Generalised parameter shift with boson sampling

Abstract

Methods are provided for configuring a boson sampler, the boson sampler comprising a light source, a reconfigurable interferometer having configurable elements, and one or more photodetectors. The met

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] Applications of Machine Learning (ML) are now ubiquitous in practically every context in science and are gaining prominence in industry. As such, deep neural network models have been a driving force, in part owing to the fact that they may act as universal function approximators. Such architectures can learn certain features from training data, so that the model will be able to predict the same features when the model is prompted with new unseen data. In training, gradient-based optimisation methods (e.g. gradient descent) are used to reduce (e.g. minimise) some objective function. Quantum Machine Learning (QML) has been proposed as a promising application for quantum computers. Similar to their classical counterparts, the vast majority of QML algorithms employ a variational approach, whereby a wide variety of models, known as Variational Quantum Algorithms (VQAs), are tuned through trainable weights in order to reproduce a desired behaviour or outcome. In order to implement QML algorithms, the computation of gradients enables a user to train these parameters so that they minimize some distance measure between the predicted output and the true output. In gate-based systems, this has led to the development of subroutines to determine gradients.

[0003] However, accurate subroutines for determining gradients in other quantum systems are not well developed. In particular, there has been no known way of efficiently and accurately determining a gradient within the linear optical quantum computing (LOQC) framework. This is particularly relevant for linear optical NISQ systems such as boson samplers. A boson sampler is a linear optical 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. By repeatedly sampling measurement outcomes, one can build up a picture of the probability distribution governing the quantum superposition. 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] A boson sampler does not process information defined in terms of qubits, as is typically required for VQAs, and no quantum logic operations are performed (e.g. CNOT gates). Instead, a boson sampler is used to sample from a non-classical probability distribution and the only information accessible to a user is that which can be gleaned from the samples collected.

[0005] 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

[0006] According to an aspect of the present disclosure, a method is provided for configuring a boson sampler. The boson sampler comprises a light source, a reconfigurable interferometer having one or more configurable elements, and one or more photodetectors. The method comprises, determining a partial derivative f of an expected value of an observable f = {0) with respect to a parameter 9k of a configurable element of the boson sampler, the parameter assigned a parameter value qk. Determining the partial derivative comprises producing a number 2R of batches of samples by, for all integers / z between 1 and R (wherein R is an integer): (i) operating the boson sampler configured with the parameter assigned a positively shifted parameter value qk + to produce a first batch of samples, the positively shifted value being the parameter value qk adjusted positively by a shift increment that depends on the integer / z; (ii) determining a first expected value of the observable f(dk+^) from the first batch of samples; (iii) operating the boson sampler configured with the parameter assigned a negatively shifted parameter value qk — to produce a second batch of samples, the negatively shifted value being the parameter value qk adjusted negatively by a shift increment that depends on the integer / z; and (iv) determining a second expected value of the observable f(dk^ from the second batch of samples. Determining the partial derivative further comprises determining the partial derivative from the first expected values of the observable and the second expected values of the observable. The method further comprises configuring the configurable element of the boson sampler based on the determined partial derivative.

[0007] Advantageously, the methods described herein enable parameters of a boson sampler to be updated efficiently based on a more accurate determination of a gradient. This means that some computational tasks, particularly those involving the determination of gradients such as gradient descent, may be performed more accurately than other known methods, while scaling well with the number of parameters.

[0008] In some examples, the configurable element may comprise a reconfigurable beamsplitter. The parameter value may correspond to an effective transmission coefficient of the reconfigurable beamsplitter.

[0009] In some examples, the configurable element may comprise a phase shifter. The parameter value may correspond to a phase imparted by the phase shifter.

[0010] For small values of R, the number of expected values (e.g. f(0^) and f(9k^y) that are evaluated in order to determine the partial derivative f of an observable f = {0) with respect to a parameter 9k are reduced while the methods may still enable a reasonably accurate update to the parameter. However, as described herein, there are a number of bounds B on the index R such that if the index R is equal to (or greater than) the bound (i.e. R >B). then the partial derivative may be determined exactly (assuming that the batches of samples obtained from the boson sampler are representative of the output distribution of the boson sampler), and accordingly the parameters may be updated accurately. The bounds may depend on one or more of: (i) a property of an input state provided to the interferometer of the boson sampler when using the boson sampler to generate a sample, for example the number of photons in the input state, (ii) a property of the interferometer, for example the connectivity of the interferometer, or (iii) a property of the observable.

[0011] In some examples, the index R may be equal to the lowest bound B that applies to the parameter.

[0012] In some examples, the observable may comprise a monomial based on a number of photons in one or more output modes of the boson sampler, in which case the degree of the monomial is a bound on the index R. The index R may be greater than or equal to the degree of the monomial. Advantageously, when the index R is greater than or equal to the degree of the monomial, the partial derivative of the expectation value of the observable with respect to the parameter may be determined exactly. This in turn means that the parameter may be accurately configured. In some examples, the index R may be equal to the degree of the monomial, in which case it is possible to determine an exact value for the partial derivative with the fewest number of expectation values (e.g.

