of a single spin flip on qubit 2 half way through the evolution period, producing [w₂]_j = ~[w₂]₂ = 1 , Here we have introduced the notation for the control propagator [Wj ], = Wj (lAt) , wit each discrete time-step indexed by 1 . Similarly, processor 202 can change the sign of the effective spin interaction from ferromagnetic to antiferromagnetic and vice versa. Processor 202 achieves this simple map through the application of π pulses on the second qubit at the beginning and end of the evolution, as Z₂e ^l0x^^xz₂ - e ^^{1 2}' ,

Figs. 4a and 4b illustrate the generation of the effective interaction Q_eff between first qubit 402 and second qubit 404 by application of timed sequences of Z₂ operators ( .7... pulses). Time moves from top to bottom of the graph.

Fig. 4a illustrates how processor 202 applies a pulse 406 to one of the tw qubits 404 halfway through, the evolution period whic leads to a net cancellation of the interaction. Fig. 4b illustrates how processor 202 applies pulses 408 and 410 at the start and end of the sequence and set Ω_≤({ -Ω .

Fig. 5 illustrates a transiationall invariant dynamical filter generation for a linear chain

502 of coupled qubits. The linear- chain 502 comprises linearly arranged qubits with distance-dependent coupling. For an N -qubit chain the largest distance is (N— l )d , where d is the primitive lattice constant. This shows ho the transformation of the interaction is discretised according to the lattice structure of the linear chain 502.

Only interaction between the first qubit 504 and others in the chair are presented for clarity. Pulse sequence and associated control propagators, 506 , for the N qubits are also shown. Time runs from top to bottom in k discrete steps. Black bars indicate π_ζ pulses, causing a sign change in the associated control propagator. The particular map depicted generates f_k , (d) . Fig. 5 further, shows a graphical representation 508 of the vector dot-products. ¹ . / between spins entering into the time evolution operator. Discrete time bins with a negative sign of the interaction are shown hatched, For example, the dot- product ₂ - W_j at . t_\ (510 in Fig. 5) equals 1 because w₂ and W3 both equal -1. At. time ta (512 in Fig. 5) wj equals 1 and W3 equals -1 and as a result, the product equals -1 , Similarly, at time t.3 (51.4 in Fig, 5) w_¾ equals 1 and $ equals 1 and as a result, the product equals 1, The overlap in this case, that is, the number of time bins where the sign of the interaction is positive, is k-2.

The sum over time bins gives the resulting pref actor in the effective Haniiltonian. All pairs of spins separated by distance d have a resulting overlap k-1 , all pairs separated by 2d have overlap k - 2 , decreasing . linearly with d until the qubits separated by the largest distance have overlap -k , indicatin a sign change. Note that thi illustration has not included proper normali ation of fi_6fF < Ω , Using these building blocks we generalise this approach. Processor 202 may determine a suite of basis pulse sequences that may be efficiently used to program an effective pairwise- spin- coupling Hamiltonian in a multi-qubit system. The basic model incorporates a finite qubit chain on a one dimensional lattice (502 in Fig. 5) with a fully connected interaction Hamiltonian

where there are N qubits in the chain and the frequency O_d characterizes the interaction strength of two qubits that are separated by d lattice spacings.

We additionally consider a homogeneous transverse field H_T - B^ .Z_; , where the magnitude of B is controllable and the total Hamiltonian is H = H_t t l _r . This form of long-range spi interaction may be realized, for example, in trapped-ion spi simulators and optical lattice simulators [24,25,26] .

Processor 202 determines an extensible set of filtering basis pulse sequences that permit an arbitrary effective Hamiltonian within this class, b modifying the native interaction so that _A→€l_d f .(d) , where the transformation f(d) (reference numeral 120 in Figs, I. and 2) depends on the choice of pulse-induced control propagators

, up to an o verall energy rescaling. Note thai both the original and effective Hamiltonians are translationaliy invariant. We begin wit the simple case where the transverse field .is set to zero, resulting in an. exact mapping to the desired effective Hamilfonian, Detailed analysis of the general case is deferred to later in this description,, but, we quote the relevant results here.

To simulate the evolution of an effective Hamiltonian H^tff ≤ H^^ff for a time T , we break T into k segments of equal duration At_k = T / k , with pulses applied as required at the transitions between time segments [Dodd2002J. This results in a famil of control propagators wf^} labelled by k , where again j indexes (he qubit being addressed. We define operators P. . which are applied both at the beginning and end o the 1 th time segment. Thus at the boundary of time segments 1 and 1 + 1 we actually apply the operator \ \ . , .

