System and method for simulating at least one solution of a stochastic differential equation and its use in generative machine learning

Abstract

An optoelectronic system and method for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process, where the stochasticity of the Lévy process is represented by the photon statistics of at least one optical mode. The system comprises at least one input port for receiving the at least one optical mode, computational and recursive components, and an output port for readout.

Description

[Technical Field]

[0001] [Reference to Related Application] This application claims priority to the following U.S. provisional applications, the entire contents of each of which are incorporated herein by reference: U.S. Provisional Application No. 63 / 378,677 (filed October 6, 2022); U.S. Provisional Application No. 63 / 380,338 (filed October 20, 2022); U.S. Provisional Application No. 63 / 479,512 (filed January 11, 2023); U.S. Provisional Application No. 63 / 501,317 (filed May 10, 2023); U.S. Provisional Application No. 63 / 509,563 (filed June 22, 2023); U.S. Provisional Application No. 63 / 514,047 (filed July 17, 2023). [Background technology]

[0002] Stochastic differential equations (SDEs) are used to describe the dynamics of random processes, allowing for the modeling of a wide variety of phenomena. The stochastic nature of these equations allows them to be exploited by a variety of quantum and classical algorithms. Stochastic differential equations are used in many applications, ranging from solving optimization problems to implementing machine learning (ML) models. For example, performing a random walk with a drift term can guide a stochastic process towards a local minimum of the objective function, reaching a global minimum for a complex non-convex objective function.

[0003] Generative machine learning is a field of research that aims to leverage the exceptional pattern recognition and representation generalization capabilities of machine learning to learn to generate structure from randomness. At least one goal of generative learning is mapped to the problem of augmenting, transforming, or generating data points. For example, given a dataset of training examples based on an underlying data distribution, a generative model can provide a mechanism for generating examples that are not present in the dataset but are indistinguishable or nearly indistinguishable from the data examples. Indistinguishable examples comprise generated data that appears "real" to humans. Generative models capture the underlying characteristics of a data distribution and provide a means for extracting (e.g., sampling) data from that distribution.

[0004] Generative machine learning has many applications. For example, generative models can be used to extend the training datasets of other machine learning models (data augmentation), for anomaly detection (such as fraud detection and disease screening), and for image-to-image transfer learning (such as transferring the style of a painting, colorizing black and white photos, or reconstructing darkened images). Generative models are also used to generate photorealistic images from text prompts (e.g., stock photo generation). Summary of the Invention

[0005] The present disclosure provides optoelectronic systems and methods for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process, represented by photon statistics of at least one optical mode, and methods for using such systems and methods for generative machine learning. The present disclosure may improve upon existing systems and methods for simulating at least one solution of a stochastic differential equation by advantageously utilizing optical quantum devices.

[0006] The disclosed systems and methods offer at least some of the following advantages: By operating at optical frequencies, the disclosed systems can operate at clock speeds orders of magnitude faster than electronic-circuit-based systems; furthermore, by exploiting quantum noise resulting from quantum measurements or injected optical states, they can efficiently generate naturally random processes essential for stochastic dynamics; optical quantum devices consume less power than electronic devices, thereby reducing the operational costs of evaluating diffusion models; and the disclosed systems and methods can be directly implemented by directly mapping continuous variables in a given optimization problem or machine learning model to continuous properties of optical quantum devices (e.g., pulse amplitude), thereby eliminating or reducing the need for binarization (e.g., no need to convert continuous variables in the problem to binary variables). This significantly reduces the number of required resources. Furthermore, optical quantum devices can operate at room temperature, eliminating or reducing the need for costly and energy-consuming cryogenic equipment. The disclosed systems and methods can exploit the internal functionality of optical devices by simulating the dynamics of stochastic differential equations using their noise and physical evolution. The systems and methods of the present disclosure, including resistive memory arrays, may utilize analog electronic circuitry to implement the drift term, which can provide lower power consumption and faster processing compared to digital approaches.

[0007] In one aspect, the present disclosure provides an optoelectronic system for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process, the system may include at least one input port for receiving at least one optical mode, where the photon statistics of the at least one optical mode represent the stochasticity of the Lévy process, and a computational and recursive component configured to perform one or more operations on one or more signals derived from the at least one optical mode to simulate at least one solution of the stochastic differential equation.

[0008] In some embodiments, the computational and recursive component is further configured to implement at least a portion of a generative machine learning model based on the at least one optical mode. In some embodiments, the system further comprises an output port configured to provide a readout, wherein the solution of at least one of the stochastic differential equations is based at least in part on the readout. In some embodiments, the computational and recursive component comprises one or more members selected from the group consisting of a digital electronic component, an analog electronic component, an optoelectronic component, and an optical component. In some embodiments, the computational and recursive component can include a network of optical components. In some embodiments, the analog electronic component comprises a resistive random access memory (RRAM) device. In some embodiments, the digital electronic component comprises a field programmable gate array (FPGA).

[0009] In some embodiments, the system further includes one or more optical waveguides, the one or more optical waveguides comprising at least one optical mode. In some embodiments, the computation and reflection component includes one or more optical components, at least one of the one or more optical components comprising at least one coupler configured to accept the optical mode via at least one input port of the one or more optical components. In some embodiments, the computation and reflection component includes a measurement module configured to perform homodyne measurements on the at least one optical mode. In some embodiments, each variable and one or more parameters of the SDE are represented by at least one of an optical signal or an electronic signal. In some embodiments, the signal encodes information using a current pattern. In some embodiments, the signal encodes information using a voltage pattern.