[0013] In some examples, the observable may comprise a polynomial based on a number of photons in one or more output modes of the boson sampler. The polynomial may be expressible as a sum of monomials, each monomial having a corresponding degree, in which case the greatest degree is a bound on the index R. The index R may be greater than or equal to the greatest degree of the monomials making up the polynomial.

[0014] In some examples, the index R may be based on a number of photons that may impinge upon the configurable element when the boson sampler is operated to produce a sample, which is a bound on the index R. Advantageously, when the index R is greater than or equal to a number of photons that may be directly influenced by the choice of parameter value, the partial derivative of the expectation value of the observable with respect to the parameter may be determined exactly. This in turn means that the parameter may be accurately configured.

[0015] In some examples, the index R may be greater than or equal to the number of photons in an input multimodal photonic state provided to the reconfigurable interferometer of the boson sampler. The total number of photons in the input multimodal photonic state provided to the reconfigurable interferometer of the boson sampler is also a bound on the index R. Advantageously, when the index R is greater than or equal to the number of photons in the input multimodal photonic state, the partial derivative of the expectation value of the observable with respect to the parameter may be determined exactly. This in turn means that the parameter may be accurately configured. In some examples, the index R may be equal to the number of photons in an input multimodal photonic state provided to the reconfigurable interferometer of the boson sampler.

[0016] In some examples, each shift increment may be given by = Advantageously, using such a shift increment provides a balanced spacing between parameter values used in determining the partial derivative. This is useful for being able to distinguish between output distributions of the boson sampler.

[0017] In some examples, determining the partial derivative from the first expected values of the observable and the second expected values of the observable may comprise determining: R M=i wherein represents the expectation value of the observable determined from the first batch of samples, and wherein represents the expectation value of the observable determined from the second batch of samples, and wherein represents a partial derivative of a Dirichlet kernel and is dependent on the shift increment .

[0018] In particular, for a parameter that may be assigned a parameter value between —n and n, (for example a phase shift imparted by a phase shifter) determining the partial derivative from the first expected values of the observable and the second expected values of the observable may comprise determining: Z ) - K0! / )) 2 sin(s / 2)’ g=i In particular, for a parameter that may be assigned a parameter value between 0 and n, (for example an effective reflection coefficient of a reconfigurable beamsplitter) determining the partial derivative from the first expected values of the observable and the second expected values of the observable may comprise determining: g=l

[0019] According to an aspect of the present disclosure, a non-transitory computer-readable medium is provided. The non-transitory computer-readable medium comprises stored instructions that, when executed by a computing device, cause the computing device to perform operations including determining a partial derivative of an expected value of an observable with respect to a parameter of a configurable element of a boson sampler, the parameter assigned a parameter value qk, and configuring the configurable element of the boson sampler based on the determined partial derivative. Determining the partial derivative comprises producing a number 2R of batches of samples by, for all integers / z between 1 and R (wherein R is an integer): (i) operating the boson sampler configured with the parameter assigned a positively shifted parameter value qk + to produce a first batch of samples, the positively shifted value being the parameter value qk adjusted positively by a shift increment that depends on the integer / z; (ii) determining a first expected value of the observable from the first batch of samples; (iii) operating the boson sampler configured with the parameter assigned a negatively shifted parameter value qk — sk 1° produce a second batch of samples, the negatively shifted value being the parameter value qk adjusted negatively by a shift increment that depends on the integer / z; and (iv) determining a second expected value of the observable f(0k^ from the second batch of samples. Determining the partial derivative further comprises determining the partial derivative from the first expected values of the observable and the second expected values of the observable.

[0020] According to an aspect of the invention, a system is provided. The system comprises a boson sampler and one or more processors. The boson sampler comprises a light source, a reconfigurable interferometer having one or more configurable elements, and one or more photodetectors. The one or more processors are configured to determine a partial derivative of an expected value of an observable with respect to a parameter of a configurable element of the boson sampler, the parameter assigned a parameter value qk. The one or more processors are further configured to configure the configurable element of the boson sampler based on the determined partial derivative. Determining the partial derivative comprises producing a number 2R of batches of samples by, for all integers / z between 1 and R (wherein R is an integer): (i) operating the boson sampler configured with the parameter assigned a positively shifted parameter value qk + to produce a first batch of samples, the positively shifted value being the parameter value qk adjusted positively by a shift increment that depends on the integer / z; (ii) determining a first expected value of the observable from the first batch of samples; (iii) operating the boson sampler configured with the parameter assigned a negatively shifted parameter value qk — to produce a second batch of samples, the negatively shifted value being the parameter value qk adjusted negatively by a shift increment that depends on the integer / z; and (iv) determining a second expected value of the observable f(0k^ from the second batch of samples. Determining the partial derivative further comprises determining the partial derivative from the first expected values of the observable and the second expected values of the observable.