Defining P_fci in this way greatly simplifies the analysis. Each is a product of Z . operators on some subset of the N different qubits in the chain. The result of. applyin these pulse sequences is a modification of the time evolution during the 1 th time segment to U (At_fc) . - P exp ~iHAt_k]P_{k i} . The P_kJ operators generate sign changes in the control propagators of individual qubits so that the sign of P^X^., is encoded by the 1 th element of w†^} = | w^^k!]_j (506 and 508 in Fig. 5). The effective time evolutio

The total evolution is therefore given b the product of evolution operators over all time periods and can be written

where H_t ^eff =^ H^ Ω_β f_k (d)l¾ is fil dynamically modified qubit coupling.

Here wc have defined the transformation function f_k(d) to be equal to the vector dot product of the control propagators f' · w^. over the discrete time segments resulting from the applied sequences.

The primary class of basis pulse sequences, A_k , which processor 202 determines has the effect of transforming Q_d by the factor These basis

functions vary as triangle waves as a function of qubit distance (as shown in Fig. 5), with period 2k and range ¾^A (d) e [-~l,lj (and (x| denotes the fractional part of x).

This is accomplished, by determining the. suite of pulse operators, that is, basis pulse sequences, on the N quhits {P_{k j}-H^, with

In this construction the exponent on the i th Paul! Z operator is always an integer so that either Z_; or the identity is applied to the i th qubit depending on the parity, and the index j denotes the relevant time bin. A basis pulse sequence is associated with a corresponding basis function if they have the same index k.

Fig. 6a illustrates accessible filters achievable using dynamical mapping basis pulse sequences, such as a basis of triangle wave functions 600 arising from f_}i ^A(d) . Traces plotted here have k = 1 (highest frequency) to k = 8 (lowest frequency).

Fig 6b is a representation of the set Q and the universal filter space contained within it. represented by the square confined in size to fit within the convex set defined along axes Ω^, and Ώ^, , the only possible values for three spins. Black points represent the external values of accessible filters, (1,1) = A₀ , (-l,i) = A, and (0₅-l) = A₂ in the two dimensions, bounding a conve set denoted by the blue line. Only A₀ and A₂ are required to construct any other triangle wave where k > 2 , since all higher order waves are a weighted average of these two maps when d < 2. and concatenation of sequences does not yield new filters - (Λ₀· + A_{)/ 2, A - A₀ and Α,Α₂· = Λ₂. Fig, 6c is a representation of the set Q and the universal filter space (cube) contained within it. Projections of the complex poly tope onto planes in the basis of pairwise qubit coupling are represented using shading. In Figs. 6a and 6c schematic representations of possible interquhU couplings (giving axes on the filter space) are presented.. Details of the derivation of C_N are presented in the later description.

The form of transformation f_k ^A(d) allows considerable flexibility in the effective coupling since it takes values over the range ±1. In particular, it is possible to switch the sign of the interaction between different sites from .ferromagnetic to antiferromagnetic as a function of distance, d , simply via single-qubit pulsed control. Fmther, since the period of the triangle wave is given by 2k , we see that for a given T , the number of time steps into which the sequence is broken determines the form of interaction. This suite of basis pulse sequences thus provides a flexible framework for realising tuneable interqubit quantum interactions, coupled with others defined later.

For the simulation of arbitrary effective Hamiltonians it may be necessary to build complex functional dependences, that is, transformations f (d) from, this set of basis pulse sequences. Triangle waves as basi functions form an orthononnal basis much like sines and cosines, meaning it is possible to achieve complex multiqubit interactions via weighted positive linear combinations of filters, that is, basis functions, e.g. f ^> and f ^m , such that Ω„ [τ' " (d) +T ΐ (d )] . This also shows that the transformation f (d ) of the simulator quantum interactio 118 to the problem quantum interaction 112 is approximated by a combination .of basis functions T^a> f^a>(d) +T ⁾ f ⁾(d) .

The relative weight of each filter, that is, basis function, in the linear combination is captured through the length of the relevant filter; ef. Eq. 3. This operatio is physically achieved via sequential application of differen basis pulse sequences, with appropriately adjusted evolution times. This means the duration of each basis pulse sequence is determined based on the weights for each basis function.

Similarly, products of filters giving a combination of basis functions of O_d τ⁽ Τ'"' f ^ll-'(d ) f ^t¾(d)J may be achieved via sequence concatenation, Interestingly, high-order concatenation of a single-triangle wave approximates a Dirac comb.

These two operations are simple but extremely powerful. With these it is possible to engineer universal quantum interactions, that is, qubit couplings, subject to the restriction that only even functions of d may be achieved.

We define the convex set Q to be the set of allowed filters f_t(d) o N spins generated by our basic filters and both linear combination and concatenation. This set may take a complex form in a multidimensional space where each dimension is defined by the strength of coupling between qubits separated by a particular number of lattice constants. However, this set contains an N - .1. -dimensional hypercube around the Origin that can yield an arbitrary transformation to an arbitrary Cl _:d)■

In Figs. 6b and 6c, we show Q and Q , including the universal set contained therein.

