Binary optimization system and method

Abstract

A system for identifying a binary sequence as a solution to a binary optimization problem is disclosed, the binary sequence comprising a number of bits. The system comprises a boson sampler and a processor. The boson sampler comprises a light source, a reconfigurable interferometer, and one or more photodetectors. The processor is configured to, for each bit of the binary sequence, select an initial probability value, the initial probability value associated with a probability of a bit flip occurring. The processor is further configured to select initial parameter values for one or more configurable elements of the reconfigurable interferometer. The processor is further configured to iteratively train the parameter values and the probability values until a stopping condition is satisfied. The processor is further configured to operate the boson sampler, configured in accordance with the trained parameters, to produce a batch of samples. The processor is further configured to, for each sample in the batch of samples, determine a candidate binary sequence using the trained probability values. The processor is further configured to identify, from amongst the candidate binary sequences, the solution to the binary optimization problem. Methods are also described.

Description

Binary Optimization System And MethodTechnical Field

[0001] The present disclosure relates to hybrid quantum-classical systems that include a boson sampler, and associated methods for use with such systems.Background

[0002] A boson sampler is a quantum system that relies on the interference of photons to generate its output. More particularly , a bo son sampler comprises a network of optical components or elements (an interferometer) in which photons interfere with one another. As photons are quantum objects, the output of this network is described by a quantum superpositionof all the possible outcomes. When a measurement is performed at the output of this network using one or more photodetectors, a single measurement outcome is realised from this superposition. For example, if photon number resolving (PNR) detectors are used, then each sample or measurement outcome may be describedby an array orstringor sequence of integers indicating how many photons were found ineach output mode of the output state; if threshold detectors are used, then each measurement outcome maybe describedby an array or string or sequence of integers, for example a binary sequence, indicating whetherphotons were present or absent in each output mode of the output state. By repeatedly sampling measurement outcomes, one can build up a picture of the probability distribution governing the quantum superposition.

[0003] The photonic superposition states output from an interferometer of a boson sampler can be highly entangled. Accordingly, the outputprobability distributions generatedby a boson sampler may have a complex structure and simulating this sampling task is understood to be intractable classically. Modern supercomputers fail to simulate boson sampler distributions generated from more than a few tens of modes.

[0004] Boson sampling has been shown to be useful in some optimization and machine learning tasks. Accordingly, improved boson sampling systems and methods are of interest.Summary

[0005] Accordingto an aspect of the present disclosure, a system is provided for identifying a binary sequence as a solution to a binary optimization problem, the binary sequence comprising a number of bits. The system comprises aboson sampler. The boson sampler comprises a light source. The boson sampler further comprises a reconfigurable interferometer. The boson sampler further comprises one or more photodetectors. The system further comprises one or more processors. The one or more processors is / are configmed to, for each bit of the binary sequence, select an initial probability value, the initial probability value associated with a probability of a bit flip occurring. The one or more processors is / are further configured to select initial parameter values for one or more configurable elements of the reconfigurable interferometer. The one or more processors is / are configured to iteratively train the parameter values and the probability values until a stopping conditionis satisfied. The one or more processors is / are further configured to operate the boson sampler, configured in accordance with the trainedparameters, to produce a batch of samples. The one or more processors is / are further configuredto, for each sample in the batch of samples, determine a candidate binary sequence using the trained probability values. The one or more processors is / are furtherconfigured to identify, from amongst the candidate binary sequences, the solution to the binary optimization problem.

[0006] Accordingto an aspect of the present disclosure, a method is provided for identifying a binary sequence as a solution to a binary optimization problem with a heterogeneous computing system comprising a boson sampler, the boson sampler comprising a light source, a reconfigurable beamsplitter, and one or more photodetectors, the binary sequence comprisinga number of bits . The method comprises, for eachbit of the binary sequence, selecting an initial probability value, the initial probability value associated with a probability of a bit flip occurring. The method further comprises selecting initial parameter values for one or more configurable elements of the reconfigurable beamsplitter. The method further comprises iteratively training the parameter values and the probability values until a stopping condition is satisfied. The method further comprises operating the boson sampler, configured in accordance with the trained parameter values, to produce a batch of samples. The method further comprises for each sample in the batch of samples, determining a candidate binaiy sequence usingthe trained probability values. The methodfurthercomprises identifying, fromamongstthe candidate binary sequences, the solution to the binary optimization problem.

[0007] Advantageously, the systems andmethods described herein utilise trainedprobabilify values to determine whetherabitflip should occur. Accordingly, allbinary sequences inthe solution space forthe binary optimization problem are reachable oraccessible to the heterogeneous system, and the system is in general less biasedtowaids one solution over another until after training. This reduces the time and memo ly resources required to identify the solution to the binary optimization problem. Furthermore, the use of probability values reduces the complexity of the physical system required to achieve meaningful solutions, for example the use of probability values means that a threshold mapping (in which the presence or absence of photons measured in each output mode of the boson sampler represents a 1 or 0 respectively) is very effective at exploring the entire solution space, which in turn means that threshold detectors, oftenoperable at room temperature, may be used instead of traditional photonnumber resolving detectors which typically require cryogenics to operate.

[0008] Many modifications and other embodiments set out herein will come to mind to a person skilled in the art in light of the teachings presented herein. Therefore, it will be understood that the disclosure herein is not to be limited to the specific embodiments disclosed herein. Moreover, although the description provided herein provides example embodiments in the context of certain example combinations of elements, steps and / or functions, it will be appreciated that different combinations of elements, steps and / or functions may be provided by alternative embodiments without departing from the spirit or scope of the disclosure.Brief Description of the Figures

[0009] Illustrative embodiments of the present disclosure will now be described by way of example only, with reference to the accompanying figures.

[0010] Fig. 1 shows a block diagram of a system according to an example.

[0011] Fig. 2 shows a diagram of a spatial boson sampler according to an example.

[0012] Fig. 3 shows a diagram of a temporal boson sampler according to an example.

[0013] Fig. 4 shows a flowchart of a method for determining a solution to a binary optimization problem according to an example.

[0014] Fig. 5 shows a flowchart of a method for determining a solutionto a binary optimization problem, in particular a quadratic unconstrained binary optimization (QUBO) problem, according to an example .

[0015] Fig. 6 shows an illustration of the way in which candidate sequences may be determined fromboson sampler outputs according to an example.

[0016] Throughout the description and the drawings, like reference numerals refer to like parts.Detailed Description

[0017] Embodiments ofthe disclosure are described with reference to the accompanying drawings. However, it shouldbe appreciated that the disclosure is not limited to the embodiments, and all changes and / or equivalents or replacements thereto also belong to the scope of the disclosure. The same orsimilar reference denotations maybe used to refer to the same or similar elements throughout the specification and the drawings.

[0018] As used herein, the terms “have”, “may have”, “include”, or “may include” afeature (e.g. a number, function, operation, or a component such as a part) indicate the existence of the feature and do not exclude the existence of other features. Throughout the description and claims of this specification, the words “comprise” and “contain” and variations of them mean “includingbut not limited to”, and they are not intended to (and do not) exclude other components, integers or steps. Throughout the description and claims of this specification, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplatingplurality as well as singularity, unless the context requires otherwise.

[0019] As used herein, the terms “A or B”, “at least one of A and / or B”, or “one or more of A and / orB” may include all possible combinations of A and B. For example, “A or B”, “at least one of A or B”, “at least one of A and B” may indicate all of (1) including at least one A, (2) including at least one B, or (3) including at least one A and at least one B.

[0020] As used herein, the terms “first” and “ second” may modify various components regardless of importance and do not limit the components. These terms are only used to distinguish one component from another. For example, reference to a first component and a secondcomponentmay indicate different components from each other regardless of the order or importance of the components.