[0021] In some examples, a photodetector of the one or more photodetectors may comprise a photon number resolving detector.

[0022] In some examples, the reconfigurable interferometer may comprise a reconfigurable temporal interferometer.

[0023] In some examples, wherein the reconfigurable interferometer may comprise a reconfigurable spatial interferometer.

[0024] In some examples, the configurable element may comprise a reconfigurable beamsplitter. The parameter may correspond to an effective transmission coefficient of the reconfigurable beamsplitter.

[0025] In some examples, the configurable element may comprise a phase shifter. The parameter may correspond to a phase imparted by the phase shifter.

[0026] 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

[0027] Illustrative embodiments of the present disclosure will now be described by way of example only, with reference to the accompanying figures.

[0028] Fig. 1 shows a block diagram of a system according to an example.

[0029] Fig. 2 shows a diagram of a spatial boson sampler according to an example.

[0030] Fig. 3 shows a diagram of a temporal boson sampler according to an example.

[0031] Fig. 4 shows a flowchart of a method for configuring a boson sampler, such as may be performed by the system of Fig. 1.

[0032] Fig. 5 shows a flowchart of a method for determining a partial derivative of an expectation value of an observable with respect to a parameter of the boson sampler, such as may be performed by the system of Fig. 1.

[0033] Fig. 6 shows a sketch of a conceptual boson sampler for explanatory purposes.

[0034] Fig. 7 shows a sketch of a conceptual boson sampler for explanatory purposes.

[0035] Throughout the description and the drawings, like reference numerals refer to like parts. Detailed Description

[0036] 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.

[0037] 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.

[0038] 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.

[0039] 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.

[0040] 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.

[0041] 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.

[0042] 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 observable may depend on the particular task being processed, or the particular parameter being updated. An observable is denoted 0 throughout this description. Furthermore, an expectation value (0) of an observable 0 is denoted f throughout this specification (e.g. f = (0)). As an example, an observable may comprise a number of photons rtj measured in an output mode j of the boson sampler, and the expectation value (nj) of the observable may be derived from a batch of samples obtained from the boson sampler.

[0043] The term “positively shifted value” of a parameter 9k as used herein is understood to mean a value of a parameter that has been adjusted from a current value qk in a first direction by some shift increment s^. Similarly, the term “negatively shifted value” of a parameter 0k as used herein is understood to mean a value of a parameter that has been adjusted from a current value qk in a second direction by some shift increment s^.

[0044] Hybrid quantum-classical systems can provide performance advantages over classical computational systems when performing some types of computational tasks including solving some optimization problems and performing some machine learning based tasks. Many of the tasks with which such systems show an advantage over classical processing systems include the determination of gradients. A consideration in such computational tasks is how to update the parameters of the quantum subsystem, and this may include determining a gradient with respect to a parameter of the quantum subsystem. However, conventional approaches to deciding how to update the parameters are typically inaccurate.

[0045] Advantageously, the methods and systems described herein enable the quantum subsystem to be configured and operated with great precision by providing a resource efficient way to determine a partial derivative exactly.

[0046] 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.

[0047] 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 communication fabric 102, which provides communications between a processor unit 104, memory unit 106, input / output unit 108, communication 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.

[0048] 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 118 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.

[0049] 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.

[0050] 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.

[0051] 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.

[0052] 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.

[0053] 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.

[0054] 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.

[0055] 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.

[0056] 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.

[0057] 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.

[0058] 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

[0059] The state generation unit 116 is configured to generate an input multimodal photonic state ^IN) 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.

[0060] 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 = 11^12,-,0M) = 4- aM- ,0M) (EQ. 1) where aj 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 and are also applicable when the number of photons N is greater than the number of input modes M.

[0061] 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.

[0062] 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 {q} 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 qa of the set of parameter values {q} 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 qb of the set of parameter values {q} may characterise the transmission coefficient of the reconfigurable beam splitter

[0063] 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.

[0064] Due to interference between photons in different 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 WOUT^q})) = ^ac\n^n^n^) (EQ'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 {q}, 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.

[0065] 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.

[0066] 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 122 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.

[0067] 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 {q} 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.

[0068] 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.

[0069] The controller 122 is further configured to communicate the response from the state detection unit 120 to the processor unit 104.

[0070] 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.

[0071] 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.

[0072] 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.

[0073] 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.

[0074] The number of single photon sources 210 may be greater than the number M of input modes of the input multimodal photonic state I'P / IV) 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.

[0075] 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.

[0076] 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 0, to 06 in this example relate to the reflection (transmission) coefficients of the reconfigurable beam splitters. The skilled person will appreciate that the interferometer 118a may comprise further reconfigurable elements.