The cost of this universality is reseaiing of the interaction strength under applicalion of the resulting filters, corresponding physically to an increase of the total required evolution time.

Remarkably, processor 202 can generate efficiently an arbitrary desired Q_d f (d) using the set of basis functions (filters) generated before, linear combination, and concatenation. This process is accomplished as a problem in linear programming; using linear algebra routines thi procedure may be implemented numerically and provides provably optimal solutions in terms of the total evolution time.

Fast, practical implementations of linear programming algorithms may be used to solve this problem at scale of interest for the implementation of useful quantum simulations in polynomial time. Moreover, we are able to prove that for the model e address, the realisation of an arbitrary interaction Hamiltonian within this Universal set may be accomplished using at most 0(log,N) concatenation steps. These are the most resource-intensive operation we employ, requiring an exponential increase in pulse number and evolution time with concatenation step.

A more detailed discussion of the complexify of our scheme, other pulse sequences we define in order to prove universality, and a discussion of the optimal! ty of linear programming is given futher below.

Some examples of Hamiltonian engineering will help to reveal the power, generality, and utility of our approach. One example is the application of the .transformation filter

w whheerree 11. iinnddiiccaatetess ffrreeee eevvoolluuiiiioonn ooff eeqquuaall dduurraattiioonn.. TThhiiss ppaarrttiiccuullaarr ffiilltteerr eexxppllooiittss tthhee ffaacctt tthhaatt ΠΠ^ΛΛ ((kk)) == --11 ,, eennssuurriinngg tthhaatt a allll q quubbiittss sseeppaarraatteedd bbyy ddiissttaannccee kk wwiillll bbee ffuullllyy ddeeccoouupplleedd ffoolllloowwiinngg aa ffrreeee eevvoolluuttiioonn ppeerriioodd ((cc..ff.. FFiiggss.. 44aa aanndd 44bb))..

UUssiinngg ccoonnccaatteennaattiioonn,, pprroocceessssoorr 220022 ccaann eelliimmiinnaattee aallll i inntteerraaccttiioonnss iinn tthhee ssppiinn cchhaaiinn eexxcceepptt ffoorr,, ee..gg..,, nneeaarreesstt--nneeiigghhbboorr iinntteerraaccttiioonnss.. TThhiiss HHaammiillttoonniiaann mmaayy bbee uusseedd nnoott oonnllyy ffoorr ssiimmuullaattiioonn ooff q quuaannttuumm mmaaggnneettiissmm,, bbuutt aallssoo mmaannyy ootthheerr qquuaannttuumm iinnffoorrmmaattiioonn pprorottooccoollss [[2299,,3300]].. Another example of a useful Hamiltonian mapping relates to problems in quantum magnetism [31 ,32,33] where long-range qubit interactions can be engineered to scale as Ω_ί( oc d ^"a , e [0,3] » ^a form of interaction that arises for instance in phonon -mediated spi simulators using trapped ions [34,35,36]. In practice, many simulators cannot reach the achievable limits of this range, or there may be a desire to induce a scaling outside of the range of this native interaction. For a system of N qubits, the interaction strengt can be mapped O_d -» !¾ / d through the sequential application of increasingly complex concatenations of f_{j +J}(d) (see below).

Finally, we describe how dynamical filtering may be applied to adiabatic simulators and preparation of entangled ground-slates of designer Hamiltonians, that is,

Hamiltonian realised by applyin a combination of basis pulses to the quantum system. Say processor 202 is required to adiabaiically evolve to the ground state of a target Hamiltonian, H_t , but can only turn on the available Hamiltonian H_;> in a particular experimental apparatus. Returning again fo the I D qubit chain, processor 202 initializes in | g_s) , which is the ground state of a simple Hamiltonian such as and allows the system t evolve under the time-dependent

i t

Hamiltonian, H (l) = 1— i f ! - 1 If the evolution is adiabatic, then I ^ _a U_a (r) g₅) I ^'» .1 at the end of the interaction.

Now processor 202 applies pulsed modulation in order to drive the system to the new ground state | g_t ) of H , repeatedly applying the sequence as the Hamiltonian generally does not commute with itself at different times. The rate of application is set such that to first order the Hamiltonian is constant over the pulse period, At , giving a iota! evolution defined by J^ ^R F [ϋ₈ (1Δί,(1 -1)Δί)] with R = r/At the number of filterin operations performed durin the adiabatic ramp, F the dynamic filter, and U_a it, ,t_>) is the time-ordered evolution from t, to t, . Fig. 7 illustrates an application of dynamic filtering to the adiabatic evolution of a quantum simulator. An inset 70 shows a schematic of the approximation of breakin a linear ramp of Hamiltonian H_a into pieecwise-constant segments during which a dynamic filter, F is applied. In the example treated here, processor 202 filters an all- to-all. interaction between four spins to give only nearest-neighbor interactions using F (II ₀) = (Λ₂.°Λ, )(U₀) . In. the main panel 704 processor 202 calculates the state fidelity of the adiabatic evolution as function of the number of filtering operations. The error, defined as 1 -|{^Ι ^ |^" ¾ Var(H ^1:')At² / iff +0(At') , where the variance is calculated with the respect to | g_t,} , decreases quadraticali with filtering step. The dashed line is a guide to the eye showing the quadratic improvement.