[0021] It will be understood that when an element (e.g. a first element) is referred to as being (physically, operatively or communicatively) “coupled with / to”, or “connected with / to” another element (e.g. a second element), it can be coupled with / to the other element directly or via a third element. In contrast, it will be understoodthat whenanelement (e.g. afirst element) is referred to asbeing“directlycoupledwith / to”or“directly connected with / to” another element (e.g. a second element), no element (e.g. a third element) intervenes between the element and the other element.

[0022] The terms as used herein are provided merely to describe some embodiments thereof, but not to limit the scope of other embodiments of the disclosure. It is to be understood that the singular forms “a”, “an”, and “the” include plural references unless the context clearly dictates otherwise. All terms including technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the embodiments of the disclosure belong. It will be furtherunderstoodthatterms, such as those defined in commonly used dictionaries, shouldbe interpreted as having a meaningthat is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealised or overly formal sense unless expressly so defined herein.

[0023] The term “bit” as used herein refers to a binary unit of information that may take one of any two complementaiy binary values. For example, a bit may be representedby a “0” or“ 1”, a bit may be represented by a “+” or“-“, abit may be representedby a “ 1” or “-1”. A “bit flip” is understood to meanthat the value of the bit is changed from one binary value to the other, for example a “0” is changed to a “ 1” or vice versa.

[0024] Many binary optimization problems are NP hard and so cannotbe reliably solved ona classical computer alone. As the number L of bits increases, the number of binary sequences of length L increases exponentially as two to the power of L. An exhaustive search through all binary sequences of length L to find a solution to the binary optimization problem may be impractical for large numbers of bits.

[0025] Described herein are methods and heterogeneous systems that utilise boson sampling to address binary optimization problems. Abosonsampleris a quantum system thatrelies onthe interference of photo ns to generate its output. More particularly, a boson sampler comprises a network of optical components or elements (an interferometer) in which light modes interfere. As photons are quantum objects, the output of this network is described by a quantum superposition of all the possible outcomes. When a measurement is performed at the output of this network using one or more photodetectors, a single measurement outcome is realised from this superposition. For example, if photon number resolving (PNR) detectors are used, then each sample or measurement outcome may be described by an array or string or sequence of integers indicating how many photons were found in eachoutput mode of the output state; if threshold detectors are used, theneachmeasurement outcome maybe described by an array or stringor sequence of integers, forexample abinary sequence, indicating whether photons were present or absent in each output mode of the output state .

[0026] To address a binary optimization problem, samples from the boson sampler (e.g. strings of integers indicating the number of photons in each mode) may be deterministically mapped to binary sequences. For example, the presence or absence of photons in a particular output mode of the interferometer may representa one or zero respectively. The skilled person will appreciate that other mappings of samples to measurement outcomes may also be utilised, for example a parity mappingbased on whether the detected number of photons in an output mode is evenor odd. However, whateverthe choice of mapping, the use of one mapping may restrict the solution space that is accessible or reachable using the boson sampler (e.g. the possible outputs of the boson sampler may not map to all two to the power of L possible binary sequences). Furthermore, some possible solutions may be greatly underrepresented or overrepresented in the solution space.

[0027] One way to address the issues describedabove is to try to solve the binary optimizationproblem multiple times using different mappings or even different input states to the boson sampler. For example, one may find a possible solution using a first threshold mapping in which zero photons in an output mode maps to a binary zero and more thanone photon in an output mode maps to a binary one. One may then find a secondpossible solution using a second threshold mapping complementary to the first threshold mapping in which zero photons in an output mode maps to a binary one and more thanone photon in an output mode maps to a binary zero. Such an approachwouldensure thatthe solution to the binaiy optimization problem is reachable by the system andmay partially mitigate the underrepresentation or overrepresentation of some binary sequences. However, such an approach may be slow and resource intensive.

[0028] Advantageously, the methods and systems described herein utilise a number of trainable probability values associated with bit flips. As detailed herein, the use of the probability values means that the entire solution space is reachable or accessible to the system and that the choice of mapping of samples to binary sequences isless restrictive. Accordingly, the system may utilise threshold detectors in place of photon number resolving detectors. Furthermore, shallow interferometers may be utilised. A shallow interferometer is understood to mean an interferometer that is not full-depth. A full-depth interferometer is an interferometer in which a photon in any input mode may be scattered to any output mode of the outputphotonic state; in a shallow interferometer, there may be some limitations on how the input photons may be scattered to the output modes.

[0029] Fig. 1 depicts ablock diagram of a heterogeneous system 100 in which illustrative embodiments maybe implemented. The heterogeneous system 100 comprises both classical processing apparatus and quantum processing apparatus. Other architectures to that shown in Fig. 1 maybe usedas willbe appreciatedby the skilled person. For example, system 100 may be distributed across multiple interconnected devices.

[0030] System 100 is anexample ofaspecialisedcomputingapparatus, in whichcomputerusable program code or instructions implementing the processes may be located. In this example, system 100 includes communications fabric 102, which provides communications between a processor unit 104, memory unit 106, input / output unit 108, communications module 110, display 112, and boson sampler 114, the boson sampler comprisinga state generation unit 116, an interferometer 118, a state detection unit 120 and a dedicated controller unit 122.

[0031] The system 100 may be implementedin any of a number of ways. For example, the system 100 may be provided as a number of hardware modules suitable for installation in a server / computer rack (for example a conventional 19 -inch server rack). For example, the processor unit 104, memory unit 106, input / output unit 108, and communicationmodule 110 may be provided in a first rack-mounted hardware module, the controller 122 may be implemented in a second rack -mounted hardware module and electronically coupled to the first hardware module, the state generation unit 116 may be implementedin a third rack-mo unted hardware module electronically coupled to the controller 122, the interferometer 116 may be implemented in a fourth rack -mounted hardware module electronically coupled to the controller 122 and opticalfibre -connected to the state generationunit 116, and the photodetectors of the state detectionunit 120 may be provided in another hardware module electronically coupled to the controller 122 and optical fibre -connected to the interferometer module and, optionally , to the state generation unit 116. In other examples, the system 100 may be implemented using one or more separate devices communicatively coupled (at least in part) over a network such as the internet.

[0032] The processor unit 104 is configured to execute instructions for software that may be loaded into the memory unit 106. Processorunit 104 may be a set of one or more processors or may be a mufti -processorcore, depending on the particular implementation. Furthermore, processor unit 104 may be implemented using one or more heterogeneous processor systems in whichamain proces sorispre sent with seco ndaiy processors on a single chip. The processor unit 104 may comprise one or more central processing units (CPUs), one or more graphics processing units (GPUs) or any combinationthereof. If the processor unit 104 comprises multiple processors, the multiple processors may operate individually or collectively.

[0033] The memory unit 106 may comprise any piece of hardware that is capable of storing information, such as, for example, data, program code in functional form, and / or other suitable information on a temporary basis and / or a permanent basis. The memory unit 106 may include, for example, a random-access memory or any other suitable volatile or non-volatile storage device. The memory unit 106 may include a form of persistent storage, for example a hard drive, a flash memory, a rewritable optical disk, a rewritable magnetic tape, or some combinationthereof. The media usedforpersistent storage may alsobe removable. Forexample, the memory unit 106 may include a removable hard drive.

[0034] Input / Output unit 108 enables the input and output of data with other devices that may be in communication with the system 100. For example, input / output unit 108 may provide a connectionfor user input through a keyboard, a mouse, and / orother suitable devices. The input / output unit 108 may provide outputs to, for example, a printer.

[0035] Communications module 110 enables communications with other data processing systems or devices. The communications module 110 may provide communications through the use of either or both physical and wireless communications links. Forexample, the communications module 110 may be configmed to communicate with other data processing systems or devices via a wired local area network connection, via WiFi or over a wide area network such as the internet.

[0036] Instructions forthe applications and / orprograms may be locatedin the memory unit 106, which is in communication with the processor unit 104 through communications fabric 102. Computer-implementable instructions may be in a functional form on persistent storage in the memory unit 106 and may be performed by processor unit 104. These instructions may sometimes be referred to as program code, computer usable program code, or computer-readable program code that may be read and executed by a processor in processor unit 104. The program code in the different embodiments may be embodied on different physical o r tangible computer- readable media.