[0077] 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.

[0078] 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.

[0079] 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.

[0080] 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.

[0081] 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 units 104. The controller 122a may receive a set of parameter values {q} 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 Qi parameter 01; the parameter value q2 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 therefore 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.

[0082] 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.

[0083] 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.

[0084] The interferometer 118b 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 photon(s) in the next temporal mode on the parametrised beam splitter 320. The time-bin interferometer may comprise further optical components including further optical switches.

[0085] The controller 122b is configured to tune the parameter value (e.g. transmittance) of the parametrised beam splitter 320 for each time interval. In other words, with a temporal interferometer such as interferometer 114b, the configurable parameters relate to a property of the parameterised beam splitter at a moment in time. 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 9,, 92 and 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 Qi to cause interference between the first temporal mode and second temporal mode (e.g. the first and second photon) of the input state I'P / IV). The controller 122b may then, as a third photon is emitted from the photon source 310, configure the reconfigurable beamsplitter 320 using parameter value q2 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 q3 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.

[0086] 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.

[0087] 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.

[0088] The skilled person would appreciate that 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.

[0089] The hybrid quantum-classical system 100 of Fig. 1 is configured to perform the method 400 depicted in the flowchart of Fig- 4 For the purposes of this example, it is expected that the boson sampler has already been configured with a set of parameter values {q}, and that the set of values {q} is in memory and accessible to the processor unit. One parameter 9k of the set of parameter values is set to current value qk. The method of Fig. 4 describes a way to update that parameter value. 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.

[0090] At 405, the method begins. At 410, the method comprises determining a partial derivative f of an expected value f = {0) of an observable 0 with respect to the parameter 9k of a configurable element of the boson sampler. At 415, the method comprises configuring the configurable element of the boson sampler based on the determined partial derivative f. At 420, the method ends.

[0091] The partial derivative f of an expected value f = {0) of an observable 0 with respect to the parameter 9k can be determined with accuracy. In particular, the partial derivative may be determined from: A (EQ- 3) zw = ( / (C)- / (^^ M=1

[0092] In (EQ.3), / '(0fc) represents the partial derivative of an expected value f = (0) of an observable 0 with respect to the parameter 9k, represents the expectation value of the observable when the parameter 9k has been adjusted from its current value (9k = qk) to a “positively shifted value” (9k = qk + sM) and f(0k^ represents the expectation value of the observable when the parameter 9k has been adjusted from its current value (0fc = qk) to a “negatively shifted value” (0fc = qk — s^. In (EQ.3) represents a derivative of a Dirichlet kernel and is based on the shift increment sM. For example, if the parameter 9k may be assigned a parameter value in the range — n to n (e.g. a phase shift imparted by a phase shifter) then the partial derivative may be determined from: O t. ((-1)^1 (eq.4) For example, if the parameter 9k may be assigned a value in the range 0 to n (e.g. an effective reflection coefficient of a reconfigurable beamsplitter), then the partial derivative may be determined from: t. ((-1)^1 (eq.5) M=1

[0093] The shift value may be expressed as: 27T / Z Sm ~ 27? + 1 (EQ. 6)

[0094] To evaluate the expression provided in (EQ.3), one may accordingly operate the boson sampler to sample from 2R distributions to obtain the expectation values / (0^) and / (0^)- The higher the number of samples in each batch of samples, the more representative that batch of samples is of the underlying sampled probability distribution. In the limit of infinite samples, the expression of (EQ. 3) is exact, and accordingly the greater the number of samples the more accurate the determined partial derivative.

[0095] Advantageously, there are several bounds on the index R, at or above which the expression of (EQ. 3) is exact. Some of these bounds are discussed further below.

[0096] Fig. 5 shows a flowchart of a method 500 which describes method stage 410 of method 400 in more detail. For the purposes of this example, it is expected that the boson sampler has already been configured with a set of parameter values {q}, and that the set of values {q} is in memory and accessible to the processor unit. Generally speaking, the method 500 may be used to determine a partial derivative f of an expected value f of an observable 0 with respect to a parameter 9k. The skilled person will appreciate that while the method 500 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 500.

[0097] At 505, the method begins. A dummy variable / z is initially set to have an integer value of one (at 510). With reference to Fig- 1, processor unit 104 may, during the course of executing instructions loaded from the memory unit 106 or elsewhere, set the value of the dummy variable. At 515, a determination is made by the processor unit 104 as to whether the value of the dummy variable / z is less than or equal to a defined index value R. If true, the method progresses to 520, otherwise, the method progresses to 545.

