Binary optimization system and method with trainable probability values

Abstract

A hybrid quantum-classical system for solving a binary optimisation problems with trainable probability values and parameters. The hybrid quantum-classical system comprises a boson sampler and a class

Description

Technical 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 boson 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 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 each output mode of the output state; if threshold detectors are used, then each measurement outcome may be described by an array or string or sequence of integers, for example a binary sequence, indicating whether photons 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 output probability distributions generated by a boson sampler may have a complex structure and simulating this sampling task is understood to be intractable classically. Modem 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] According to 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 a boson 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 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 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 condition is satisfied. The one or more processors is / are further configured to operate the boson sampler, configured in accordance with the trained parameters, to produce a batch of samples. The one or more processors is / are further configured to, 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 further configured to identify, from amongst the candidate binary sequences, the solution to the binary optimization problem.

[0006] According to 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 comprising a number of bits. The method comprises, for each bit 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 binary sequence using the trained probability values. The method further comprises identifying, from amongst the candidate binary sequences, the solution to the binary optimization problem.

[0007] Advantageously, the systems and methods described herein utilise trained probability values to determine whether a bit flip should occur. Accordingly, all binary sequences in the solution space for the binary optimization problem are reachable or accessible to the heterogeneous system, and the system is in general less biased towards one solution over another until after training. This reduces the time and memory 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, often operable 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 solution to 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 from boson sampler outputs according to an example.

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

[0017] Embodiments of the disclosure are described with reference to the accompanying drawings. However, it should be 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 or similar reference denotations may be 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” a feature (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 “including but 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 contemplating plurality 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 / or B” 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 second component may 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 understood that when an element (e.g. a first element) is referred to as being “directly coupled with / 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 further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that 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 complementary binary values. For example, a bit may be represented by a “0” or “1”, a bit may be represented by a “+” or a bit may be represented by a “1” or “-1”. A “bit flip” is understood to mean that 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 cannot be reliably solved on a 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. A boson sampler is a quantum system that relies on the interference of photons 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 each output mode of the output state; if threshold detectors are used, then each measurement outcome may be described by an array or string or sequence of integers, for example a binary 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 represent a 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 mapping based on whether the detected number of photons in an output mode is even or odd. However, whatever the 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 described above is to try to solve the binary optimization problem 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 than one photon in an output mode maps to a binary one. One may then find a second possible 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 than one photon in an output mode maps to a binary zero. Such an approach would ensure that the solution to the binary optimization problem is reachable by the system and may 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 is less 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 output photonic 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 a block diagram of a heterogeneous system 100 in which illustrative embodiments may be implemented. The heterogeneous system 100 comprises both classical processing apparatus and quantum processing apparatus. Other architectures to that shown in Fig. 1 may be used as will be appreciated by the skilled person. For example, system 100 may be distributed across multiple interconnected devices.

[0030] System 100 is an example of a specialised computing apparatus, in which computer usable 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 comprising a state generation unit 116, an interferometer 118, a state detection unit 120 and a dedicated controller unit 122.

[0031] The system 100 may be implemented in 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 communication module 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 implemented in a third rack-mounted 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 optical fibre-connected to the state generation unit 116, and the photodetectors of the state detection unit 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. Processor unit 104 may be a set of one or more processors or may be a multi-processor core, depending on the particular implementation. Furthermore, processor unit 104 may be implemented using one or more heterogeneous processor systems in which a main processor is present with secondary 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 combination thereof. 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 combination thereof. The media used for persistent storage may also be removable. For example, 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 connection for user input through a keyboard, a mouse, and / or other 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. For example, the communications module 110 may be configured 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 for the applications and / or programs may be located in 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 or tangible computer-readable media.

[0037] The program code may contain instructions which, when processed by the processor unit 104, cause the 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 transferred to system 100 from computer-readable storage medium 124 through a 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 memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), 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 can contain or 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 over a 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 detection unit 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 \VIN) 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 ^IN) 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-photon boson sampler), the input state can without loss of generality be expressed as \VIN)sp = ,lN,0N+ll- ,0M) = 4- a^O.,- ,0M) (EQ. 1) where is the bosonic creation operator in the kth mode. The skilled person will appreciate that the methods and systems described herein are also applicable when one or more input modes comprise more than one photon.