[0010] In some embodiments, the computational and recursive component comprises a feedback loop including at least one of an electronic component or an optical component. In some embodiments, the computational and recursive component comprises a signal aggregator configured to sum signals. In some embodiments, the summer comprises one or more optical components including a circuit node, a summing amplifier circuit, or at least one coupler. In some embodiments, the at least one optical mode comprises at least one member selected from the group consisting of a squeezed state, a vacuum state, and a coherent state, and the at least one optical mode is injected through at least one input port. In some embodiments, the optical mode comprises a vacuum state that has been converted to a squeezed state using the computational and recursive component.

[0011] In some embodiments, the stochasticity is controlled by quantum measurements. In some embodiments, one of the quantum measurements changes the photon statistics of one or more of the at least one optical mode. In some embodiments, a quantum measurement on a particular optical mode of the at least one optical mode generates an optical mode with different photon statistics than the quantum measurement of the measured optical mode. In some embodiments, when operating in a low-photon regime, the quantum measurement comprises a weak quantum measurement and is performed by the environment. In some embodiments, the at least one coupler is at least one of a tunable coupler or a fixed coupler. In some embodiments, the one or more optical components comprise at least one of the group consisting of a beam splitter, a phase shifter, an amplifier, and a squeezer. In some embodiments, the at least one optical component comprises at least one nonlinear optical component. In some embodiments, the at least one optical component comprises a Mach-Zehnder interferometer (MZI) including two 50:50 beam splitters and at least one phase shifter. In some embodiments, the computation and recursion component comprises an MZI mesh including a network of MZIs, the MZI mesh adapted to implement an analog-optical matrix-vector product (MVM) computation unit, hi some embodiments, the computation and recursion component comprises a free-space analog-optical matrix-vector product unit.

[0012] In some embodiments, the computational and recursive component comprises a processing network, the output of which at least partially represents the drift term of the stochastic differential equation. In some embodiments, the computational and recursive component comprises a processing network, the output of which at least partially represents the stochasticity, the stochasticity being generated using results of a homodyne measurement of at least one optical mode. In some embodiments, the measurement module is operably optically connected to at least one coupler using one or more optical waveguides, the one or more optical waveguides being in a closed loop configuration, and an output of the homodyne measurement being fed back to the at least one coupler. In some embodiments, the system further comprises one or more optical cavities including one or more optical waveguides arranged in a closed loop connecting the at least one coupler and the measurement module, and the output of the homodyne measurement being fed back to the one or more optical cavities. In some embodiments, the one or more optical waveguides are optical fibers.

[0013] In some embodiments, the computation and recursion component is configured to perform an encoding to represent a latent variable, the latent variable comprising a portion of the machine learning (ML) model. In some embodiments, the computation and recursion component is configured to simulate a solution to at least one of the forward stochastic differential equations, the solution being used to train the generative machine learning (ML) model. In some embodiments, the computation and recursion component comprises a function approximator that takes as input one or more parameters and one or more variables. In some embodiments, the function approximator is trained to approximate a score function of the machine learning (ML) model. In some embodiments, the drift term of the stochastic sub-equation comprises a drift term of an inverse stochastic differential equation representing inverse diffusion. In some embodiments, the function approximator is trained to approximate a function that represents the logarithm of a probability distribution of the data of the machine learning (ML) model. In some embodiments, the function approximator is trained to approximate a gradient of a function that represents the logarithm of the probability distribution of the data of the machine learning (ML) model.

[0014] In some embodiments, the stochastic differential equation represents Langevin dynamics. In some embodiments, the function approximator comprises a neural network. In some embodiments, the neural network comprises an optical neural network or an analog electronic neural network. In some embodiments, the stochastic differential equation represents an optimization problem. In some embodiments, the optimization problem comprises at least one member selected from the group consisting of a box-constrained quadratic programming problem, a weighted maximum independent set optimization problem, and a two-dimensional assignment optimization problem.

[0015] In another aspect, the present disclosure provides a method for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process in an optoelectronic system, the method including the steps of: (a) generating, in the optoelectronic system, signals representing initial conditions of variables and one or more parameters of the stochastic differential equation (SDE); (b) receiving, at at least one input port, one or more optical modes having photon statistics that at least partially represent the stochasticity of the Lévy process driven by the stochastic differential equation (SDE); and (c) performing, at a computational and recursive component operatively connected to the input port, one or more operations on the signals representing the variables and one or more parameters of the stochastic differential equation (SDE) and one or more signals derived from the at least one optical mode to simulate at least one solution of the stochastic differential equation (SDE).

[0016] In some embodiments, the method further comprises performing recursive processing in a computation and recursion component by repeating steps (b) and (c) at least once. In some embodiments, the method further comprises providing a readout at a readout port of the optoelectronic system. In some embodiments, the stochastic differential equation (SDE) represents the optimization problem. In some embodiments, the drift term in the stochastic differential equation (SDE) represents at least one of the group consisting of a gradient of an objective function, a function based on the gradient of the objective function, or a time-dependent function based on the gradient of the objective function that depends on past time points. In some embodiments, the optimization problem is at least one of the group consisting of a box-constrained quadratic programming problem, a maximum weight independent set optimization problem, and a two-dimensional assignment optimization problem.