To test, this adiabatic simulation, we numerically integrate the Schrodiiiger equation for a system of 4 qubits undergoing adiabatic evolution, starting from a distance- independent spin coupling, and filtered to produce only nearest-neighbor coupling. We calculate the error accrued due to deviation from the assumption of a piecewise- constant. Hamiltonian during the filtering operations as the overlap of

with a state whose evolution includes a first-order time-dependence of the Hamiltonian during filtering (see below).

In this case, the overlap between the ground state of the target Hamiltonian and that of the unfOtered Hamiltonian is l (g_t I g_a ) i²- 0.33 , but with the application of dynamic filters the infidelity dencreases towards zero approximately proportional to Δ , surpassing 10 ^* with sk filtering steps.

These results continue to hold when the transverse field is present, with the cavea that processor 202 needs to use additional pulses in a Suzuki-Trot ter-type decomposition of the non-commuting terms in the Hamiltonian, The additional error incurred from the Suzuki-Trotter approximation can be bounded as a polynomial function of T and N (see below), so the protocol can achieve a bounded error without introducing unreasonable additional resource overhead.

In conclusion, it is shown how processor 202 determine a generalized framework of basis functions and associated basis pulse sequences for dynamic filtering, that is, transformation, of Hamiltonians in the time domain that enables programmable quantum simulators using single-qubit Pauli operations and a native long-range spin coupling.

The basis of filters enables the processor 202 to perform numerical decomposition of a universal class of realizable couplings into the basis of available filters. This means the processor 202 approximates the transformation 120 and this approximation is a decomposition of the transformation .120 into a combination of basis functions .124, Further, there are provided explicit examples of how one might tune the power- law of a long-range spin coupling or cancel undesired spin couplings on a I D lattice,

We have additionally showed how this technique can be applied by processor 202 to augment the adiabatie evolution of a spin Harniltonian in a form of hybrid adiabatic quantum simulator, The following description provides further mathematical details on the- above methods. Explicit Forms for Filters of Interes

For a chain of N spins, we now define the physical operation for the filter, that is, transformation, A_k that we described above. Recall that we divide the total evolution time T into k bins and associate the j th bin with an operator P. . Each of these operators act on the spins individually. We can thus represent the action of the filter as follows;

The exponent on the i th Pauli Z operator is always an integer so that either Z-. or the identity is applied to the i th qubit depending on the parity. As discussed above beneath Eq. 3, this filter modifies the system's evolution by

f_k ^A(d) = (~l)*(l -2{ }), (6)

k

where { xj denotes the fractional, part of x .

In addition t this filter, we make use of another filter defined as

Π¾ ' (?)

The filter T_I: is capable of providing a relative enhancement to the interaction strengt only of qubits separated by integer multiples of k lattice spacings. The _K map is implemented in a similar fashion as above and modifies the system's evolution according to

Because of the selectivity of the filter's action in providing a periodic (in d ) relative enhancement of quhit coupling, we refer to this as a boost filter. Inclusion of a Transverse Field

In the above description, the transverse Field was set to zero so that ail of the term in. the HamiUonian would commute with one another. This resulted in an exact mapping from an initial Hamiltonian of the fo

to another Hamiltonian of the same form, but where the distance function is replaced with Ώ_ά f (d ) . The functional dependence f (d ) can take an arbitrary form, up to an overall .re-scaling factor. To incorporate a homogeneous transverse-field term of the form

H_T = B∑Z_p (10) j

we use a Suzuki-Trotter- type decomposition of the evolution operator [38,39,40]. We assume that the overall strengt B of the transverse field is tunable parameter so that we can match its resealed- strengt relative to the reseated coupling strength £¾ by the same overall scale factor.

Given this assumption, there are two natural ways to analyze this situation: an a!ways- on field and a field which can be switched on and off rapidly.

In some implementations, it i further reasonable to expect that the interactions can be switched on and off rapidl (e.g., some trapped ion experiments [IslamlOI l ]). We will also analyze this case, since in fact it is the easiest to analyze and gives insight into the other results, We wish to bound the deviation between the exact and the approximate (Suzuki- Trotter) evolution, assuming of course that all of our pulse sequences are accurate and perfectly timed. Let's denote the ideal evolution operator by

ex [^~-i(H_T. + H_f )t |, (11) where (^P_jH_jP^-M; = H_tt . The approximate evolution is defined in terms of the second-order integrator (the "s lit-ste " method), i ven b the following formula:

(12) Here the H , are the allowed evolutions over a given time interval, and their exact definition depends o which resources we allow ourselves in the protocol but they are related to the ideal effective Hamiltonian (here called simply H ) by

H = H_T + H _[ =∑H_r (13)

j-i

We will discuss the specific choices below. We have also introduced a parameter r , the Trotter number, which counts the number of step in our Suzuki -Trotter expansion and hence controls our error.