[0037] The program code may contain instructions which, when processed by the processor unit 104, causethe processor unit 104 to communicate with the boson sampler to sample a bosonic probability distribution.

[0038] In Fig. 1, computer-readable instructions 126 are located in a functional form on computer -readable storage medium 124 that is selectively removable and may be loaded onto or transferred to system 100 for execution by processor unit 104. Alternatively, computer-readable instructions 126 may be transferredto system 100 from computer-readable storage medium 124 througha communications link to communications module 110 and / or through a connection to input / output unit 108. The communications link and / or the connection may be physical or wireless.

[0039] A computer-readable storage medium may be, for example but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or any suitable combination thereof. More specific examples of the computer -readable medium include the following: a portable computer diskette, a hard disk, a random-access memoiy (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flashmemo ly), a portable compact disc read-only memory (CDROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a computer-readable storage medium may be any tangible medium that cancontainor store a program for use by or in connection with an instruction execution system, apparatus, or device.

[0040] In some illustrative embodiments, computer-implementable instructions 126 may be downloaded overa network to the memory unit 106 from a remote device for use with system 100. For instance, computer- implementable instructions stored in a remote server may be downloaded over a network from the server to the system 100.

[0041] The boson sampler 114 comprises a state generation unit 116, a linear interferometer 118, a state detectionunit 120 and a dedicated (classical) control unit 122. Example boson sampler architectures are described further below in relation to Fig. 2 and Fig. 3.

[0042] The state generation unit 116 is configured to generate an input multimodal photonic state \l!N} comprising a plurality of input modes. The input multimodal photonic state is a product state (in other words, there is no quantum entanglement between input modes) comprising a plurality of N non-vacuum optical inputs distributed across a plurality of M input modes.

[0043] In some examples, the state generation unit 116 may be configured to generate an input multimodal photonic state \l!N) comprising N photons distributed across the plurality of M modes. When the number of photons N is less than or equal to the number of input modes M, and one photon is provided in each of the populated input modes (in which case the boson sampler may be referred to as a single -photonboson sampler), the input state can without loss of generality be expressed aswhere akis the bosonic creation operator in the / c th mode. The skilledperson will appreciate that the methods and systems described herein are also applicable when one or more inputmodes comprise more thanone photon.

[0044] In other examples, the state generation unit 116 may be configured to generate an input multimodal photonic state |¥ / / lV)comprisingaGaussianpho tonicinput ineachofthe N inputmodes(inwhichcase the boson sampler may be referredto as a Gaussianboson sampler). For example, a single mode squeezed state (SMSS), also referred to as a squeezed coherent state, may be input into each input mode.

[0045] In other examples, the state generationunit may be configured to generate an input multimodal photonic state \l!N) comprising a non-Gaussian photonic input in at least one of the N input modes (in which case the boson sampler may be referredto as a non-Gaussianboson sampler). For example, a photonmay be subtracted froma single mode squeezed state and detected, thereby heraldingthe generationof a non-Gaussian state forinput into an input mode.

[0046] In some examples, the state generation unit 116 may be configured to generate an input multimodal photonic state \l!N) thatcomprises single photons in some modes and squeezed coherent states in other modes.

[0047] The state generation module 116 may comprise one or more light sources. For example, in a singlephoton boson sampler, the state generation module 116 may comprise a non-linear photonic material (such as periodically -poled lithium niobate (PPLN) or potassium titanyl phosphate (KTP)) configured to receiveapump beam from a pump laser and to probabilistically generate pairs of entangled photons, and may further comprise a photodetector configmed to detect a photon of the entangled pair, thereby heralding the presence of the other photon of the pair. For example, in a Gaussianboson sampler, the state generation module 116 may comprise PPLN waveguides configmed to generate two entangled modes of light, and a 50 : 50 beamsplitter for interfering the two modes of light, thereby generating two independent single mode squeezed Gaussian states. In a non- Gaussian boson sampler, the state generation module 116 may comprise PPLN waveguides and a 50:50 beamsplitter, and then a 99 : 1 beamsplitter and a dedicated photodetector, which together may herald a photon subtracted state.

[0048] The reconfigurable interferometer 118 comprises one or more optical elements arranged to interfere the modes of the input multimodal photonic state, thereby transformingthe input multimodal photonic state to produce an output multimodal photonic state. The interferometer 118 is configured to receive the input multimodal photonic state, to transform the input multimodal photonic state to an output multimodal state, and to output the output multimodal photonic state to the state detectionunit 120. One or more optical elements of the interferometer118 are configurable, and each configurable optical element has one or more configurable parameters that influence the operation of that element when the boson sampler is used. Accordingly, the transformation from input multimodal photonic state to output multimodal photonic state is dependent on a set of parameter values {r } that define the function of one or more configurable optical elements. One or more of the configurable parameters may characterise a single mode operation. For example, a configurable optical element may comprise a phase shifter, and a parameter value raof the set of parameter values {r} may characterise the phase shift impartedby the phase shifter. One or more of the configurable parameters may characterise a multimodal operation. For example, a configurable optical element may comprise a reconfigurable beamsplitter having a tuneable transmission (or equivalently, a reflection) coefficient, and a parameter value rbof the set of parameter values {r} may characterise the transmission coefficient of the reconfigurable beamsplitter.

[0049] The interferometer 118 may be designed and manufactured in any suitable and desired way e.g. depending on the modes of electromagnetic radiation to be transformed by the interferometer 118. Thus, for example, when the electromagnetic radiation has an optical or infrared wavelength (e.g. between 400nm and 700nm or between 700nm and 1600nm), the optical paths through the interferometer 118 may be implemented at least partially using optical fibres. In some examples, the interferometer 118 may be implemented in bulk optics. However, in other examples, the interferometer 118 may comprise (i.e. is designed and manufactured using) an integrated circuit. In the integrated circuit, the optical paths may be implemented with, for example, a plurality of etched waveguides and plurality of couplinglocations arrangedinthe integrated circuit. At each coupling location, tuneable elements may be arranged (e.g. EOM phase shifters) that are configured to control the coupling interactionbetweenthe waveguides. The integrated circuitmay be implementedin silicon nitride (Si3N4) or any other suitable material.

[0050] Due to interference betweenphotons in different temporal modes, in operation the boson sampler 114 transforms the input multimodal photonic state into an output multimodal photonic state that may be expressed as a superposition of the different possible configurations of the photons in the output modes aswhere C is a configuration, n is the number of bosons in the jth output mode in configmation C, and acis the probability amplitude associated with configuration C . The skilled person would appreciate that while the state of (EQ. 2) is expressed as a pure state, this is for illustrative purposes only - photonloss may , for example, mean that the output state canbe expressed only as a mixed state. By tuning the parameter values {r}, the probability amplitudes associated with each configuration may be changed. Accordingly, a measurement of the number of photons in each output mode canyield a measurement outcome representable as a stringof integers corresponding to a configuration C. By operating the boson sampler 114 a number of times to produce abatch of samples S, it is possible to establishan empirical probability distribution of the bosonic configurations of the outputstate. One can expect that with many samples, the probability pcof obtaining a measurement outcome corresponding to configuration C is approximately given by pc= |ac|2.

[0051] The state detectionunit 120comprises anarrangementofoneormorephotodetectors configured to detect photons output from the interferometer 118 and produce correspondingdetection event signals . In some examples, the photodetectors may comprise photon number resolving (PNR) detectors, capable of determining how manyphotons are received. For example, the detectors may comprise superconducting nanowire detectors that generate an output signal intensity proportional to the (discrete) number of photons that strike a detector. The PNR detectors may comprise transition edge sensors (TESs). In other examples, the photodetectors may comprise threshold detectors, also known as on / olf detectors. Threshold detectors are not capable of determining how many photons are received but are capable of determining the presence / absence of photons in an output mode.