[0044] In other examples, the state generation unit 116 may be configured to generate an input multimodal photonic state ^IN) comprising a Gaussian photonic input in each of the N input modes (in which case the boson sampler may be referred to as a Gaussian boson 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 generation unit may be configured to generate an input multimodal photonic state ^IN) comprising a non-Gaussian photonic input in at least one of the N input modes (in which case the boson sampler may be referred to as a non-Gaussian boson sampler). For example, a photon may be subtracted from a single mode squeezed state and detected, thereby heralding the generation of a non-Gaussian state for input into an input mode.

[0046] In some examples, the state generation unit 116 may be configured to generate an input multimodal photonic state ^IN) that comprises 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 receive a pump beam from a pump laser and to probabilistically generate pairs of entangled photons, and may further comprise a photodetector configured to detect a photon of the entangled pair, thereby heralding the presence of the other photon of the pair. For example, in a Gaussian boson sampler, the state generation module 116 may comprise PPLN waveguides configured 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 nonGaussian 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 transforming the 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 detection unit 120. One or more optical elements of the interferometer 118 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 ra of the set of parameter values {r} may characterise the phase shift imparted by 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 rb of 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 coupling locations arranged in the integrated circuit. At each coupling location, tuneable elements may be arranged (e.g. EOM phase shifters) that are configured to control the coupling interaction between the waveguides. The integrated circuit may be implemented in silicon nitride (Si3N4) or any other suitable material.

[0050] Due to interference between photons 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 as ^OUT^r})) = ^ac\n^,n^,n^) 2) c where C is a configuration, rij is the number of bosons in the j th output mode in configuration C, and ac is 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 - photon loss may, for example, mean that the output state can be 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 can yield a measurement outcome representable as a string of integers corresponding to a configuration C. By operating the boson sampler 114 a number of times to produce a batch of samples S, it is possible to establish an empirical probability distribution of the bosonic configurations of the output state. One can expect that with many samples, the probability pc of obtaining a measurement outcome corresponding to configuration C is approximately given by pc = lacl2.

[0051] The state detection unit 120 comprises an arrangement of one or more photodetectors configured to detect photons output from the interferometer 118 and produce corresponding detection event signals. In some examples, the photodetectors may comprise photon number resolving (PNR) detectors, capable of determining how many photons are received. For example, the detectors may comprise superconducting nano wire 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 / off 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 coupled to the processor unit 104, the state generation unit 116, the interferometer 118 and the state detection unit 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 processing unit. For example, the controller 122 may comprise an application-specific integrated circuit (ASIC) or an application-specific standard product (ASSP) or another domain-specific architecture (DSA). Alternatively, the controller 122 may be implemented in adaptive 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 processor unit 104. More particularly, the controller is configured to, if so directed by the processor unit 104, configure the interferometer 118 according to a set of parameter values {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 beam splitter or the phase imparted by a phase shifter. The controller 122 is further configured to, if so directed by 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 configured to 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 detection unit 120. In other words, the controller is configured to sample from the output distribution of the configured boson 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 configured to receive parameter values from the processor unit 104; to produce a batch (in other words, a plurality) of samples by configuring the interferometer 118 according to the received parameter values, generating an input photonic state and measuring an output photonic state; and to communicate the batch of samples to the processor unit 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 each output 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 person would appreciate that other architectures may be 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, a non-linear crystal is pumped with a laser and, probabilistically, entangled photons are emitted (the “signal” and the “idler”). A photodetector (not shown in Fig. 2) is arranged to detect the presence of the idler photon which, due to the entanglement, heralds the presence of a photon in the signal mode. Other photon 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 \lPIN)Sp in order to 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 118a comprises M input ports, M output ports, and a plurality of waveguides 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 between pairs 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 700nm 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 and plurality of coupling locations arranged in the integrated circuit. The integrated circuit may be implemented in silicon nitride (Si3N4) or any other suitable material, for example thin-film lithium niobate.

[0062] A reconfigurable beam splitter 230 is arranged at each of the coupling locations such that at 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 reflection coefficient (transmission coefficient). The configurable parameters (denoted with a theta in the figure) in this example relate to the reflection (transmission) coefficients of the reconfigurable beamsplitters. The skilled person will appreciate that the interferometer 118a may 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 coupling two 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 -for example a parametrised beam splitter may comprise a Mach-Zehnder type interferometer containing a 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 an external phase shifter on one external path of the Mach-Zehnder interferometer to control the relative phases of the two modes 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 recoupling the 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 118a couples the single photons received at the 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 a plurality 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 120a comprises one PNR detector for each of the M output modes and accordingly the measurement outcomes are representative of the number of photons measured in all output modes of the output multimodal photonic state. The PNR detectors may comprise nanowire photodetectors.