Note that it is difficult, to use the higher-order integrators which provide faster convergence because they require evolution for negative values of t [Suzukil 991j. If rapidly flipping the sign of the interactions and the transverse field is feasible in a given system, then these higher-order integrators could be used to give sharper error bounds than the ones given below.

Given this ver general framework, we can upper bound the error as follows [Berry20(⁾7J

PV(t) -U_r (t)I¾16m^J PH P t³7rA (14)

Here the error is in terms of the operator norm (largest singular value). Note that PH P≤ N² /2 + BN by the triangle inequality, so this quantity is always polynomial in the number of spins. This bound on the -norm of the difference has an operational meaning: it givens an upper bound on the trace distance between the evolved states, namely

PV™U P ^Tf IV^ -U EJ * I (15) for any initial state p and unitaries U and V . To make the error bound in Eq. (14) more concrete, we need to deteimine what m is for a given protocol. We see from the form of the bound that the fewer different. Hamiltonian types we use, the better will be our error scaling. The simplest ease to consider is that where both the interactions and the transverse field can be switched on and off rapidly. Then we can use the above bound with m = 2 : we simply switch between two types of Hamiltonians, the exact evolution of the effective Hamiltonian with zero transverse field and just the transverse field with no interactions. That is, we have

H_} = H_t and H₂ = H_T . (16)

Note that H_t can be implemented exactly by subdividing each Suzuki-Trotter step into smaller segments, as discussed in the mai text.

Next, consider the case where the interactions are always on, but the transverse field can be switched on and off rapidly. Here a good strategy is to turn on the transverse field in short, strong bursts, and turn it off while applying the filter functions. The control Hamiltonians during each of the m = 2 Suzuki-Trotter steps are given b

H^ H_j and H_2. = £(H_t ^■ +fH_T). (17)

Here we introduce an additional parameter S ~ I, and we incur a small error in the final Hamiltonian,

I I _,, nr H . + H_t + H,₍, (18) We can choose 3 as small as possible to be compatible with the assumption of accurate and rapid switching of the transverse field. Thus, the error bound is nearly the same as the previous m = 2 bound, except that we add a small error from the inexact decomposition.

PV(t) -U, (t) M?V(t) -V, (t )P+ PVj (t) -U, (t ) P< 128 PH , P t ³ f r + 0(δΡΉ _l Pt). ( 1 )

Lastly, consider the most pessimistic case where both the interactions and the transverse field are not rapidly switehable. In this case the best, strategy seems to be to use a different Suzuki-Trotter step for each increment in a pulse sequence. In this case, we have no choice but to use the bound in Eq. (14) by setting m equal to the total number of pulses in the pulse sequence. For a chain of length N , we can alway bound the maximum number of pulses required by m < N , which follows from Caratheodory's theorem. This give a pessimistic but nonetheles polynomial bound on the required Trotter number to achieve a fixed error. Construction of Q. For the general ease of N qubits, the interaction space is N -l dimensional and the existence of a universal filter space is not as clear as in the examples provided in Fig 2, In order to construct a suitable set of extreme points to ensure that their conve hull contains the origin and a small ball around it, we show that it is possible to generate a complete set of effective of interactions such that only qubits that are separated by a specific distance interact with one another.

These basis interactions, to which we refer as the Kronecker delta interaction vectors, would be of the form , 0,... ,0, ±1,0, ..., 0) where is a positive number whose magnitude is irrelevant for the existence of d e universal filter space, (Of course, larger a are preferred since these control the strength of the interaction,) The ± sign is important, so that we can generate both ferromagnetic and antiferromagnetic interactions. We now show how to construct both a positive and negative Kronecker delta interactio vector for all d < N - l. The argument proceeds in four steps, beginnin with the special cases d - 1,2,3,4 which must be treated separately and then the generic case of d > 4. Most of the insight into why the argument works is available in the d =l case, and the other eases largely reduce to this argument. However, they require special treatment because of some of the unique properties of the filter functions that we use.

The basis pulse sequence related to the d = 1 negative Kronecker delta interaction is associated with Λ, as its basis function, which is -1 at d = 1. Similarly, the other basis pulse sequences are also associated with their respectiv basis function.