[0052] The controller 122 is communicatively coupledto the processorunit 104, the state generation unit 116, the interferometer 118 and the state detectionunit 120. The controller 122 may be any suitable classical computing resource for controlling the operation of the boson sampler 114. In some examples, the controller 122 is implemented in a dedicated, application-specific processingunit. For example, the controller 122 may comprise an application-specific integrated circuit(ASIC) or an application-specific standard pro duct (ASSP) or another domain-specific architecture (DSA). Alternatively, the controller 122 may be implemented inadaptive computing hardware (in other words, hardware comprising configurable hardware blocks / configurable logic blocks) that has been configured to perform the required functions, for example in a configured field programmable gate array (FPGA). The controller may have a dedicated random-access memory or other memory element for temporarily logging data. In some examples, the functionality of the controller 122 may be incorporated into the functionality of the processor unit 104.

[0053] The controller 122 is configured to receive instructions from the processorunit 104. More particularly, the controller is configured to, if so directedby the processorunit 104, configure the interferometer 118 according to a set of parametervalues {r } and thereby control the transformation of the input multimodal photonic state that is implemented by the interferometer 118. For example, the controller 122 may directly send control signals that tune the reflectivity / transmittance of a reconfigurable beamsplitter orthe phase impartedby a phase shifter. The controller 122 is further configured to, if so directedby the processor unit 104, generate one or more control signals to cause the state generation unit 116 to produce an input multimodal photonic state. The controller 122 may optionally be able to control which input multimodal photonic state is input into the boson sampler, for example by generating one or more control signals to control a number of photons in each input mode. For example, in a photonic boson sampler in which the state generation module comprises a plurality of single photon sources, the controller 122 may be able to generate one or more control signals to cause a selected number of photons to be emitted at a particular time point.

[0054] The controller 122 is further configuredto receive a response from the state detection unit 120. More particularly, the controller 122 is configured to receive measurement outcomes from the photodetectors of the state detectionunit 120. In other words, the controlleris configuredto sample from the output distribution of the configuredboson sampler. For example, the measurement outcomes may comprise an electrical signal from each photodetector at which a detection event occurs. In examples wherein the photodetectors are PNR detectors, the electrical signals may further be indicative of the number of photons received. In examples wherein the photodetectors are threshold detectors, the electrical signals may be indicative of the presence of photons in particular output modes.

[0055] The controller 122 is further configured to communicate the response from the state detection unit 120 to the processor unit 104.

[0056] In operation, the boson sampler 114 is configuredto receive parametervalues from the processor unit 104; to produce abatch (in other words, a plurality) of samples by configuring the interferometer 118 accordingto the received parameter values, generating an input photonic state and measuring an output photonic state ; and to communicate the batch of samples to the processorunit 104. Each sample may indicate the presence or absence of photons in each output mode of the output multimodal photonic state. In the event that one or more number resolving detectors are used, each sample may indicate the number of photons in eachoutput mode of the output multimodal photonic state.

[0057] The skilled person would appreciate that the architecture described above in relation to Fig. 1 is not intended to provide limitations on the computing devices with which the methods described herein may be implemented. Instead, the skilled personwouldappreciate thatotherarchitectures maybe applicable. For example, the computing device may include more or fewer components.

[0058] The boson sampler 114 of Fig. 1 may comprise a spatial mode interferometer, such as the single-photon boson sampler 114a illustrated in Fig.2. In the boson sampler 114a of Fig.2, the modes of the input multimodal photonic state are spatial modes - that is, the state is defined by the number of photons in each of a plurality of spatially distinct paths. The boson sampler 114a may be implemented, at least in part, in a photonic integrated circuit.

[0059] The state generation unit 116a of Fig. 2 comprises a plurality of single-photon sources 210 configured to produce single photons. One suitable photon source technology is spontaneous parametric down -conversion (SPDC). In SPDC, anon-linear crystal is pumpedwithalaserand,probabilistically,entangledphotons are emitted (the “signal” and the “idlef’). A photodetector (not shown in Fig. 2) is arrangedto detect the presence of the idler photonwhich, due to the entanglement, heralds the presence of a photonin the signal mode. Otherphoton sources may also be used, for example solid state photon sources and quantum dots.

[0060] The number of single photon sources 210 may be greater than the number M of input modes of the input multimodal photonic state \VIN)SPin orderto account for the fact that single photons may be generated only probabilistically. The state generation unit 116a of Fig. 2 comprises a multiplexer 220 to route successfully generated single photons to N input ports of the M input ports of the interferometer 118a. In the example shown in Fig. 2, the number of single photons N is equal to the number of input ports M of the interferometer 118a.

[0061] The interferometer 118acomprises M inputports, M outputports, and a plurality of wave guides arranged to pass through the interferometer 118a to connect the M input ports to the M output ports. The plurality of waveguides are arranged to provide a plurality of coupling locations betweenpairs of the plurality of waveguides. The interferometer 118a may be designed and manufactured in any suitable and desired way e.g. depending on the modes of electromagnetic radiation to be transformed by the interferometer. Thus, for example, when the electromagnetic radiation has an optical or infrared wavelength (e.g. between 400nm and 700nm or between 7 OOnm and 1600nm), the waveguides may comprise optical fibres. In some examples, the interferometer may be implemented in bulk optics. However, in other examples the interferometer comprises (i.e. is designed and manufactured using) an integrated circuit, with the plurality of waveguides andplurahty of coupling locations arranged in the integrated circuit. The integrated circuit may be implemented in s ilicon nitride (Si3N4) or any other suitable material, for example thin-film lithium niobate.

[0062] A reconfigurablebeamsplitter230 is arranged at eachof the couplinglocations suchthatat each coupling location the two modes of electromagnetic radiation carried by the two respective waveguides are capable of coupling with each other with a reconfigurable reflectioncoelficient (transmission coefficient). The configurable parameters (denoted with a theta in the figure) in this example relate to the reflection (transmission) coefficientsof the reconfigurable beamsplitters. The skilled personwill appreciate that the interferometer 118amay comprise further reconfigurable elements.

[0063] A parametrised / reconfigurable beam splitter is understood to mean any tuneable element or device or tuneable collection of elements / devices capable of couplingtwo modes of electromagnetic radiation with each other with a reconfigurable reflection / transmission coefficient and optionally a reconfigurable phase shift coefficient (not indicated in Fig.2). The parametrised beam splitters may be implemented in any suitable way - forexample aparametrisedbeam splittermay comprise aMach-Zehndertype interferometercontaininga variable phase shifter in one internal path for controlling the effective beam splitter reflection coefficient of the Mach- Zehnder interferometer. The Mach-Zehnder interferometer may further comprise anexternal phase shifter on one external path of the Mach-Zehnder interferometer to control the relative phases of the two mode s acted upon. For example, when the interferometer 118a is implemented in an integrated circuit, a reconfigurable beamsplitter may comprise a first waveguide coupling region for coupling the electromagnetic radiation modes in each waveguide, an electro-optical phase shifting element for adjusting the phase in one of the outgoing waveguides from that coupling region, and a second waveguide coupling region for recouplingthe two electromagnetic modes output from the first waveguide coupler. For example, when implemented in bulk or fibre optics, a reconfigurable beamsplitter may comprise two 50 / 50 beamsplitters and a phase shifter element arranged therebetween.

[0064] The interferometer 118a may further comprise reflective elements (e.g. mirrors) and other passive photonic elements (not shown). Accordingly, the interferometer 118acouples the single photons receivedatthe M input ports to the plurality of M output ports based on operations defined by a set of parameter values.

[0065] The interferometer 118a of Fig. 2 is suitable for transforming an input multimodal photonic state comprising M input spatial modes to an output multimodal photonic state comprising M output spatial modes. The skilled person will appreciate that other architectures for the interferometer 118a may be utilised. Of course, while in the illustration the number of input and output modes is M = 4, an interferometer 118a may be provided to operate on a greater number of spatial modes. Of course, the interferometer 118a may comprise any number of reconfigurable / parametrised elements and in any configuration that leads to interference between spatial modes.