[0067] The controller 122a is coupled to each of the state generation unit 116a, the interferometer 118a and the state detection unit 120a. The controller 122a is further communicatively coupled to the processor unit 104. The controller 122a may receive a set of parameter values {r} from the processor unit 104, and may generate control signals to configure the tuneable elements 230 of the interferometer 118a in accordance with those parameter values. For example, the controller 122a may assign the parameter value rx to parameter 01; the parameter value r2 to parameter 02 and so on. For example, each reconfigurable beam splitter 230 may comprise a Mach-Zehnder interferometer comprising two 50 / 50 beam splitters and a phase shifter located in each of one or both of its internal optical paths. The phase shifter may be implemented using an electro-optical modulator. The control signals may comprise an electric field for controlling the phase shift imparted by the internal phase shifters and therefor the coupling strength of the reconfigurable beam splitter. The controller 122a may further generate a control signal to cause the single photon sources 210 to begin generating single photons, for example the control signal may cause a pump laser to pump light into the non-linear material of the 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 singlephoton boson 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, which means that the state is defined by the number of 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 114a of Fig. 2, the state generation unit 116a may comprise further single photon sources and a multiplexer in order to reliably ensure that a single photon is generated in each time period t.

[0070] The interferometer 116b comprises a temporal mode coupling 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, optical fibre. The delay line 330 has a length ct where c is the speed of light in the fibre. In this way, the field of the photon in one 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 can be used to implement the equivalent operations of the three beam splitters defined by parameters 02 and 03 shown in Fig- 2 For example, the controller 122b may, as a first photon is emitted from the photon source 310, configure the reconfigurable beamsplitter 320 to loop the first photon into 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 r± to 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 r2 to 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 r3 to cause interference between the third temporal mode and fourth temporal mode. This may continue until a predetermined transformation has been performed on the input photon sequence of M time bins.

[0072] The state detection unit 120b comprises a photon number 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 skilled person would appreciate that the architecture of the temporal mode boson sampler 114b of Fig-3may be varied in several ways. For example, the boson sampler 114b may comprise further reconfigurable beamsplitters 320 and further delay lines 330 in order to generate more complicated interference between temporal modes. The skilled person would further appreciate that delay lines of different lengths may be used to 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 skilled person would appreciate that the spatial mode boson sampler of 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 generation unit 116a / l 16b. Furthermore, the PNR detector(s) of the state detection modules 120a / 120b may be replaced with threshold detectors, in which case the measurement outcomes output from the state detection unit are indicative of the presence or absence 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[soq as a solution to a binary optimization problem, the binary sequence comprising a number L of bits. The skilled person will appreciate that while the method 400 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} 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 beamsplitter and 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 sought bit bj of the sought binary sequence b[soq, a corresponding probability value qj which will be associated with a probability of the corresponding bit bj being flipped. As the binary sequence 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 instructions loaded from the memory unit 106 or elsewhere, select initial parameter values and probability values.

[0077] At 420, the method comprises iteratively training the parameter values {r} and probability values {q} until a stopping condition is satisfied. An example of this stage will be described further below 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 been updated a threshold number of times, for example that a threshold number of epochs or iterations has been reached. The threshold number of iterations may be selected in advance by a user of 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 a batch of samples S. With reference to Fig. 1, processor unit 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 according to the received parameter values, generate control signals to cause the state generation unit 116 to produce an input multimodal photonic state, and receive detection 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 using the 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;im]. For each preliminary binary sequence, the processor unit may then probabilistically flip each bit bt of that preliminary binary sequence with a probability associated with the corresponding trained probability value qj, to produce a candidate binary sequence b[candy In other words, the processor 104 may use a random process weighted in 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, from among the candidate binary sequences, a solution b[soq to the binary optimization problem. With reference to Fig. 1, 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 a binary optimization problem that the system 100 may solve using the method of Fig. 4 is a quadratic unconstrained binary optimization (QUBO) problem. The task is to find a binary sequence b[soq comprising a number L of bits that minimizes a target function: (EQ. 3) where W is an L X L symmetric matrix and b is a binary sequence of length L determined from samples from a 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 photons nk measured in each corresponding output mode k of the 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 that the length L of the sought binary solution ^[soi] is equal to the number of output modes M of the boson sampler (L = M), and that each measured photon number nk is mappable to a corresponding bit bkof a binary sequence. However, the skilled person will appreciate that, 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 describing the 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 U({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 number of photons ng measured in a particular output mode g is greater than zero then a corresponding bit of the preliminary binary sequence bg is 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 binary sequence (1,0,1,0) under this threshold mapping. 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 instead of PNR detectors to determine the preliminary binary sequence. By training the parameter values of the boson sampler, the output distribution of the boson sampler may be adapted until the solution binary 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 problem being addressed. For example, when a single-photon boson sampler is used, due to energy conversation principles one would expect that the number of photons input to the interferometer of the boson sampler should be the same as the number of photons output from the interferometer and measured -for example, four photons input into the interferometer in four input modes (e.g. (1,1,1,1)) 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 turn means that the preliminary binary sequence (0,0,0,0) can not be represented. That is, the entire solution space of two to the power of L binary sequences may not be 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 binary 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 optimization problem. One may then restart the process to find a second possible solution with a complementary threshold mapping in which if the number of photons ng measured in a particular output mode g is greater than zero then a corresponding bit of the preliminary binary sequence bg is 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 each bit bj of the sought solution to the binary optimization problem, a corresponding probability value qj associated with a probability of flipping that bit. The binary optimization problem may then be represented by an objective function that is based on the output of the boson sampler and the set of probability values, and then the set of parameter values {r} and the set of probability values {q} 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 binary sequences is reduced. Once a stopping condition is 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 optimization problem. Furthermore, the system is typically able to find the solution faster than a heterogeneous system that has to optimize using multiple mappings.