[0017] In some embodiments, the stochastic differential equation (SDE) represents Langevin dynamics. In some embodiments, the one or more operations comprise at least one of the group consisting of: (i) an analog optical operation; (ii) a digital operation; (iii) an analog electronic operation; (iv) a conversion from a digital signal to an optical mode; (v) a conversion from an analog electronic signal to an optical mode; (vi) a conversion from an analog electronic signal to a digital signal; (vii) a conversion from a digital signal to an analog electronic signal; (viii) a conversion from an optical mode to a digital signal; and (ix) a conversion from an optical mode to an analog electronic signal. In some embodiments, the conversion from an optical mode to a digital signal comprises a conversion from an optical mode to an analog electronic signal and a conversion of the analog electronic signal to a digital signal, and the conversion from a digital signal to an optical mode further comprises a conversion from a digital signal to an analog electronic signal and a conversion of the analog electronic signal to an optical mode.

[0018] In another aspect, the present disclosure provides a method for generating approximate samples from a distribution of training data, the method may include the steps of: (a) obtaining values ​​for a plurality of parameters of a trained function approximator, the trained function approximator at least partially representing a machine learning (ML) model; (b) implementing the trained function approximator on an optoelectronic system using at least the values ​​of the plurality of parameters, the optoelectronic system including at least one input port configured to receive at least one optical mode and computational and recursive components; (c) simulating, on the optoelectronic system, at least one solution to a stochastic differential equation (SDE) driven by a Lévy process representing the distribution of training data of the machine learning (ML) model; and (d) generating approximate samples based at least in part on at least one solution from the distribution of training data.

[0019] In some embodiments, the method further includes, before step (a), training a function approximator to obtain parameter values. In some embodiments, the training includes the following steps: (A) obtaining an iteration number t between 0 and a maximum iteration number; (B) initializing the optoelectronic system with samples of training data for a machine learning (ML) model; (C) operating the optoelectronic system to simulate at least one solution of a stochastic differential equation (SDE) driven by a Lévy process over t iterations; and (D) outputting a state of the optoelectronic system after the t iterations. In some embodiments, the method further includes updating parameter values ​​of the function approximator. In some embodiments, steps (A) through (D) and updating the parameters of the function approximator are repeated at least once.

[0020] In another aspect, the present disclosure provides an optical system for simulating at least one solution of a stochastic differential equation, the optical system comprising one or more optical components that generate controllable noise representative of randomness in the stochastic differential equation, the noise being generated using weak quantum measurements.

[0021] In some embodiments, the one or more optical components further generate a drift term in the stochastic differential equation. In some embodiments, the weak quantum measurement is a measurement on a quantum mode. In some embodiments, the system further comprises a feedback loop, in which an output of the weak quantum measurement is used to generate a feedback signal. In some embodiments, the system operates in a low-photon regime. In some embodiments, the system operates in a low-photon regime, and the weak quantum measurement is indicated by the environment. In some embodiments, the feedback loop may include one or more optical cavities.

[0022] In some embodiments, the drift term is programmable. In some embodiments, the drift term is generated using a digital computing device. In some embodiments, the drift term is generated using a field programmable gate array (FPGA). In some embodiments, the drift term is generated by an optical component. In some embodiments, the one or more optical components comprise at least one of the group consisting of a beam splitter, a phase shifter, an amplifier, and a squeezer. In some embodiments, the one or more optical components include at least one nonlinear optical component.

[0023] In some embodiments, the system is used to execute a forward stochastic differential equation for training a generative machine learning (ML) model. In some embodiments, the drift term comprises a function approximator having one or more parameters and receiving one or more variables as input. In some embodiments, the drift term of the stochastic differential equation may represent a score function of the machine learning diffusion model. In some embodiments, the function approximator is trained to approximate the score function of the machine learning diffusion model. In some embodiments, the drift term of the stochastic differential equation is a drift term of back-diffusion. In some embodiments, the function approximator is trained to approximate the gradient of a function representing the logarithm of the probability distribution of data of the machine learning model.

[0024] In some embodiments, the stochastic differential equation represents Langevin dynamics. In some embodiments, the function approximator comprises a neural network. In some embodiments, the neural network comprises an optical neural network. In some embodiments, the system comprises multiple function approximators with identical parameters, connected using one or more optical cavities.

[0025] In another aspect, the present disclosure provides a method for simulating at least one solution of a stochastic differential equation, the method comprising: (a) generating, in an optical system including a network of optical components having one or more optical components, a plurality of optical pulses representing random variables that serve as initial conditions for dynamics described by a stochastic differential equation; (b) performing one or more quantum measurements on the plurality of optical pulses; and (c) performing one or more optical operations on the plurality of optical pulses using the network of optical components.

[0026] In some embodiments, the light pulses comprise a vacuum state or a coherent state. In some embodiments, the method further comprises digital manipulation of the plurality of light pulses, comprising converting one or more light pulses of the plurality of light pulses to a digital signal, performing digital processing on the one or more light pulses, and converting the one or more light pulses back into light pulses. In some embodiments, the method further comprises repeating steps (b) and (c) one or more times.