[0066] The state detection unit 120a comprises aplurahty of photon number resolving (PNR) photodetectors 240, each arranged to receive any photons output from a corresponding output port of the interferometer 118a. The state detection unit 120acomprises one PNR detectorforeachofthe M output modes and accordingly the measurement outcomes are representative of the number of photons measured in all outputmodes of the output multimodal photonic state. The PNR detectors may comprise nanowire photodetectors.

[0067] The controller 122a is coupled to each ofthe state generation unit 116a, the interferometer 118aandthe state detection unit 120a. The controller 122a is further communicatively coupled to the processor unit 104. The controller 122a may receive a set of parametervalues {r} fromthe processorunit 104, and may generate control signals to configure the tuneable elements 230 of the interferometer 118a in accordance with those parameter values. Forexample, the controller 122amay assign the parameter value rxto parameter / ^, the parametervalue r2to parameter 02andsoon. For example, each reconfigurable beam splitter 230 may comprise a Mach-Zehnder interferometer comprisingtwo 50 / 50 beamsplitters andaphase shifter locatedineachofoneorbothof its internal opticalpaths. The phase shiftermay be implemented usingan electro -optical modulator. The control signals may comprise an electric fie Id for controlling the phase shift impartedby the internal phase shifters and therefor the coupling strength of the reconfigurable beam splitter. The controller 122a may further generate a control signal tocause the single photon sources 210 to begin generating single photons, for example the control signal may cause a pump laserto pump light into the non-linear materialofthe single-photon sources 210. The controller 122a may further receive signals from each of the PNR detectors 240 indicative of the number of photons detected at each of the PNR detectors 240, which may be interpreted as a sample of the output probability distribution produced by the boson sampler 114a. The controller 122a may then communicate the sample information to the processor unit 104.

[0068] The boson sampler 114 of Fig. 1 may comprise a temporal mode interferometer, such as the singlephotonboson sampler 114b illustrated in Fig. 3. In the boson sampler 114b of Fig. 3, the modes of the input multimodal photonic state are temporal modes, whichmeans that the state is defined by the numberof photons in each of a plurality of temporal modes or time bins.

[0069] The state generation unit 116b of Fig. 3 comprises a single-photon source 310 operable to produce a single photon in each of a plurality of time bins, so that each photon enters the time -bin interferometer 118b separated from the next by a duration t. As in the boson sampler 114aof Fig. 2, the state generationunit 116a may comprise further single photon sources and a multiplexer in order to reliably ensure that a single photonis generated in each time period t.

[0070] The interferometer 116b comprises a temporal mode coup ling device. In particular, in Fig.3, a temporal mode coupling device comprises a reconfigurable beam splitter 320 and a delay line 330. The delay line 330 is arranged to connect one input port of the reconfigurable beam splitter 320 with one output port of the reconfigurable beam splitter 320. The delay line may comprise, for example, opticalfibre. The delay line 330 has a length ct where c is the speedof light in the fibre. In this way, the field ofthe photon inone temporal mode may be coupled, at least partially, into the delay line 330 so as to interfere with photons in the next temporal mode on the parametrised beam splitter 320. The time -bin interferometer may comprise further optical components including further optical switches.

[0071] The controller 122b is configured to tune the parameter value (e.g. transmittance) of the parametrised beam splitter 320 for each time interval. For example, for four input modes, the temporal mode coupling device canbe used to implement the equivalent operations of the three beam splitters defined by parameters 0 02and 03shown in Fig. 2. For example, the controller 122b may, as afirstphotonis emitted fromthe photon source 310, configure the reconfigurable beamsplitter 320 to loop the first photoninto the delay line 330. The controller 122b may then, as a second photon is emitted from the photon source 310, configure the reconfigurable beamsplitter 320 using parameter value rxto cause interference between the first temporal mode and second temporal mode (e.g. the first and second photon) of the input state | 'PIN). The controller 122b may then, as a third photon is emitted from the photon source 310, configure the reconfigurable beamsplitter 320 using parameter value r2to cause interference between the second temporal mode and third temporal mode. The controller 122b may then, as a fourth photon is emitted from the photon source 310, configure the reconfigurable beamsplitter 320 using parameter value r3to cause interferencebetweenthe third temporal mode and fourth temporal mode. This may continue until a predetermined transformation has beenperformed on the input photon sequence of M time bins.

[0072] The state detectionunit 120b comprises a photonnumber resolving (PNR) photodetector 340 configured to detect the number of photons in each temporal mode. By measuring the number of photons in each of M time bins output from the interferometer 118b, the boson sampler 114b takes a sample of the output distribution.

[0073] The skilledpersonwould appreciatethatthe architecture ofthe temporal mode bosonsampler 114b of Fig. 3 may be varied in several ways. For example, the boson sampler 114b may comprise further reconfigurable beamsplitters 320and further delay lines 330inorderto generate mo recomplicatedinterferencebetween temporal modes. The skilledpersonwouldfurtherappreciate that delay lines of different lengths maybe usedto vary which temporal modes are interfered with one another. In other temporal mode boson samplers, the temporal mode coupling device may comprise a quantum memory that may be controlled to selectively interfere photons in different temporal modes.

[0074] The skilledperson would appreciate thatthe spatialmodebosonsamplerof Fig.2 and the temporal mode boson sampler of Fig. 3 may further be operable as Gaussian boson samplers or non-Gaussian (e.g. photon- subtracted) boson samplers with a suitable substitution of the state generationunit 116 a / 116b. Furthermore, the PNR detector(s) ofthe state detection modules 120a / 120b may be replaced with threshold detectors, in which case the measurement outcomes outputfromthe state detectionunit are indicative of the presence or ab sence of photons in each output mode but not the number of photons in output modes. The PNR detector(s) may be replaced with pseudo-threshold detectors.

[0075] The hybrid quantum-classical system 100 of Fig. 1 is configured to perform the method 400 depicted in the flowchart of Fig.4 for identifying a binary sequence b [so / ] as a solution to a binary optimization problem, the binary sequence comprising a number L of bits. The skilledperson will appreciate that while the method400 is discussed with reference to the system 100 specifically, other hybrid quantum-classical architectures comprising a (classical) processor unit and a configurable boson sampler may also be configured to perform the method 400.

[0076] At 410, the method comprises selecting a set of initial parameter values {r} forone or more configurable elements of the interferometer 118. For example, if the interferometer 118 is a spatial interferometer such as 118a, the initial parameter values may each relate to a correspondingreconfigurablebeamsplitterand characterise the effective reflection coefficient of that corresponding reconfigurable beamsplitter. For example, if the interferometer 118 is a temporal interferometer such as 118b, the initial parameter values may each relate to a single reconfigurable beamsplitter and characterise the effective reflection coefficient of that reconfigurable beamsplitter at different points in time. At 410, the method further comprises selecting, for each soughtbit bj of the soughtbinaiy sequence b[so;j,acorrespondingprobabilify value qj which willbe associated with a probability of the correspondingbit bj being flipped. As the binary sequence b[so / ] comprises a number L of bits, the set of probability values {q } comprises a corresponding number L of probability values. The probability values may be selected in any manner. For example, the probability values may be selected deterministically (e.g. all probability values are initially set to a value of 0.5) or may be randomly selected from a random distribution. The initial probability values may all be the same or may be different from one another. With reference to Fig. 1 , processor unit 104 may, during the course of executing instmctions loaded from the memory unit 106 or elsewhere, select initial parameter values and probability values.

[0077] At 420, the method comprises iteratively trainingthe parameter values {r} and probability values {q} until a stopping condition is satisfied. An example of this stage will be describedfurtherbelow in relation to Fig. 5. Training the probability values and parameter values may comprise for example reducing a cost function / objective function in a gradient based process or method e.g. gradient descent.