Processo 202 then eliminates all other couplings through a judicious choice of concatenations which are implicated through the expression,

· ^'· V - ,<^■

This statement can be proved through an inductive argument with the base case of m = 2 being easily checked by hand. For the case of an arbitrarily long chain of qubits. we proceed with the inductive step. Assuming that we have zeroed out all couplings up to 2" ^{f i} -l using the concatenation scheme described above, we need to show that

1

subsequent concatenation wit (Ά^, +Λ_ρ) and (A^,._} + (1 ~— )Λ₀) will extend the decoupling range out to d < 2^{n' : '} -l . Using the explicit form for f_k' (d) , it is trivial to show that these additional concatenations zero out the 2"^{< !} -l , 2^{tl< !} and 2" ' +l couplings.

5 Additionally, every filter used in the proceeding concatenations is equal to 1 at d = 2ⁿ⁺¹. (since they all have a periodicity equal to a power of 2 ), implying that their behavior from 1 < d < 2^!" ' will be repeated for 2" " < d < 2'" ². Given the assumption of the inductive step, this implies that the concatenation has extended the full decoupling to d < 2'¹⁺²— 1. Constructing the positive Kronecker Delta interaction for 10 d = 1 is done in the same manner, only beginning with A₀ as the base function. In this rubric, a chain of N qubits requires 2m- i— 2[log₂(N + l) -3 total concatenations for the Kronecker delta interaction at d = 1 . rFhe negative d = 2 Kronecker delta interaction begins with A₂ as the base function and we then concatenate with (A_j +A₀) to eliminate all inieraciions where d is odd.

The remaining task maps directly onto the construction of the d = 1 Kronecker delta interaction if all the distances are reseated by a factor of 2 and eliminate the unwanted interactions using the fillers implicated by the following expression,

>0

This scheme requires a total of 2 log_?(N) -4 concatenations for the d = 2 Kronecker delta interaction so that 2m-2 = flog₂(_.N -l)^~| -2 ,

The d = 3 Kronecker delta interaction is constructed by starting with A₃ and concatenating with (Γ_¾ + A₀ / 3) to decouple all qubit except those that are separated by multiples of three lattice spacings. The remaining task can be accomplished by a similar sequence of filters indicated by the expression,

0

The d = 4 Kronecker delta interaction construction starts with A₄ and proceeds with concatenation with Γ₄ to decouple qubits that are not separated by a multiple of four lattice spacings. The decoupling is accomplished via the concatenation of the filters i dicated by,

N + 3

requiring 2m - 3 = 2 iog . i + 1 concatenations.

12

The final construction is for the Kronecker delta interaction at d - Q where

4

4 < Q < N -1. We employ F_Q which is positive for all d . For d < Q , f_Q (d) = 1 .

Q

4

Applying .((1 )A^_n + T_Q) for m = 0,1,2,3...,. < log₂N decouples all qubits except

Q ²

those that are separated by multiples of Q lattice spacings. The next task of decoupling all qubits separated by multiples of Q can, again, be mapped onto our original construction of the d - 1 Kronecker delta interaction by repealing the distance by Q .

The entire decou ling sequence is indicated by the expression,

N -l

resulting i a total of log₂(N -l)] + 2 - 4 filtering operations.

Q

We thus see that it is possible to construct the Kronecker delta interaction with 0(log,(N)) concatenations. Accordingly we know that it is possible to define a convex hull that includes the origin and the extremal points defined by the Kronecker deltas above, provin universalit for an N -qubit chain as defined herein.

Linear Programming for Efficiently Compiling Pulse Sequences Here we show that, subject to some mild caveats, the problem of compiling pulse sequences can be solved efficiently in N , the number of spins. While the existence of the universal filtering space guarantees that some pulse sequence exists, the algorithm presented here actually finds valid pulse sequence (if one exists given the input filters) and minimizes d e total amount of time spent applyin pulses.

Consider a set of elementary filter functions f_t f_w , which we think of as vectors of length N -l . These could come from e.g. the basic A and Γ filters defined above, together with k levels of concatenation for some fixed k . We can also add the O(N) specific "basis" filters which we constructed in the previous section at k = 0(log N) levels of concatenation to form the universal filtering space. We restrict, to a polynomial upper bound on m , i.e. m = 0(N^c ) for some constant c independent of

N , in order to ensure that the algorithm below runs in polynomial time.

Given this set of filter functions, we wish to compile a positive linear combination of pulse sequences that will generate given Hamiltonian coupling. This problem can be cast as a linear program, which is an efficient method for finding optima of linear objective functions subject to linear equality and inequality constraints, importantly, the runtime of a linear program with It input variables and poly(n) constraints is polynomial in n .

To see that our problem is a linear program, we make the following observations. Given a desired vector of couplings O_eff , processor 202 can determine the transformation 120 as Ω_ί(Γ / Ω . Processor 202 can find time steps t.. > 0 to realise the transformation such that

Here we actually have a vector equation, and the division on the righthand side is done element wise.