[0027] In another aspect, the present disclosure provides a method for generating approximate samples from a distribution of training data, the method may include: (a) obtaining parameter values ​​of a trained function approximator that approximates a score function of a machine learning diffusion model; (b) implementing the trained function approximator on an optical system using the obtained parameter values, the optical system including one or more optical components; (c) operating the optical system to perform dynamics described by a stochastic differential equation that represents the distribution of training data of the machine learning diffusion model; and (d) reporting results to generate approximate samples from the distribution of training data.

[0028] In some embodiments, the method further comprises, prior to step (a), training the function approximator to obtain parameter values. In some embodiments, the training comprises the following steps: (A) sampling time t between 0 and a maximum time; (B) initializing the optical system with samples of training data of the machine learning diffusion model; (C) implementing dynamics described by a stochastic differential equation representing the diffusion process of the machine learning diffusion model to operate the optical system for t iterations; and (D) outputting a state of the system after the t iterations. In some embodiments, the method further comprises updating the parameter values ​​of the function approximator.

[0029] In another aspect, the present disclosure provides an optical system for simulating at least one solution described by a stochastic differential equation, the optical system comprising: (a) at least one coupler including one or more optical components, the at least one coupler accepting an optical quantum mode through at least one input port of the one or more optical components; and (b) a measurement module performing a homodyne measurement on the optical quantum mode, the measurement module being optically operatively connected to the at least one coupler using one or more optical waveguides; wherein controllable noise in the optical quantum mode is generated using at least the quantum measurement of the optical quantum mode, the controllable noise representing at least a portion of the randomness of the stochastic differential equation.

[0030] In some embodiments, the system further comprises a processing network operatively connected to the at least one coupler, the processing network generating at least one programmable optical mode that at least partially represents the drift of the stochastic differential equation. In some embodiments, the processing network is also operatively connected to a measurement module, the generated at least one programmable optical mode representing at least a portion of the randomness of the stochastic differential equation and generated using homodyne measurements of optical quantum modes, and the generated at least one programmable optical mode is fed back to the system. In some embodiments, the processing network comprises at least one of a digital electronic component, an analog electronic component, an optoelectronic component, and an optical component. In some embodiments, the analog electronic component is a resistive random access memory (RRAM) device. In some embodiments, the digital electronic component is a field programmable gate array (FPGA). In some embodiments, the at least one coupler is at least one of a tunable coupler or a fixed coupler. In some embodiments, the optical quantum mode is a squeezed state, and an amount of squeezing of the squeezed state controls the controllable noise of the optical quantum mode, and the squeezed state is generated using a processing network. In some embodiments, one or more optical waveguides are arranged in a closed loop to connect the at least one coupler and the measurement module, and an output of the homodyne quantum measurement is fed back to the at least one coupler.

[0031] In some embodiments, the system further includes one or more optical cavities, the optical cavities including one or more optical waveguides arranged in a closed loop to connect the at least one coupler and the measurement module, and the output of the homodyne quantum measurement is fed back to the one or more optical cavities. In some embodiments, the one or more optical waveguides are optical fibers. In some embodiments, the system operates in a low-photon regime. In some embodiments, the system operates in a low-photon regime, and the quantum measurement is performed in a weak quantum measurement environment. In some embodiments, the one or more optical components comprise at least one of the group consisting of a beam splitter, a phase shifter, an amplifier, and a squeezer. In some embodiments, the one or more optical components comprise at least one nonlinear optical component.

[0032] In some embodiments, the system is used to encode representations of latent variables used in machine learning techniques. In some embodiments, the system is used to execute forward stochastic differential equations to train generative machine learning (ML) models. In some embodiments, the processing network comprises a function approximator having one or more parameters and receiving one or more variables as input. In some embodiments, the function approximator is trained to approximate a score function of the machine learning model. In some embodiments, the drift term of the stochastic differential equation comprises a drift term of dediffusion. In some embodiments, the function approximator is trained to approximate a function representing the logarithm of a probability distribution of data of the machine learning model. In some embodiments, the function approximator is trained to approximate a gradient of a function representing the logarithm of a probability distribution of data of the machine learning model. In some embodiments, the stochastic differential equation represents Langevin dynamics. In some embodiments, the function approximator comprises a neural network. In some embodiments, the neural network comprises an optical neural network.

[0033] In another aspect, the present disclosure provides a method for implementing dynamics described by a stochastic differential equation, comprising: (a) injecting, into an optical system including at least one coupler including one or more optical components, a plurality of optical quantum modes representing random variables as initial conditions for the dynamics described by the stochastic differential equation; (b) performing one or more quantum measurements on the plurality of optical quantum modes; and (c) performing one or more operations on the plurality of optical quantum modes using the one or more optical components.

[0034] In some embodiments, the one or more operations comprise at least one of the group consisting of: (i) analog optical operations; (ii) digital operations, including converting one or more of the plurality of photon modes into a digital signal, digitally processing the digital signal, and converting the digital signal back to the photon mode; and (iii) analog electronic operations, including converting one or more of the plurality of photon modes into an analog electronic signal, analog electronic processing the analog electronic signal, and converting the analog electronic signal back to the photon mode. In some embodiments, converting one or more of the plurality of photon modes into a digital signal includes converting one or more of the plurality of photon modes into an analog electronic signal and converting the analog electronic signal into a digital signal, and further, converting the digital signal back to the photon mode includes converting the digital signal into an analog electronic signal and converting the analog electronic signal back to the photon mode. In some embodiments, the method further includes repeating steps (b) and (c) one or more times.