[0078] Any suitable stopping condition may be utilised. For example, determining that a stopping condition has been satisfied may comprise determining that the set of parameter values or probability values have beenupdateda threshold number of times, for example that a threshold number of epochs or iterations has been reached. The thresholdnumberof iterations maybe selected in advance by auserof system 100. As another example, a stopping condition may comprise a convergence criterion. Determining that a convergence criterion has been met may comprise determining that a cost function / objective function has not changed more than a threshold amount between updates of the parameter values or probability values.

[0079] At 430, the method comprises operating the boson sampler 114 in accordance with the trained parameter values to produce abatchof samples S. With reference to Fig. 1, processorunit 104 may instruct the controller 122 to operate the boson sampler in accordance with the trained parameter values. The controller 122 may accordingly configure the interferometer 118 accordingto the received parameter values, generate control signals to cause the state generation unit 116 to produce an input multimodal photonic state, and receive det ection event signals from the state detection unit 120 which can be interpreted as integer strings representative of samples or measurement outcomes. The controller 122 may communicate the samples to the processor unit 104.

[0080] At 440, the method comprises, for each sample s of the batch of samples S, determining a candidate binary sequence b [cand] usingthe trained probability values. For examples, and with reference to Fig.1 , processor unit 104 may determine from each sample a preliminary binary sequence b[pre;imj. For each preliminary binary sequence, the processor unit may thenprobabilistically flip eachbit btof that preliminaiy binary sequence with a probability associated with thecorrespondingtrained probability value q7. to producca candidate binary sequence b[cand]. In otherwords, the processor 104 may use a random process weightedin accordance with the trained probability value to determine whether or not to flip the corresponding bit of the preliminary sequence. Flipping the bit may comprise bit manipulation, that is algorithmic manipulation of the binary digit. Flipping the bit may comprise performing a bitwise NOT operation on the bit.

[0081] At 450, the method comprises determining, fromamongthe candidate binary sequences, a solution b [so / ] to the binary optimization problem. With reference to Fig. l,the processor 104 may for example trial each of the determined candidate binary sequences and determine, from amongst the candidate binary sequences, which candidate binary sequence provides the best result to the binary optimization problem.

[0082] An example of abinary optimization problem that the system 100 may solve using the methodof Fig. 4 is a quadratic unconstrained binary optimization (QUBO) problem. The task is to find a binary sequence b^ol^ comprising a number L of bits that minimizes a target function: b ' Wb (EQ. 3) where W is an L x L symmetric matrix and b is abinary sequence of length! determined from samples froma configured boson sampler.

[0083] Fig. 5 shows a flowchart of a method 500 for determining a binary sequence as a solution to a QUBO problem using the system 100. In this example, it is assumed that the one or more photodetectors of the state detection unit 120 comprise photon number resolving (PNR) detectors and so each sample collected from the boson sampler 114 indicates a number of pho tons nkmeasured ineach corresponding output mode k ofthe boson sampler 114. However, the skilled person will appreciate that threshold detectors may be used instead of photon number resolving detectors. In this example it is further assumed thatthe length L ofthe sought binary solution b[S0 / ] is equal to the number of output modes M ofthe boson sampler (L = M), and that each measured photon numbernfeis mappable to a correspondingbit bfeof abinary sequence. However, the skilledperson will appreciatethat, for example, the length of the binary sequence L may be of a different length to the number of output modes M and that the transformation described further below may be varied.

[0084] Before describingthe method 500 in further detail, the idea behind the method is described at a high level with reference to Fig. 6. Parameter values {r} are initially selected that define a substantially unitary transformation ! / ({r}) of input modes to output modes in the boson sampler. A photon number resolving measurement of the number of photons in each output mode yields a string of integers. A preliminary binary sequence may then be deterministically determined from the measured sample. In the illustration of Fig. 6, a “threshold mapping” is used e.g. if the numberof photons ngmeasured in a particular output mode g is greater than zero then a corresponding bit of the preliminary binary sequence bgis set to one while if the number of photons measured in that particular output mode is zero then the corresponding bit of the preliminary binary sequence is set to zero. Multiple measurement outcomes may yield the same preliminary binary sequence, for example the strings (3 ,0, 1 ,0) and (2 ,0,2,0) lead to the same binaiy sequence (1,0,1 ,0) underthis thresholdmapping. The skilled person will appreciate that while in Fig. 6 a sample comprises a string of integers that are algorithmically mapped to a preliminary binary sequence due to the use of PNR detectors, threshold detectors may be used insteadof PNR detectors to determine the preliminary binary sequence. By trainingthe parameter values of the boson sampler, the output distribution of the boson sampler may be adapted until the solution binaiy sequence is more likely to be found.

[0085] However, the preliminary binary sequences may not themselves represent the entire space of solutions to the binary optimization problembeing addressed. For example, when a single -photonboson sampleris used, due to energy conversation principles one would expectthatthe numberof photons input to the interferometer of the boson sampler shouldbe the same as the numberof photons output from the interferometer and measured - for example, four photons input into the interferometer in four input modes (e.g. (1,1,1,!)) should not (in the absence of photon loss) be able to yield an output state (0 ,0,0,0) in which zero photons are output from the interferometer, and this in turnmeans that the preliminary binary sequence (0 ,0,0,0) can notbe represented. That is, the entire solution space of two to the power of L binary sequences may notbe reachable or accessible to the boson sampler when the threshold mapping alone is used. Furthermore, for similar reasons the threshold mapping may bias some binary sequences over other binary sequences, which may influence the training of the parameter values. Similar problems may arise even when using a Gaussian boson sampler or non-Gaussian (e.g. photon- subtracted) boson sampler in place of a single-photon boson sampler.

[0086] One option to overcome the issues described above is to use different mappings to produce binaiy sequences from the outcomes of the boson sampler. For example, one may train the parameter values of the boson sampler with the threshold mapping to find a possible solution to the binary optimizationproblem. One may then restart the process to find a second possible solution with a complementaiy threshold mapping in which if the number of photo ns ngmeasured in a particular output mode g is greater than zero then a corresponding bit of the preliminaiy binary sequence bgis set to zero while if the number of photons measured in that particular output mode is zero then the corresponding bit of the preliminary binary sequence is set to one. However, such an approach is time consuming and energy inefficient.

[0087] With reference again to Fig.6, a more resource efficient approach is to initially select, for eachbit bj of the sought solution to the binary optimizationproblem, a correspondingprobability value associated with a probability offlippingthatbit. Thebinaiy optimizationproblem may thenbe represented by an objective functionthat is based on the output of the boson samplerand the set of probability values, and then the set of parameter values {r} and the set of probability values { <7 } can both be trained. In this way , the entire solution space for the binary optimization problem is reachable by the boson sampler and inherent overrepresentation of underrepresentation of particular binaiy sequences is reduced. Once a stoppingconditionis satisfied, the boson sampler is operated to produce a batch of samples, and each sample is deterministically mapped to a preliminary binary sequence using the threshold mapping. However, each bit of each preliminary binary sequence is then probabilistically flipped in accordance with the corresponding trained probability value to determine a corresponding bit of a candidate binary sequence.

[0088] By providing trainable probability values, the heterogeneous system is given more flexibility to find the solution to the binary optimizationproblem. Furthermore, the system is typically able to find the solutionfaster than a heterogeneous system that has to optimize using multiple mappings.