[0098] At 520, the method comprises operating the boson sampler configured with the parameter 0fc having a positively shifted value qk + to produce a first batch of samples. With reference to Fig. 1, processor unit 104 may, during the course of executing instructions loaded from the memory unit 106 or elsewhere, sample a bosonic probability distribution. The processor unit 104 may for example communicate a set of parameter values {q} including the positively shifted value qk + for assigning to parameters of the boson sampler 114 to controller 122 to operate the boson sampler 114 with that set of parameter values. The controller 122 may accordingly configure the interferometer 114 in accordance with the received set of 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 an integer string representative of samples or measurement outcomes. The controller 122 may communicate the first batch of samples to the processor unit 104.

[0099] The batch of samples may contain any number of samples suitable for representing the output distribution of the boson sampler. The number of samples may accordingly be selected based on, for example, the number of input modes or output modes of light processed by the boson sampler, or the number of configurable elements in the boson sampler. For example, the batch of samples may comprise more than one hundred samples, more than one thousand samples, or more than ten thousand samples.

[0100] At 525, the method comprises operating the boson sampler configured with the parameter 9k having a negatively shifted value qk — to produce a second batch of samples. This may be achieved in much the same way as 520.

[0101] At 530, the method comprises determining a first expected value of the observable 0 from the first batch of samples obtained at 520. The first expected value f(0k+^ is the expected value of the observable as obtained from operating the boson sampler with the parameter 9k having a positively shifted value qk + s^. For example, the processor 104 may evaluate the first batch of samples to determine the first expected value and store that first expected value. At 535, the method comprises determining a second expected value f(0k^ of the observable 0 from the second batch of samples obtained at 525. The second expected value f(9k^ is the expected value of the observable as obtained from operating the boson sampler with the parameter 9k having a negatively shifted value qk — s^. For example, the processor 104 may evaluate the second batch of samples to determine the second expected value and store that second expected value.

[0102] The skilled person will appreciate that the steps 520, 525, 530 and 535 may be performed in a different order. For example, step 525 may be performed before 520 or after 530.

[0103] At 540, the integer / z is updated to + 1 and the method 500 returns to 515. If the integer / z is no longer less than or equal to R, then the method proceeds to 545.

[0104] At 545, the method comprises determining the partial derivative from the first expected values of the observable and the second expected values of the observable obtained previously. In particular, the processor unit may determine the partial derivative from computing the expression provided above in (EQ. 3). At 550, the method ends. Referring again to Fig. 4, the method 400 may proceed to step 415 described further above.

[0105] An example in which methods 400 and 500 may be particularly useful is in a gradient descent method. Consider, for example, an algorithm that may be assigned to the system 100 which involves reducing (e.g. minimising) an objective function denoted Q. To reduce Q using gradient descent may involve determining a partial derivative of Q with respect to a parameter 9k. Determining the partial derivative may comprise, for each observable Oj of a set of observables {0}, a corresponding first value V^Oj) representative of a partial derivative of the objective function Q with respect to the expectation value of the observable f = {Oj). Determining the partial derivative may further comprise determining, for each observable Oj of the set of observables {0}, a corresponding partial derivative of the expectation value fj of the observable Oj with respect to the parameter 9k. The partial derivative of the objective function Q with respect to the parameter 9k may then be determined from: (EQ. 7) where Cj is a coefficient. In some examples, the coefficients Cj may be equal and the sum may be unweighted. The partial derivatives fj {9k) of the expectation values with respect to the parameter fj {9k) may be determined using a method such as method 500. The values 14 (0;) may be obtained in any of a number of ways apparent to the skilled person. The parameter 9k may then be updated according to: dQ (BQ. s) The gradient descent process may continue until a stopping condition is satisfied.

[0106] While in the example described above in relation to Fig. 4, at (EQ. 3) the derivative of a Dirichlet kernel was used, the skilled person will appreciate that other expressions may be used. For example, consider the “modified Dirichlet kernel”: = (EQ. 9) 1 k) 2 / ?tan(0fc / 2) The inventors have recognised that using the derivative of the modified Dirichlet kernel in (EQ. 3), with a modified shift increment s^, enables one to accurately determine the partial derivative / '(0fc) with smaller batches of samples (that is, fewer samples collected). In particular, (EQ. 4) may be replaced with: (eq.10) / W = £ ( / CO + / (^))-—tty 4R sin2 (½) where / (07^) represents the expectation value of the observable when the parameter 9k has been adjusted from its current value (0fc = qk) to a positively shifted value (9k=qk + s^) and f(9''k0 represents the expectation value of the observable when the parameter 9k has been adjusted from its current value (9k = qk) to a negatively shifted value (9k=qk — s’^j and wherein the shift value is expressed as: (2 / z — 1)tt (EQ. 11) S / i “ 2R

[0107] In the discussion provided herein, to determine a partial derivative of an expected value of an observable, a number 2R of expected values of observables are obtained. The index R may be selected according to any of a number of methods, and up to a point the greater the value of R the greater the accuracy of the determined partial derivative. A value of the index R may be determined, for example, based on a consideration of one or more of: the input multimodal photonic state, the type of interferometer in the boson sampler, or a property of the configurable element to which the parameter relates.