[0035] In another aspect, the present disclosure provides a method for generating approximation samples from a distribution of training data, the method comprising: (a) obtaining parameter values ​​of a trained function approximator that at least partially represents a machine learning model; (b) implementing the trained function approximator on an optical system including one or more optical components using the obtained parameter values; (c) operating the optical system to perform dynamics described by a stochastic differential equation that represents the distribution of training data for the machine learning model; and (d) outputting results to generate approximation samples from the distribution of training data.

[0036] In some embodiments, the method further comprises, prior to step (a), training a function approximator to obtain parameter values. In some embodiments, the training comprises the following steps: (A) obtaining t ranging from 0 to a maximum number of iterations; (B) initializing the optical system with samples of training data for a machine learning model; (C) operating the optical system for t iterations to exercise the dynamics described by the stochastic differential equation representing the machine learning model up to t; and (D) outputting a state of the system after the t iterations. In some embodiments, the method further comprises updating parameter values ​​of the function approximator.

[0037] In another aspect, the present disclosure provides a system comprising one or more computer processors and a computer memory coupled thereto, said computer memory comprising machine-executable code that, when executed by said one or more computer processors, implements any of the methods described above or elsewhere herein.

[0038] Further aspects and advantages of the present disclosure will become apparent to those skilled in the art from the following detailed description, of which only illustrative embodiments are shown and described. As will be realized, the present disclosure is capable of other and different embodiments, and its details can be modified in various obvious respects without departing from the disclosure. Accordingly, the drawings and description are to be regarded as illustrative and not restrictive. [Cited by reference]

[0039] All publications, patents, and patent applications mentioned herein are incorporated by reference in their entirety as if each such publication, patent, and patent application was expressly incorporated by individual reference. To the extent that the publications, patents, and patent applications incorporated by reference conflict with the disclosure of this specification, the contents of this specification control, supersede, and are intended to control for the conflicting content. [Brief explanation of the drawings]

[0040] The novel features of the invention are set forth with particularity in the appended claims. The features and advantages of the present invention will be better understood by reference to the following detailed description and accompanying drawings (hereinafter also referred to as "Figures" or "FIG.") that set forth illustrative embodiments, in which the principles of the invention are utilized.

[0041] [Figure 1] 1 is a diagram of an optoelectronic system for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process, the stochasticity of which is represented by the photon statistics of at least one optical mode.

[0042] [Figure 2] 1 is a flowchart of a method for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process, the stochasticity of which is represented by the photon statistics of at least one optical mode.

[0043] [Figure 3A] 1 is a diagram of an optoelectronic system for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process, where the stochasticity of the Lévy process is expressed by the photon statistics of at least one optical mode, in a configuration where the optoelectronic system comprises optical components.

[0044] [Figure 3B] FIG. 1 is a diagram of an optoelectronic system for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process, where the stochasticity of the Lévy process is expressed by the photon statistics of at least one optical mode, in a configuration where the optoelectronic system includes a processing network.

[0045] [Figure 3C] FIG. 1 is a diagram of an optoelectronic system for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process, where the stochasticity of the Lévy process is expressed by the photon statistics of at least one optical mode, in a configuration where the optoelectronic system comprises two processing networks.

[0046] [Figure 4] 1 is a flowchart of a method for generating approximate samples from the distribution of training data.

[0047] [Figure 5] 1 is a flowchart of a method for training a function approximator.

[0048] [Figure 6] 1 is a flowchart of a method for generating samples from a distribution of training data for a generative machine learning model using a hybrid optoelectronic system having an optical device coupled to an electronic score function approximator.

[0049] [Figure 7]1 is a flowchart of a method for generating samples from a distribution of training data for a generative machine learning model using an optoelectronic system with an optical score function approximator.

[0050] [Figure 8] 1 is a flowchart of a method for simulating dynamics described by a stochastic differential equation (SDE). DETAILED DESCRIPTION OF THE INVENTION

[0051] While various embodiments of the present invention have been shown and described herein, it will be apparent to those skilled in the art that these embodiments are provided by way of example only. Numerous changes, modifications, and substitutions will occur to those skilled in the art without departing from the scope of the invention. It should be understood that various alternatives may be used in place of the embodiments described herein.

[0052] In a string of two or more numbers, when the first number is preceded by the words "at least," "greater than," or "greater than or equal to," the words "at least," "greater than," or "greater than or equal to" apply to all numbers in the string. For example, "one, two, or three or more" is equivalent to "one or more, two or more, or three or more."

[0053] In a column of two or more numbers, when the words "no more than," "less than," or "less than or equal to" appear before the first number, the words "no more than," "less than," or "less than or equal to" apply to all numbers in the column. For example, "3, 2, or 1 or less" is equivalent to "3 or less, 2 or less, or 1 or less."

[0054] Some inventive embodiments herein contemplate numerical ranges. When a range is stated, the endpoints of the range are included, and all subranges and values ​​within that range are treated as if they were explicitly stated.

[0055] The terms "about" or "approximately" mean within an acceptable error range for a given value, which error range depends in part on how the value is measured or determined, e.g., the limitations of the measurement system. For example, "about" can mean within one standard deviation or more than one standard deviation, based on common practice in the art. Alternatively, "about" can mean within 20%, 10%, 5%, or 1% of the given value. When a specific numerical value is described in the present specification and claims, unless otherwise specified, "about" should be interpreted as meaning within an acceptable error range for the given value.