Suppose that each filter function ΐ has a cost e. associated to it. (For simplicity, we can imagine that all of these costs are equal,) Then a linear program which will compile a given pulse sequence to generate the effective coupling Ω_ϊΛ. is

By adding slack variables with additional equality constraints for each of the components of the vector equation, the above formulation can be converted to the standard form. Als note that this might not have a solution if our filte set isn't chosen to be universal Regardless, the program will terminate in time polynomial in m and N and will minimize the total amount of evolution time required to implement the desired coupling.

Further improvements might be desired, such as having only a few nonzero values for t _j (tha is, requiring only a few different filter functions). By Caratheodory's theorem , only N of the filter functions need to be nonzero for any fixed value of Ω_Είί / Ω in order to guarantee a solution. However, finding such sparse solutions is in general NP- hard. so one would likely have to resort to heuristic methods to make improvements along these lines.

Power- Law Interaction Engineering

As described earlier, many spi simulator exhibit pairwise interactions scaling as- a power-law in the qubit distance. Here we provide details of how, using pulse sequences introduced in our method, one may adjust the power-law scali g beyond the accessible bounds native to the underlying hardware.

As an example of how this method works, consider a system whose qubit interact throug a Coulomb interaction so that Ω. = Ω, / d . In order to use this system to simulate a many-body system whose particles interact through a 1 / d² potential, we must construct filter with functional dependence f ^'(d) « 1/d so that D_d— O_d / d .

We now show that for a system of N qubits, the interaction strength. D_d can be mapped to one that is approximately < i_d /d through the sequential application of increasingly complex concatenations of Λ , thereby increasing the power law dependence by a single power. Any Λ_}: filter can be used as long a k is larger than the number of qubits i the system, but k = N + 1 minimizes the number of pulses in the primiti ve filter. I a system of N qubits, the map generated by A_k is Linear over the entire chain when d

k > N so that it can be described as l^A(d) ~ l - 2— . Rewriting this expression as

k

— j hel s to highlight that this is equal to the linear term in the Taylor k l_„ 2 J

k / 2

expansio of around d -.k/2 , with a radius of convergence of k / 2, namely d

We now define the evolution operator U which is generated hy an interaction that obeys a power law of

id" , Consider the following filtered evolution operator V ,

- V_mn. (t)A_N+l(U _yd„ (t L, (U i ,/_Μd_Λ" (0}Λ⁵ _ν+, (U 1„/d _(<1 (t)) ... (28) ex

Here we denote a concatenation of Λ up to k level by A^k , and the last line follows by sutami^'rig the geometric series. As can be seen, the effective interaction strength obeys a new inverse power law where the exponent has been incremented by a single power. More generally, one can choose different timings for the various filtering operations in order to construct, a more general dynamical mapping function,

The function g(d,{ar_s }) is restricted by the fact that the coefficients {or. j are necessarily positive. This implies that the function f must have Taylor expansion coefficients that alternate in sign, meaning that the technique is capable of producing any inverse power law filter but is incapable of producing polynomials with positive exponents.

Although the construction above requires an infinite sequence of pulses to implement exactly, one can bound the error of a finite sequence using the techniques explained earlier. Another possibility to realize these power laws is to use the linear program described above together with a family of pulse sequences to try to find the most time- efficient power law across all N of the spins. Adiabatie Evolutions Using the filtering approach in adiabatic simulators to reach a target ground state \ & ) can be done by repeatedl applying the sequence at a fast enough rate so that the

Hamiltonian is approximately constant over the time it takes t apply the pulse sequence. Defining a filter F that maps the evolution operator generated by H₃ -» H₍ gives a piece wise-constant filtered adiabatic evolution operator

where R - τ/Δΐ is the number of filtering operations performed during the adiabatic ramp. The state then ev

Here H )^5> is the filtered Hamiltonian. at t - j Ai ,

The error accrued due to the Hamiltonian changing during the filtering operations c cai~n, be calculated as the overlap of

whose evolution includes first t-- order time-dependence of the Hamiltonian during filtering operations. In the adiabatic

H(t')dt'] which ca an i ^■

be approximated by exp[-i(H(t)At +~ H(t)A )] when At is small. In eases like ours - d d - where, P— H.(t)P_f =— P,H (t)P_; we can write

3 di

which is meant to capture the detrimental effect on the filtering operation due to the changing Hamiltonian to lowest order in At . The convergence of the filtered adiabatic protocol can defined as the overlap of |^ and j^) as defined in Eqs. (34) and (35).

Thus we have

by th ^■Baker-Campbell-Hausdorff formula. The sum H^tJ)At can be approximated by an integral when At is small enough so that the overlap is.