[0089] Referring again to Fig. 5, at 505, the method comprises selectinga set of initial parameter values {r} for one or more configurable elements of the interferometer 118. For example, if the interferometer 118 is a spatial interferometer such as 118a, the initial parameter values may each relate to a corresponding reconfigurable beamsplitterandcharacterisethe effective reflectioncoefficient of thatcorrespondingreconfigurablebeamsplitter. For example, if the interferometer 118 is a temporal interferometer such as 118b, the initial parameter values may each relate to a single reconfigurable beamsplitter and characterise the effective reflection coefficient of that reconfigurable beamsplitter at different points in time. The parameter values may be selected in any suitable manner, for example the process or unit 104 may utilise a random number generator to produce a random selection of parameter values. At 505, the method further comprises selecting, for each soughtbit bj of the sought binaiy sequence b[so / ], a corresponding probability value qj which will be associated with a probability of the corresponding bit bj being flipped. As the binary sequence b^ol^ comprises a number L of bits, the set of probability values {q } comprises a corresponding number L of probability values. The probability values may be selected in any manner. For example, the probability values may be selected deterministically (e.g. all probability values are initially set to a value of 0.5) or may be randomly selected from a random distribution. The initial probability values may all be the same or may be different from one another. With reference to Fig. 1 , processor unit 104 may, during the course of executing instmctions loaded from the memory unit 106 or elsewhere, select initial parameter values and probability values.

[0090] Steps 510, 515, 520, 525 and 530 may togetherprovide an iterative process to train the set of parameter values {r} and the set of probability values {q } until a stopping condition is reached. In particular, through a process of gradient descent, parameter values and probability values are iteratively updated to reduce (e.g. minimise) an objective function. The objective function to be reduced is given by:represents a matrix element of the symmetric matrix W of (EQ. 3 ), btand bj represent the ith and jth bits of a binary sequence respectively and (b^bj) represents an expectationvalue of the product of the bits b^bj derivedfromabatchS of samples s andis dependent on the ith and jthprobability values q;and q;and observables determined from the batch of samples.

[0091] Under the threshold mapping, the quantity (b^bj) can be expressed as:where mi7is the probability, determined from a batch S of samples, of zero photonsbeing detected in the ith mode and zero photons being detected in the jth mode (probability that n, = 0 and n7= 0); is the probability, determined from a batch S of samples, of zero photons being detected in the ith mode (probability that ni= 0); and rrij is the probability , determined from a batch S of samples, of zero photons being detected in the jth mode (probability that rtj = 0). The quantity in (EQ. 5) may reflect that bits of a candidate binary sequence may be determined from the probabilistic flipping of corresponding bits of a preliminary binary sequence deterministically determined from samples.

[0092] At 510, the method 500 comprises operatingthe boson sampler 114 to produce abatchof samples S. The processor unit 104 may instruct the controller 122 to operate the bo son sampler 114 in accordance with the set of parametervalues {r} to produce abatchof samples S representativeof the output distribution ofthe boson sampler 114. Each sample s may comprise, for example, a string of integers indicative of the number of photons received in each output mode of the output multimodal photonic state (EQ. 2).

[0093] At 515, the method 500 comprises, for at least one probability value qk. determining from the batch of samples S. an estimate of a partial derivative or gradient ofthe objective function with respectto that probability value qk. The processor unit 104 may determine the estimate of the partial derivative in any suitable way, for example through a classical back-propagation algorithm.

[0094] At 520, the method 500 comprises, for at least one parameter value rkof the set of parameter values, determining, from the batch of samples S. an estimate of a gradient or partial derivative of the objective function (EQ. 4) with respect to the parameter value rk.

[0095] Accordingto a first example method for using the boson sampler 114 to determine the gradient of the objective function with respectto the selected parameter, the processor 104 determines from the batch of samples S the objective function value fQUB0({r, q}) (EQ. 4). The processorunit 104 may then instruct the controller 122 to configure the interferometer 118 suchthat the selected parameter rkis shifted by a small value e while all other parameter values are maintained, and to produce a second batch of samples. The processor unit 104 may then determine from the second batch of samples the objective function value+e) - The argument“{r, q].rk+ e” has been used to denote that all current parameter values have been unchanged except for the selected parameter rk. The processor unit 104 may then determine a numerical estimate for the gradient of the cost function with respect to the selected parameter from:

[0096] Accordingto a second example methodforusingthebosonsampler 114 to determine the gradient of the objective function with respectto the selected parameter, determining the gradient comprises determiningfrom the batch of samples S, for each observable Oj of a set of observables {O}, a correspondingfirst value Vi(O7) representative of a partial derivative of the objective function with respect to the expectation value of the observable Oj). In other words, the firstvalue Vj(O7) for the observable Oj is approximately:

[0097] Accordingto the second example method, determining the gradient further comprises determining, for each observable Oj of the set of observables {0}, a corresponding second value V2Oj) representative of a partial derivative of the expectationvalue of the observable (O;) with respect to the parameter value rk. In other words, the second value F2(O7) for the observable Oj is approximately:

[0098] Accordingto the second example, determining the gradient further comprises determining the gradient fromthe firstvalues V1and the secondvalues V2. For example, the estimate of the partial derivative of the objective function with respect to the parameter value rkmay be determined as a weighted or unweighted sum over the product of the first values and second values, for example:where Cj is a coefficient. In some examples, the coefficients Cj may be equal and the sum may be unweighted.

[0099] The term “observable” as used herein is understood to meana quantity determinable directly or indirectly (for example using a deterministic function) from a sample of the output of a boson sampler. The choice of the set of observables {0} may, however, dependontheobjective functionused,forexample, howwell the objective functioncanbe decomposed into orrepresentedby the set of observables {O}. The choice of the set of observables may depend onthe configurable parametertobe optimised, for example on how quickly the secondvalues V2may be calculated. In this particular example, a sensible observable Ogmay be the measured number of photons ngin a corresponding output mode g of the interferometer 118.

[0100] The first values V1may be determinedinany ofanumberofways. Forexample, determiningthevalues V1may comprise determining, from the batch of samples S. an affine fit of the observables to the objective function. For example, the processor unit 104 may perform a linear regression.

[0101] The secondvalues V2may also be determined in any of a number of ways. In particular, the choice of observables may be made to make determination of the secondvalues a computationally easy task. In some examples, the second values may be determined fully analytically or from an analytic function of the relationship between the expectation value of the observable (O;) and the configurable parameter.

[0102] The second example methodfordeterminingthe gradient described above may be advantageous overthe first method described above, because using the second example method a single batch of samples is used to determine estimates of the gradients of an objective function with respect to multiple parameters of the boson sampler.

[0103] Accordingto a third example method, the partial derivative may be determined using simultaneous perturbation stochastic approximation (SPSA).

[0104] The skilled person will appreciate that the order of steps 515 and 520 may be reversed, or that steps 515 and 520 may take place in parallel.

[0105] At 525, the method 500 comprises updatingthe set of parametervalues {r} and set of probability values {q} based on the determined estimated gradients. In particular, the processor unit 104, may establish an updated set of parameter values {r'} and updated set of probability values {q'} using:where q and / are coefficients known as learning rates. The processor unit 104 may instruct the controller 122 to configure the boson sampler 114 according to the updated set of parameter values.

[0106] At 530, the method 500 comprises determining whether a stopping condition has been satisfied. Any suitable stopping condition may be utilised. For example, determining that a stopping conditionhasbeen satisfied may comprise determining that the set of parameter values or the set of probability values have been updated a threshold number of times, for example that a threshold number of epochs or iterations has been reached. The thresholdnumberof iterations maybe selected in advance by auserof system 100. As another example, a stopping condition may comprise a convergence criterion. Determining that a convergence criterion has been met may comprise determining that the objective function has notchanged more thana threshold amountbetweenupdates of the parametervalues or probability values. If the stopping condition has not been satisfied, the method returns to step 510. If the stopping condition has been satisfied then the method proceeds to step 535.

[0107] At 535, the method 500 comprises operating the boson sampler 114 with the trained set of parameter values to produce a new batch of samples.