[0056] The title and abstract should not be construed as limiting the scope of the disclosed invention in any way. The title of this application and the headings of its sections are for convenience only and are not intended to limit the contents of this disclosure. high performance computing equipment

[0057] The computing device disclosed herein may be capable of interacting with the optical computing device disclosed herein, such as a coherent optical network, and may be a high-performance computing (HPC) device. The HPC device may include one or more of the following: a graphics processing unit (GPU), a tensor processing unit (TPU), a field programmable gate array (FPGA), an application-specific integrated circuit (ASIC), and a tensor streaming processor (TSP). Any other suitable processing unit capable of performing matrix multiplication may also be used. Certain computing devices may be more efficient than others at operations such as matrix multiplication. These computing devices may provide additional efficiency improvements compared to other computing systems. For example, a matrix multiplication device may be a GPU. A GPU is a high-throughput, dedicated electronic circuit optimized to perform the same operation on many blocks of data at once in parallel. For example, a matrix multiplication device may be a TPU. A TPU is a type of ASIC developed for low-bit precision processing, such as that developed by Google Inc. See patent application US 2016 / 0342891 A1, which is incorporated herein by reference in its entirety. Another example of a matrix multiplication device is an FPGA. An FPGA may be an integrated circuit chip with configurable logic blocks and programmable interconnects, which can be programmed to execute custom algorithms after fabrication. Another example of a matrix multiplication device may be an ASIC. An ASIC is an integrated circuit chip customized to execute a specific algorithm. In some cases, ASICs cannot be reprogrammed after fabrication. Another example of a matrix multiplication device may be a TSP. A TSP is a domain-specific programmable integrated chip designed for linear algebra operations performed in artificial intelligence applications; examples can be found, for example, in Gwennap, Linley, "Groq rocks neural networks," The Linley Group, Microprocessor Report, Technical Report, January 2020, which is incorporated herein by reference in its entirety. Optical calculation device / quantum optical device

[0058] The computational tasks described herein may be performed by a quantum optical device. A quantum optical device may be a non-classical computer that is an optical computing device. A quantum optical device may be a device with optical elements that act as quantum gates for photons. In some cases, the optical elements maintain quantum coherence between different quantum states. In some cases, the inputs and outputs of a quantum optical device are quantum modes (qumodes). In contrast, a classical computer, classical computation, or classical operation refers to a computation performed using binary or analog values ​​with discrete bits without quantum mechanical superposition or entanglement. A classical computer may be a digital computer using discrete bits (e.g., 0 and 1) without quantum mechanical superposition or entanglement, or an analog computer using continuous variables without quantum mechanical superposition or entanglement.

[0059] A nonclassical computer, nonclassical computation, or nonclassical operation refers to any method or system that performs computations outside the framework of classical computation. In some cases, a nonclassical computer is a quantum computer. A quantum device may be a device or system that performs computations using quantum mechanical phenomena, such as quantum superposition and quantum entanglement. A quantum computation, quantum processing, quantum operation, or quantum computer is a method or system that performs computations using quantum mechanical operations (e.g., unitary transformations or completely positive trace-preserving (CPTP) maps on quantum channels) on a Hilbert space represented by a quantum device.

[0060] A quantum bit (qubit) is a unit of quantum information processing whose quantum state is a two-dimensional complex unit vector. These two dimensions are commonly referred to as "0" and "1." When quantum error correction is used, a logical quantum bit is one of a set of physical quantum bits that encode a single fault-tolerant quantum bit. A quantum mode (qumode) is a quantum state represented as a quantum optical mode. A quantum optical mode is an optical mode expressed as a superposition of a set of quantum photon number states. The optical mode may be a single quantum optical mode of a quantum optical device described herein. The superposition may be an infinite superposition of all quantum photon number states. Quantum modes are sometimes used as an alternative to representing quantum information in quantum bits (qubits) or quantum dits (qudits). A quantum bit is a discrete unit of information that represents a superposition of binary values, such as zero and one. A quantum gate is a set of operations in a gate-model quantum computer. A quantum gate may be a physical device that transforms the quantum state of an input according to a unitary transformation that describes the specific behavior of the quantum gate.

[0061] The quantum optical device may prepare initial optical quantum states (e.g., quantum modes) and perform a series of quantum gates on these optical quantum states. These gates may use quantum optical elements that perform appropriate quantum unitary transformations. The quantum optical device may also perform quantum measurements, such as homodyne measurements, to introduce quantum noise or detect the mean-field amplitude of the optical pulses. This detection may be performed by measuring the amplitude and phase of the optical pulses and comparing them to a reference amplitude and phase, using methods similar to those disclosed in patent US10139703 B2 or the paper by McMahon et al., "A fully programmable 100-spin coherent Ising machine with all-to-all connections," Science 354, no. 6312: 614-617, 2016, all of which are incorporated herein by reference.