From this last expression, we see that the error can he thought of as the probability of the Bamiltonian -^(H(t_.t . ) ~ H^'(t_j)) evolving the system out of the state | g in a time

At , which, to lowest order in At , is Var(H (t, ) ~ H (t_j))At² /4 ^z where the variance is calculated with respect to | g^ . In the adiabatic evohition described in this article, | g, is the ground state of H (t. ) , meaning that the formula for the error can be further simplied to be approximately Var(H (t_r ))Ar / 4¾² ,

It will be appreciated by persons skilled in the ait that numerous variations and/or modifications may be made to the specific embodiments without departing from the scope as defined in the claims. It should be understood that the techniques of the present disclosure might be implemented using variet of technologies. For example, the methods described herein may be implemented by a series of computer executable instructions residing on a suitable computer readable medium. Suitable computer readable medi may include volatile (e.g. RAM) and/or non-volatile (e.g. ROM, disk) memory, carrier waves and transmission media. Exemplary carrier waves may take the form of electrical, electromagnetic or optical signals conveying digital data steams along a local network or a publically accessible network such as the internet. it should also be understood that, unless specifically stated otherwise as apparent from the following discussion, it is appreciated that throughout the description, discussions utilizing terms such as "estimating" or "processing" or "computing" or "calculating", "optimizing" or "determining" or "displaying" or "maximising" or the like, refer to the action and processes of a computer system, or similar electronic computing device, that processes and transforms data represented as physical (electronic) quantities within the computer system's registers and memories into other data similarly represented as physical quantiiies within the conipuier system memories or registers or other such information storage, transmission or display devices.

The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive,

REFERENCES:

[Dodd2002] J, L. Dodd, M. A. Nielsen, M, J. Bremner, and R. T. Thew, Phys. Rev. A 65, 040301 (2002).

[BiereukzOI 1] M. J. Biercuk. A. C. Doherty, and H. Uys, J. Phys. B 44, 154002/1 (2011).

[Suzukil 91] M. Suzuki, J. Math. Phys, 32. 400 (1991),

[Islam 2011 ] R, Islam, E, E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Cannichael, G.-D. Lin, L.-M. Duan, C.-C. J. Wang, J. K. Freerieks, and C. Monroe, Nat. Commun. 2 (2011), 10. i038/ncommsl374.

[Berry2007] D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, Communications in Mathematical Physics 270, 359 (2007).

Claims

CLAIMS:

1. A method for simulating a first quantum system characterised by a first quantum, interaction, the method comprising:

determining a set. of stimulation pulse sequences to stimulate a second quantum system characterised by a second quantum interaction, such that the second quantum system evolves in substantially the same way as the first quantum system,

wherein the step of determining the set of stimulation pulse sequences comprises combining multiple basis pulse sequences according to a combination of basis functions, the combination of basis functions approximating a transformation of the second quantum interaction to the first quantum interaction,

2. The method of claim 1 , wherein approximating the transformation comprises decomposing the transformation into the combination of the basis functions. 3. The method of claim 1 or 2, wherein the combination of the basis functions comprises a weighted sum of the basis functions.

4, The method of claim 3, wherein combining multiple basis pulse sequences comprises determining a duration of one or more of the multiple basis pulse sequences based on the weights of the weighted sum of the basis functions.

5. The method of any one of the preceding claims, wherein the combination of the basis functions comprises a multiplication of basis functions, 6. The method of claim 5, wherein combining multiple basis pulse sequences comprises detemiining a concatenation of one or more of the multiple basis pulse sequences.

7. The method of any one of the preceding claims, further comprising determining the approximation of the desired behaviour using linear programming.

8. The method of claim 7, wherein the linear programming comprises minimising the duration of stimulation pulse sequences. 9, The method of any one of the preceding claims, wherein the basis functions comprise triangular functions.

.1.0. The method of any one of the preceding claims, further comprising applying the stimulation pulse sequences to the second quantum system. 1.1. The method of claim 10, wherein the second quantum system comprises multiple qubits and applying the stimulation pulse sequences comprises applying one or more stimulation pulse sequences to each qubit,

12. The method of any one of the preceding claims, wherein the transformation is discretised according to a lattice structure of the second quantum system.

13. The method of any one of the preceding claims, further comprising determining the transformation of the second quantum interaction to the first quantum interaction. 14. The method of any one of the preceding claims, further comprising determining the basis pulse sequences and the basis functions based on the second quantum interaction of the second quantum system.

15. The method of any one of the preceding claims, wherein the first quantum interaction is an interaction during adiahatic simulation.

16. The method of any one of the preceding claims, further comprising performing a Suzuki-Trotter-type decomposition where a transverse magnetic field is present. 17. Software that when installed on a computer causes the computer to perform the method of any one or more of the claims 1 to 16.

18. A computer system for simulating a first quantum system characterised by a first quantum interaction, the system comprising:

a processor to generate timing data defining a set of stimulation pulse sequences to stimulate a second quantum system characterised by a second quantum interaction, such that the second quantum system evolves in substantially the same way as the first quantum system,

wherein the timing data is generated by combining multiple basis pulse sequences according to a combination of basis functions, the combination of basis functions approxtniating a transformation of the second quantum interaction to the first quantum interaction; and

an output data port to send the timing data to the second quantum system.