[0089] Referring again to Fig. 5, at 505, the method comprises selecting a 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 beamsplitter and 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. The parameter values may be selected in any suitable manner, for example the processor 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 sought bit bj of the sought binary sequence b[soq, a corresponding probability value qj which will be associated with a probability of the corresponding bit bj being flipped. As the binary sequence b[soq 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 instructions loaded from the memory unit 106 or elsewhere, select initial parameter values and probability values.

[0090] Steps 510, 515, 520, 525 and 530 may together provide 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: fQUBO ({A Q}) = Wtjibtbj) 4) i,j where wtj represents a matrix element of the symmetric matrix W of (EQ. 3), bt and bj represent the ith and jth bits of a binary sequence respectively and {bjbj) represents an expectation value of the product of the bits b[bj derived from a batch S of samples s and is dependent on the ith and jth probability values q^ and qj and observables determined from the batch of samples.

[0091] Under the threshold mapping, the quantity {bjbj) can be expressed as: (btbj) = - 2( / ,)(1 - 2qy) + m^l - 2q7) (EQ. 5) + mjqj{l - 2qt) + q^j where is the probability, determined from a batch S of samples, of zero photons being detected in the ith mode and zero photons being detected in the jth mode (probability that n, = 0 and rtj = 0); m, is the probability, determined from a batch S of samples, of zero photons being detected in the ith mode (probability that n, = 0); and mj 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 operating the boson sampler 114 to produce a batch of samples S. The processor unit 104 may instruct the controller 122 to operate the boson sampler 114 in accordance with the set of parameter values {r} to produce a batch of samples S representative of the output distribution of the 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 of the objective function with respect to 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 rk of 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] According to a first example method for using the boson sampler 114 to determine the gradient of the objective function with respect to the selected parameter, the processor 104 determines from the batch of samples S the objective function value fQUBo({r> Qi) (EQ. 4). The processor unit 104 may then instruct the controller 122 to configure the interferometer 118 such that the selected parameter rk is 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 fQUBO^{r>cl}- rk + ■ 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: dfQUBo(.{r,q^ _ fQUBo({r,q}: rk + e) - fQUB0({r, q}) (EQ. 6) drk e

[0096] According to a second example method for using the boson sampler 114 to determine the gradient of the objective function with respect to the selected parameter, determining the gradient comprises determining from the batch of samples S, for each observable Oj of a set of observables {0}, a corresponding first value 14(0^) representative of a partial derivative of the objective function with respect to the expectation value of the observable {Oj). In other words, the first value {Oj) for the observable Oj is approximately: TO) dfQUBo({r’ Q}) TO (EQ. 7)

[0097] According to the second example method, determining the gradient further comprises determining, for each observable Oj of the set of observables {0}, a corresponding second value F2(O7) representative of a partial derivative of the expectation value of the observable (Oj) with respect to the parameter value rk. In other words, the second value F2(O7) for the observable Oj is approximately: d(Oj) (EQ. 8)

[0098] According to the second example, determining the gradient further comprises determining the gradient from the first values and the second values V2. For example, the estimate of the partial derivative of the objective function with respect to the parameter value rk may be determined as a weighted or unweighted sum over the product of the first values and second values, for example: dfQUBO^r.q}) y y df d(Oj) (EQ. 9) i j 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 mean a 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, depend on the objective function used, for example, how well the objective function can be decomposed into or represented by the set of observables {0}. The choice of the set of observables may depend on the configurable parameter to be optimised, for example on how quickly the second values V2 may be calculated. In this particular example, a sensible observable 0g may be the measured number of photons ng in a corresponding output mode g of the interferometer 118.