[0108] However, the inventors have determined a number of bounds on the index R such that, if a value of the index R is selected that is greater than or equal to the bound, then the expression of (EQ.3) is exact. Accordingly, if a value of the index R is selected that is greater than or equal to the bound, then gradient descent may be performed with accuracy. Preferably, the value of the index R is equal to but not greater than a bound, as this means that the number of expected values of the observable (e.g. 6k+^ and 9k ^) is the minimal number that may be required to ensure the accuracy of the determined partial derivative. Several bounds on the index R are described. The skilled person will appreciate that which bound is lower will be determined by a combination of features which may include (the structure of or number of photons in) the input multimodal photonic state, the type of interferometer in the boson sampler, the type of observable, or a property of the configurable element to which the parameter relates (e.g. a position of the reconfigurable element in the boson sampler). The skilled person will appreciate that accordingly the bounds described herein may take different values, depending on the above. In such a case, the index R may be the selected to be the lowest of the bounds while still maintaining the accuracy of the determined partial derivative.

[0109] A first bound is the number of photons in the input multimodal photonic state. That is, in some examples, the index R may be equal to the number of photons N in the input multi-modal photonic state, irrespective of the number of input modes of the input multimodal photonic state. Fig. 6 shows a sketch of the functional operation of a boson sampler, which may be a spatial boson sampler (that is, a boson sampler having a spatial interferometer), a temporal boson sampler (that is, a boson sampler having a temporal interferometer) or any other kind of boson sampler. Referring to Fig. 6, an input state comprises a plurality of input modes, each input mode having a corresponding photon occupation nr Each mode is represented by a horizontal line in the figure. The interferometer of the boson sampler performs a unitary transformation on the input modes to produce an output multimodal photonic state, and the photon occupation of the output modes is measured. The unitary transformation comprises a first unitary transformation (denoted W in the figure), followed by a second unitary transformation provided by the configurable element to which the parameter 0k pertains, followed by a third unitary transformation (denoted by V in the figure). It is possible to prove mathematically that the total number of photons N of the input multimodal photonic state is a bound on the index R.

[0110] A second bound is the number of photons relevant to the parameter 0k that may impinge upon the configurable element to which the parameter 0k relates. In other words, in some examples, the index R may be based on the number of photons that may impinge upon the configurable element to which the parameter 0k relates when the boson sampler is operated to produce a sample. This may be understood in the context of Fig. 7. Referring to Fig. 7, an input state comprises a plurality of input modes, each input mode having a corresponding photon occupation. Each mode is represented by a horizontal line in the figure. The interferometer of the boson sampler performs a unitary transformation on the input modes to produce an output multimodal photonic state, and the photon occupation of the output modes is measured. The unitary transformation comprises a first unitary transformation (denoted W = WA®IVB), followed by a second unitary transformation provided by the configurable element to which the parameter 0k pertains, followed by a third unitary transformation (denoted by V in the figure). As can be seen in the figure, the first transformation is formed of two smaller unitary transformations WA and WB that operate on distinct subsets of the modes of the input multimodal photonic state. Accordingly, a first subset of the input modes comprising a number NA of the N input photons (NA <N) is subject to the first smaller unitary transformation WA, and a complementary second subset of the input modes comprising a number NB of the N input photons (NB <N) is subject to the second smaller unitary transformation WB. As can be seen in the figure, the parameter 0k in this example may only influence the output of the first smaller unitary transformation WA and has no influence on the output of the second smaller unitary transformation WB. For the purposes of the present discussion, only the NA photons of the first subset of input modes may (through some tuning of the parameters that may influence the operation WA) impinge upon the configurable element to which the parameter 0k relates when the boson sampler is operated to produce a sample.

[0111] For example, if the input multimodal photonic state takes the form prescribed in (EQ. 1), the index R may be equal to the number of photons N in the input multimodal photonic state. Depending on the structure of the interferometer, the index R may be less than N. For example, with reference to Fig. 2, when producing a sample with interferometer 118a, the number of input photons may be N = 4, but only two photons may impinge on the reconfigurable beamsplitter 230 to which particular parameter 0, corresponds, and so R may be two instead of four if the task is to determine a derivative of the observable with respect to that parameter value 0V Similarly only three photons may impinge on the reconfigurable beamsplitter 230 to which particular parameter 02 corresponds, and so R may be three instead of four if the task is to determine a derivative of the observable with respect to that parameter 02. For example, with reference to Fig. 3, when producing a sample with interferometer 118b, the number of input photons of the input multimodal photonic state may be e. g. N = 4, but only two photons may impinge on the reconfigurable beamsplitter 320 while it is configured with particular parameter 01; and so R may be two instead of four if the task is to determine a derivative of the observable with respect to that parameter 0i.