[0062] The quantum optical device may further perform quantum measurements, such as photon-number-resolved (PNR) measurements or homodyne detection, on the quantum modes to detect the final quantum state. This detection is performed by statistically determining the probability distribution of the final quantum state in a quantum number basis, which may be detected by repeated measurements. Further details are described in patent application US 2019 / 0325589 and the paper "Applications of near-term photonic quantum computers: software and algorithms" (Bromley et al., Quantum Science and Technology 5, no. 3: 034010, 2020), all of which are incorporated herein by reference. For example, in Xanadu quantum optical devices, the initial state may be a squeezed vacuum state obtained by squeezing an on-chip vacuum state with an optical crystal having high second-order nonlinearity. In Nippon Telegraph and Telephone Corporation (NTT) quantum optical devices, the initial state may be a quantum vacuum state or a squeezed vacuum state obtained by squeezing a vacuum state with an optical crystal having high second-order nonlinearity. An example of such a crystal is periodically poled lithium niobate (PPLN), which has been used in quantum optical devices. These squeezed vacuum states pass through modules in the systems disclosed herein, implemented using optical elements or quantum optical gates that can perform desired operations. For example, phase shifters may use materials with temperature-dependent refractive indices and be tuned using voltage-controlled heating plates. In some cases, quantum optical gates may also use on-chip or fiber-optic directional couplers or beam splitters to generate quantum superposition between quantum modes. In some cases, the superposition occurs between two or more quantum modes.

[0063] Measurements performed on quantum modes may include quantum PNR measurements and homodyne measurements. By measuring the intensity of an optical field and comparing its phase to a reference phase, the results of homodyne measurements can be used to estimate the results of quantum measurements in the position or momentum basis. Homodyne measurements are also used to introduce quantum noise into a system. While homodyne measurements are continuous and can be performed with high precision, quantum PNR measurements require more technical sophistication to perform due to the difficulty of detecting single photons. Examples of photon number detectors include, but are not limited to, single-photon avalanche detectors (SPADs), which can detect whether the photon count is zero or not, and photon-number-resolving detectors (PNRDs), which can report photon counts down to a few photons. In some examples, a PNRD may output at least two, at least three, at least four, at least five, at least six, at least seven, at least eight, at least nine, or at least ten photons. The control voltage signals for the squeezer and quantum operations, as well as the quantum measurement results, may be controlled and acquired by a classical computer such as a field programmable gate array (FPGA), which may convert between values ​​that are meaningful in terms of the mathematical description of the computational task and voltages appropriate for optical gates and detectors. Coherent Ising Machine

[0064] In some cases, a computational device such as a matrix multiplication device may be connected to the photonic computational device. The photonic computational device may be a coherent Ising machine (CIM). A CIM is an optical device with powerful optimization capabilities due to its high connectivity, such as full coupling, between optical pulses. Furthermore, by operating at optical frequencies, it may be faster than classical computers in solving optimization problems. CIMs are adapted to solving Ising problems, but can represent binary or continuous variables, or both simultaneously, using optical pulses. This modification is sometimes called a coherent optical network.

[0065] The coherent optical network may be of various types, such as those described herein. In some cases, the coherent optical network may include an optical computing device. In some cases, the coherent optical network may include a group of optical instruments, such as beam splitters, nonlinear optical elements such as periodically poled lithium niobate (PPLN), free-space lasers operating on an optical table, fiber optics and fiber ring resonators including fiber optic elements, phase-sensitive amplifiers (PSAs), second-harmonic generators (SHGs), intensity and phase modulators, photodetectors, analog-to-digital converters (ADCs), digital-to-analog converters (DACs), and field-programmable gate arrays (FPGAs). In some cases, the coherent optical network may include integrated photonics.

[0066] For example, a coherent optical network or other optical device may include a beam splitter. A beam splitter is an optical element with two inputs and two outputs that can combine input signals to generate a single output signal, generate two optical signals from a single input signal, or generate two output signals from two input signals that are a superposition (classical or quantum) of the input optical signals. The beam splitter may be a general-purpose tabletop optical beam splitter or an on-chip optical directional coupler that performs similar functions to a beam splitter.

[0067] For example, a coherent optical network or other optical device may include an optical line. The optical line may be an optical waveguide, such as an optical fiber, a silicon nanophotonics waveguide, or free space through which optical pulses can propagate. The optical line may be terminated at one or both ends, or may be configured in a loop to form an optical cavity. The optical cavity may be a closed optical path consisting of an optical line and mirrors that can form a standing wave. The optical cavity may be used to maintain the coherence of the optical pulses within the system.

[0068] For example, a coherent optical network or other optical device may include a Mach-Zehnder interferometer. A Mach-Zehnder interferometer (MZI) is an optical element that includes two beam splitters and at least one phase shifter. By adjusting the phase difference between the optical modes in the two arms between the two beam splitters, the phase shifter can control the degree of addition or subtraction of the input signal at the output. Furthermore, adding a phase shifter to either the input or output arm of the optical element can also adjust the overall phase. Integrated photonic coherent Ising machine

[0069] Another example of an optical computing device is an integrated photonic coherent Ising machine, as disclosed, for example, in "Coherent Ising machines with error correction feedback" by Satoshi Kako, Timothee Leleu, Yoshitaka Inui, Farad Khoyratee, Sam Reifenstein, and Yoshihisa Yamamoto, Advanced Quantum Technologies, Volume 3, Issue 11, 2000045, all of which are incorporated herein by reference.

[0070] In some cases, an integrated photonic coherent Ising machine is a combination of nodes and a connection network to solve a specific Ising problem. In some cases, this combination of nodes and a connection network may form an adiabatic optical computer. That is, the combination of nodes and a connection network can solve an Ising problem nondeterministically when the values ​​stored in the nodes reach a steady state that minimizes the energy of the nodes and the connection network. The values ​​stored in the nodes in the minimum-energy state may be associated with a value that is a solution to the specific Ising problem.