[0100] The first values 14 may be determined in any of a number of ways. For example, determining the values 14 may 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 second values F2 may also be determined in any of a number of ways. In particular, the choice of observables may be made to make determination of the second values 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 (Oj) and the configurable parameter.

[0102] The second example method for determining the gradient described above may be advantageous over the 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] According to 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 updating the set of parameter values {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: , 9fQUB0({r,q}) (EQ. 10) = ---- , 9fQUB0Q{r,q}) (EQ. 11) Qk ~ Qk — H 3¾ where q and / z 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 condition has been 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 threshold number of iterations may be selected in advance by a user of 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 not changed more than a threshold amount between updates of the parameter values 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 preliminary binary sequence b^prelim^. In this example, each sample comprises a string of integers representing the number of photons detected in each output mode, and each sample can be mapped to a binary sequence according to the threshold mapping (described further above 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 corresponding bit of the preliminary sequence as zero, while if the integer of the string indicates that one or more photons were detected in the corresponding output mode of the boson sampler, then the processor unit 104 may set the value of the corresponding bit of the preliminary sequence to one. Multiple integer sequences may map to the same preliminary binary sequence.

[0109] At 545, the method 500 comprises determining a plurality of candidate binary sequences from the plurality of preliminary binary sequences. In particular, determining a candidate binary sequence comprises for each bit bj of the preliminary binary sequence, probabilistically flipping the bit according to the trained probability value qj associated with that bit.

[0110] At 550, the method 500 comprises determining from 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 binary sequence provides the lowest value. The candidate binary sequence providing the lowest value may then be determined to be the solution to the QUBO problem.

[0111] Variations of the described embodiments are envisaged.

[0112] As will be appreciated by one skilled in the art, the present disclosure may be embodied as 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, microcode, etc.) or an embodiment 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 and block 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, each block in the flowchart or block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block 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] Each feature disclosed in this specification (including any accompanying claims, abstract or drawings), may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise. Thus, unless expressly stated otherwise, each feature 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 or process so disclosed. The claims should not be construed to cover merely the foregoing embodiments, but also any embodiments which fall within the scope of the claims.

Claims

1. 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; andone or more photodetectors; andone 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; andidentify, 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 system according to any preceding claim, wherein a photodetector of the one or more photodetectors comprises a photon number resolving detector.

6. A system according to any preceding claim, wherein a photodetector of the one or more photodetectors 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 according to 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 system according to any preceding claim, wherein at least part of the system 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; andfor each bit 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 training the 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, from the 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; andupdating the selected parameter values and selected probability values based on the determined gradients;wherein the objective function associates a cost to a candidate solution to the binary optimization problem.

14. A system according to any preceding claim, wherein the binary optimization problem comprises a quadratic unconstrained binary optimization (QUBO) problem.

15. A system according to claim 14 as dependent on claim 13, wherein the objective function takes the form:Wi^bibj)wherein is a matrix element defined by the QUBO problem;wherein bt represents the ith bit of a binary sequence; andwherein {bjbj) represents an expectation value of the product of bits bt and bj and is dependent on the ith and J th probability values and observables determined from the batch of samples.

16. A system according to claim 15:wherein(bibj) = nty(l - 2( / ,)(1 - ^qj) + ^(^(1 - 2q7) + mjqj(l - 2qi') + q^jwherein m, is the probability of finding zero photons in mode i as determined from the batch of samples;wherein mi; is the probability of finding zero photons in both the ith mode and the jth mode as determined from the batch of samples; andwherein q^ represents 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 binary sequence, selecting an initial probability value, the initial probability value associated with a probability of a bit flip occurring;selecting initial parameter values 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; andidentifying, from amongst the candidate binary sequences, the solution to the binary optimization problem.

18. A method according to claim 17, wherein determining a candidate binary sequence for a sample using the trained probability values comprises:determining a preliminary binary sequence from the sample; andfor each bit of the preliminary binary sequence, probabilistically flipping that bit according to 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; andupdating 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.

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