[0112] A third bound is dependent on the form of the observable. In particular, in some examples, the observable may comprise a monomial based on a number of photons in one or more output modes of the boson sampler, and the index R may correspond to the degree of the monomial. In other words, the observable 0 may be a so-called number-ordered observable of the form: 0 = n^n?2 ...n^N (EQ. 12) where nj is the number of photons measured on mode j and Pj is a corresponding exponent. The degree p of the monomial is then given by: <EQI3) j

[0113] For example, an observable may comprise a marginal output distribution of the boson sampler 114, determined from the photon number in each output mode of the boson sampler. For example, an observable may comprise a number of photons in an output mode g of the output multimodal photonic state (f = {ng}} as determined from a batch of samples output from the boson sampler, in which case, the index R may be one. For example, an observable may comprise a product of a number of photons in an output mode g and a number of photons in an output mode h of the output multimodal photonic state (f = {ngnh}) as determined from a batch of samples output from the boson sampler, in which case, the index R may be two.

[0114] In some examples, the observable may comprise a polynomial based on a number of photons in one or more output modes of the boson sampler. The polynomial may comprise a sum of monomials, each having a corresponding degree. In such circumstances, the greatest degree may be a bound on the index R.

[0115] Variations of the described embodiments are envisaged.

[0116] 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.

[0117] 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.

[0118] Each feature disclosed in this specification (including any accompanying claims, abstract or drawings), 5 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 10 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 method for configuring a boson sampler, the boson sampler comprising a light source, a reconfigurable interferometer having one or more configurable elements, and one or more photodetectors, the method comprising:determining a partial derivative of an expected value of an observable with respect to a parameter of a configurable element of the boson sampler, the parameter assigned a parameter value qk, wherein determining the partial derivative comprises;producing a number 2R of batches of samples by, for all integers / z between 1 and R: operating the boson sampler configured with the parameter assigned a positively shifted parameter value qk + to produce a first batch of samples;determining a first expected value of the observable from the first batch of samples;operating the boson sampler configured with the parameter assigned a negatively shifted parameter value qk — to produce a second batch of samples;determining a second expected value of the observable from the second batch of samples;wherein index R is an integer; anddetermining the partial derivative from the first expected values of the observable and the second expected values of the observable; andconfiguring the configurable element of the boson sampler based on the determined partial derivative.

2. A method according to claim 1, wherein:the observable comprises a monomial based on a number of photons in one or more output modes of the boson sampler.

3. A method according to claim 2, wherein index R is greater than or equal to the degree of the monomial.

4. A method according to claim 1, wherein:the observable comprises a polynomial based on a number of photons in one or more output modes of the boson sampler.

5. A method according to claim 4, wherein the polynomial comprises a sum of monomials, each having a corresponding degree, and wherein the index R is greater than or equal to the greatest degree.

6. A method according to claim 1 or claim 2, wherein:the index R is based on a number of photons that may impinge upon the configurable element when the boson sampler is operated to produce a sample.

7. A method according to claim 1, wherein the index R is greater than or equal to the number of photons in an input multimodal photonic state provided to the reconfigurable interferometer of the boson sampler.

8. A method according to any preceding claim, wherein each shift increment is given by =9. A method according to any preceding claim, wherein determining the partial derivative from the first expected values of the observable and the second expected values of the observable comprises determining:RM=iwherein represents the expectation value of the observable determined from the first batch of samples,and wherein represents the expectation value of the observable determined from the second batch of samples, and wherein D'^ comprises a derivative of a Dirichlet kernel and is dependent on the shift increment s^.

10. A method according to any of claims 1 to 7, wherein each shift increment is given by = andwherein determining the partial derivative from the first expected values of the observable and the second expected values of the observable comprises determining:RM=1wherein represents the expectation value of the observable determined from the first batch of samples,and wherein represents the expectation value of the observable determined from the second batch of samples, and wherein comprises a derivative of a modified Dirichlet kernel and is dependent on the shift increment11. A method according to any preceding claim, wherein the configurable element comprises a reconfigurable beamsplitter, and wherein the parameter value corresponds to an effective transmission coefficient of the reconfigurable beamsplitter.

12. A method according to any preceding claim, wherein the configurable element comprises a phase shifter, and wherein the parameter value corresponds to a phase imparted by the phase shifter.

13. A non-transitory computer-readable medium comprising stored instructions that, when executed by a computing device, cause the computing device to perform operations including:determining a partial derivative of an expected value of an observable with respect to a parameter of a configurable element of a boson sampler, the parameter assigned a parameter value qk, wherein determining the partial derivative comprises;producing a number 2R of batches of samples by, for all integers / z between 1 and R: operating the boson sampler configured with the parameter assigned a positively shifted value (¾ + sg) to produce a first batch of samples;determining a first expected value of the observable from the first batch of samples;operating the boson sampler configured with the parameter assigned a negatively shifted value (qk — to produce a second batch of samples;determining a second expected value of the observable from the second batch of samples;wherein index R is an integer; anddetermining the partial derivative from the first expected values of the observable and thesecond expected values of the observable; andconfiguring the configurable element of the boson sampler based on the determined partial derivative.