[0108] At 540, the method 500 comprises determining, for each sample s of the batch of samples S, a corresponding pre liminary binary sequence bprelimIn this example, each sample comprises a string of integers representing the number of photons detected in eachoutput mode, and each sample canbe mapped to a binaiy sequence accordingto the threshold mapping (described furtherabove in relation to Fig. 6). For example, if an integer of the string indicates that no photons were detected in the corresponding output mode of the boson sampler, then the processor unit 104 may set the value of the correspondingbitof the preliminaiy sequence as zero, while if the integerofthe stringindicatesthatone ormore photons were detectedinthe corresponding output mode of the boson sampler, then the processor unit 104 may set the value of the corresponding bit of the preliminaiy sequence to one. Multiple integer sequences may map to the same preliminary binary sequence.

[0109] At 545, the method 500 comprises determininga plurality of candidate binary sequences b^cand^ from the plurality of preliminary binary sequences. In particular, determining a candidate binary sequence comprises for each bit bj of the preliminary binaiy sequence, probabilistically flipping the bit accordingto the trained probability value q7associated with that bit.

[0110] At 550, the method 500 comprises determiningfrom amongst the candidate sequences a solution to the binary optimization problem. In this example, the processor unit 104 may evaluate the QUBO function (EQ. 3) for each distinct candidate binary sequence to determine which candidate binaiy sequence provides the lowest value. The candidate binary sequence providing the lowest value may thenbe determined to be the solution to the QUBO problem.

[0111] Variations of the described embodiments are envisaged.

[0112] As will be appreciatedby one skilled in the art, the present disclosure may be embodiedas a system, method, or computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro - code, etc.) oranembodime nt combining software and hardware aspects that may all generally be referred to herein as a “circuit,” “module” or “system.” Furthermore, aspects of the present disclosure may take the form of a computer program product embodied in any one or more computer-readable medium / media having computer usable program code embodied thereon.

[0113] The flowchart andblock diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present disclosure. In this regard, eachblock in the flowchart orblockdiagrams may represent a module, segment, or portionof code, which comprises one or more executable instructions for implementing the specified logical functions). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the ordernoted in the figures. For example, two blocks shown in succession may, infact, be executedsubstantially concurrently, ortheblocks may sometimesbe executedin the reverse order, depending upon the functionality involved. It will also be noted that eachblock of the block diagrams and / or flowchart illustration, and combinations of blocks in the block diagrams and / or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions.

[0114] Eachfeature disclosed in this specification (including any accompanyingclaims, abstract or drawings), may be replaced by alternative features servingthe same, equivalent or similar purpose, unless expressly stated otherwise. Thus, unless expressly stated otherwise, eachfeature disclosed is one example only of a generic series of equivalent or similar features. The invention is not restricted to the details of any foregoing embodiments. The invention extends to any novel one, or any novel combination, of the features disclosed in this specification (including any accompanying claims, abstract and drawings), or to any novel one, or any novel combination, of the steps of any method orprocess so disclosed. The claims should notbe constmed to cover merely the foregoing embodiments, but also any embodiments which fall within the scope of the claims.2b

Claims

CLAIMS1. A system for identifying a binary sequence as a solution to a binary optimization problem, the binary sequence comprising a number of bits, the system comprising: a boson sampler comprising: a light source; a reconfigurable interferometer; and one or more photodetectors; and one or more processors configured to: for each bit of the binary sequence, select an initial probability value, the initial probability value associated with a probability of a bit flip occurring; select initial parameter values for one or more configurable elements of the reconfigurable interferometer; iteratively train the parameter values and the probability values until a stopping condition is satisfied; operate the boson sampler, configured in accordance with the trained parameters, to produce a batch of samples; for each sample in the batch of samples, determine a candidate binary sequence using the trained probability values; and identify, from amongst the candidate binary sequences, the solution to the binary optimization problem.

2. A system according to claim 1, wherein the boson sampler is a single -photon boson sampler.

3. A system according to claim 1, wherein the boson sampler is a Gaussian boson sampler.

4. A system according to claim 1, wherein the light source is configured to produce at least one photon - subtracted squeezed state.

5. A systemaccordingto any precedingclaim,whereinaphotodetec tor of the one ormo re pho to detectors comprises a photon number resolving detector.

6. A systemaccordingto any precedingclaim,whereinaphotodetec tor of the one ormo re pho to detectors comprises a threshold detector.

7. A system according to any preceding claim, wherein a configurable element comprises a reconfigurable beamsplitter and wherein a parameter value is indicative of an effective reflection coefficient of the reconfigurable beamsplitter.

8. A system accordingto any preceding claim, wherein a configurable element comprises a phase shifter and wherein a parameter value represents a phase shift imparted by the phase shifter.

9. A system according to any preceding claim, wherein the reconfigurable interferometer comprises a reconfigurable temporal interferometer.

10. A system according to any preceding claim wherein the reconfigurable interferometer comprises a reconfigurable spatial interferometer.

11. A sy stem according to any preceding claim, wherein at least part of the sy stem is implemented using a photonic integrated circuit.

12. The system according to any preceding claim, wherein determining a candidate binary sequence for a sample using the trained probability values comprises: determining a preliminary binary sequence from the sample; and for eachbit of the preliminary binary sequence, probabilistically flipping the bit according to the trained probability value associated with that bit.

13. A system according to any preceding claim, wherein iteratively trainingthe parameter values and probability values comprises iteratively: operating the boson sampler, configured in accordance with the parameter values, to produce a batch of samples; for at least one selected parameter value determining, fromthe batch of samples, a gradient of an objective function with respect to the parameter value; for at least one selected probability value determining, from the batch of samples, a gradient of the objective function with respect to the probability value; and updating the selected parameter values and selected probability valuesbasedon the determined gradients; wherein the objective function associates a cost to a candidate solution to the binary optimization problem.

14. A systemaccording to any preceding claim, wherein thebinary optimization problem comprises a quadratic unconstrained binary optimization (QUBO) problem.

15. A system accordingto claim 14 as dependent on claim 13 , wherein the objective function takes the form:wherein wi;is a matrix element defined by the QUBO problem; wherein btrepresents the ith bit of a binary sequence; andwherein {b^bj) represents an expectation value of the productof bits bLand bj and is dependent on the ith and jth probability values and observables determined from the batch of samples.

16. A system according to claim 15: whereinwherein is the probability of finding zero photons in mode i as determined from the batch of samples; whereinis the probability of finding zero photons inboththe ith mode and the jthmodeas determined from the batch of samples; and wherein qirepresents the ith probability value associated with a probability of a bit flip of the ith bit of the binary sequence.

17. A method for identifying a binary sequence as a solution to a binary optimization problem with a heterogeneous computing system comprising a boson sampler, the boson sampler comprising a light source, a reconfigurable beamsplitter, and one or more photodetectors, the binary sequence comprising a number of bits, the method comprising: for each bit of the binaiy sequence, selecting an initial probability value, the initial probability value associated with a probability of a bit flip occurring; selecting initial parametervalues for one or more configurable elements of the reconfigurable beamsplitter; iteratively training the parameter values and the probability values until a stopping condition is satisfied, operating the boson sampler, configured in accordance with the trained parameter values, to produce a batch of samples; for each sample in the batch of samples, determining a candidate binary sequence using the trained probability values; and identifying, from amongst the candidate binary sequences, the solution to the binaiy optimization problem.

18. A method accordingto claim 17, wherein determininga candidatebinary sequence for a sample using the trained probability values comprises: determining a preliminary binary sequence from the sample; and for eachbit of the preliminary binary sequence, probabilistically flipping th at bit accordingto the trained probability value associated with that bit.

19. A method according to claim 17 or claim 18, wherein iteratively training the parameter values and probability values comprises: operating the boson sampler, configured in accordance with the parameter values, to produce a batch of samples;for at least one parameter value determining, from the batch of samples, a gradient of an objective function with respect to a parameter value; for at least one probability value determining, from the batch of samples, a gradient of the objective function with respect to the probability value; and updating the parameter values and probability values based on the determined gradients; wherein the objective function associates a cost to a candidate solution to the binary optimization problem.

20. A method according to any of claims 17 to 19, wherein the binary optimization problem comprises a quadratic unconstrained binary optimization (QUBO) problem.

Images