14. A system comprising:a boson sampler comprising:a light source;a reconfigurable beamsplitter having one or more configurable elements, andone or more photodetectors; andone or more processors configured to:determine a partial derivative of an expected value of an observable with respect to a parameter of a configurable element of the boson sampler, the parameter assigned a parameter value qk, wherein determining the partial derivative comprises;producing a number 2R of batches of samples by, for all integers q between 1 and R:operating the boson sampler configured with the parameter assigned a positively shifted value qk + to produce a first batch of samples;determining a first expected value of the observable from the first batch of samples;operating the boson sampler configured with the parameter assigned a negatively shifted value qk — to produce a second batch of samples;determining a second expected value of the observable from the second batch of samples;wherein index R is an integer; anddetermining the partial derivative from the first expected values of the observable and the second expected values of the observable; andconfigure the configurable element of the boson sampler based on the determined partial derivative.

15. A system according to claim 14, wherein a photodetector of the one or more photodetectors comprises a photon number resolving detector.

16. A system according to claim 14 or claim 15, wherein the reconfigurable interferometer comprises a reconfigurable temporal interferometer.

17. A system according to claim 14 or claim 15, wherein the reconfigurable interferometer comprises a reconfigurable spatial interferometer.

18. A system according to any of claims 14 to 17, wherein the configurable element comprises a reconfigurable beamsplitter, and wherein the parameter corresponds to an effective transmission coefficient of the reconfigurable beamsplitter.

19. A system according to any of claims 14 to 17, wherein the configurable element comprises a phase shifter, and wherein the parameter corresponds to a phase imparted by the phase shifter.T +44(0)30 0300 2000Search report under Section 17 of the Patents Act 1977Application No.: GB2413523.8Claims searched: 1-19Date search completed: 18 December 2025International classificationSubclass and subgroup Valid from G06N10 / 20 01 / 01 / 2022 G06N10 / 40 01 / 01 / 2022 G06N10 / 60 01 / 01 / 2022 G06N3 / 084 01 / 01 / 2023Field of searchWorldwide search of patent documents classified in the following areas of the IPC: G06NDatabases used in the preparation of this search report:INTERNET; SEARCH-NPL; SEARCH-PATENTDocuments considered to be relevantPatent literatureCategory Relevant to claims Document of relevance X 1-2, 4, 8-19 WO 2023 / 012375 A1 (QU&CO) - see figures 5B, 11B / 12B / 16B and page 28, line 3o, to page 29, line 13; page 34, line 29, to page 35, line 7; page 38, lines 18-21; and page 39, lines 3-15 Non-patent li Category terature Relevant to claims Document of relevanceT +44(0)30 0300 2000Category Relevant to claims Document of relevance X 1-2, 4, 8-19 Arxiv.org, 22 / 11 / 2022, Hoffman T and Brown D, "Gradient Estimation with Constant Scaling for Hybrid Quantum Machine Learning", https: / / arxiv.org / pdf / 2211.13981, (HOFFMAN), see figure 1, equations (8)-(12) and section 1. Introduction X 1-2, 4, 8-19 Arxiv.org, 24 / 04 / 2024, Periyasamy M et al, "Guided-SPSA: Simultaneous Perturbation Stochastic Approximation assisted by the Parameter Shift Rule", https: / / arxiv.org / pdf / 2404.15751v1, (PERIYASAMY), see Algorithm 2 and section III.B X 1-2, 4, 8-19 Arxiv.org, 20 / 06 / 2024, Goldsmith D and Day-Evans J, "Beyond QUBO and HOBO formulations, solving the Travel Salesman Problem on a quantum boson sampler", https: / / arxiv.org / pdf / 2406.14252, (GOLDSMITH), see Table 1 and page 8 A - IEEE Transactions on Knowledge and Data Engineering, 35, 2, 02 / 2023, Shi J et al, "Quantum Circuit Learning With Parametrized Boson Sampling", 1965-1976, doi: 10.1109 / TKDE.2021.3095103 (SHI) A - Arxiv.org, 24 / 09 / 2024, Facelli G et al, "Exact gradients for linear optics with single photons", https: / / arxiv.org / pdf / 2409.16369v1, (FACELLI)CategoriesLetter or symbol Description X Document indicating lack of novelty or inventive step. Y Document indicating lack of inventive step, if combined with another document of the same category. & Member of the same patent family. A Document indicating technological background. P Document published on or after the priority date but before the filing date of the present application. E Earlier application published on or after the filing date of the present application.

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