[0071] In one case, the system includes a plurality of ring resonator photonic nodes, each of the plurality of ring resonator photonic nodes storing a value, a pump coupled to each of the plurality of ring resonator photonic nodes via a pump waveguide to provide energy to each of the plurality of ring resonator photonic nodes, a connectivity network including a plurality of 2×2 building blocks, each component of the 2×2 building blocks including a plurality of phase shifters for adjusting the connectivity network with parameters related to encoding an Ising problem, the connectivity network processing the values ​​stored in each of the plurality of ring resonator photonic nodes, the Ising problem being solved, and the value stored in each of the plurality of ring resonator photonic nodes at a minimum energy state representing a solution to the Ising problem. Analog Electronic Circuits

[0072] The computing device disclosed herein may be an analog electronic device that interacts with the photonic device disclosed herein, such as a coherent optical network, and may be, for example, a resistive random-access memory (RRAM) device, such as a cross-structured resistive memory array, capable of performing matrix-vector multiplication (MVM). Such computing devices may offer improved efficiency compared to other computing systems. An RRAM device implemented as a cross-structured resistive memory array is an integrated or discrete element circuit in which each resistive memory has a programmable conductivity state to represent the numerical value of an element of a two-dimensional matrix. The RRAM device can perform a matrix-vector multiplication based on Ohm's law and Kirchhoff's laws for electronic circuits between an input vector comprising electronic voltage signals and a matrix mapped to the programmable conductivity states of the resistive memory arranged in the cross-structure. The resistive memory may represent numerical values ​​as either binary or continuous values. Other types of analog electronic elements, such as analog multiplication circuits, can also be used for the matrix-vector multiplication.

[0073] Analog electronic circuits process signals with values ​​on a continuous spectrum and proportionally represent external signals (such as sound, light, temperature, displacement, and current) as electronic voltages or currents. Basic electronic elements of these analog circuits include, but are not limited to, transistors, diodes, resistors, capacitors, and inductors. Higher-order building blocks include operational amplifiers (op-amps), which perform functions including, but not limited to, buffers, differential amplifiers, integrators, and comparators. Analog electronic circuits can be used to perform a variety of operations, including addition, subtraction, multiplication, division, integration, inversion, exponential operations, logarithmic operations, and division. Analog electronic circuits can also be used to perform nonlinear operations on electronic signals. Electronic components that make up such analog electronic circuits may include one or more of diodes, transistors, differential amplifiers such as operational amplifiers, and passive linear elements such as resistors, inductors, and capacitors. Nonlinear functions can also be generated by the cross structure of memristor devices (or other RRAM devices).

[0074] A memristor device, resistive random access memory, or resistive random access memory (RRAM) device may comprise an electronic element having two terminals or electrodes, the conductive state of which is non-volatile and can be programmed by applying a program voltage or current pulse to the electrodes and read by applying a read voltage or current pulse. A memristor crossbar, crossbar array of memristor devices, or crossbar array of resistive random access memory may comprise a two-dimensional grid-like arrangement of multiple memristor devices, where memristor devices in the same row may share a common electrode, e.g., a "top" or "input" electrode, and memristor devices in the same column may share a common but different electrode, e.g., a "bottom" or "output" electrode. digital computer

[0075] In some cases, the systems, media, networks, and methods described herein may include or use a classical computer (e.g., a digital computer). In some cases, a classical computer may include a digital computer. In some cases, a classical computer may include one or more hardware central processing units (CPUs) that perform the functions of the classical computer. In some cases, a classical computer may further include an operating system (OS) configured to execute executable instructions.

[0076] In some cases, the classical computer is connected to a computer network. In some cases, the classical computer is connected to the Internet and has access to the World Wide Web. In some cases, the classical computer is connected to one or more computer servers, which may enable distributed computing such as cloud computing infrastructure. In some cases, the classical computer is connected to an intranet or extranet, or an intranet or extranet connected to the Internet. In some cases, the classical computer is connected to a data storage device. In some cases, the network is a communications network and / or a data network. In some cases, the network is a peer-to-peer network, allowing computing devices connected to the system to function as clients or servers.

[0077] In accordance with the teachings herein, suitable classical computers include, by way of non-limiting example, server computers, desktop computers, laptop computers, notebook computers, subnotebook computers, netbooks, netpads, set-top computers, media streaming devices, handheld computers, Internet appliances, smartphones, tablet computers, personal digital assistants (PDAs), video game consoles, vehicles, etc. Smartphones are suitable for use with the systems and methods described herein. Selected televisions, video players, and digital music players, in some cases with network connectivity, are suitable for use with the systems and methods described herein. Suitable tablet computers include booklets, slates, and convertibles.

[0078] In some cases, a classical computer includes an operating system configured to run executable instructions. The operating system may be, for example, software that manages the device's hardware and includes programs and data that support the execution of applications. Suitable server operating systems include, by way of non-limiting example, FreeBSD, OpenBSD, NetBSD®, Linux®, Apple® Mac OS X Server®, Oracle® Solaris®, Windows Server®, and Novell® NetWare®. Operating systems for personal computers include Microsoft® Windows®, Apple® Mac OS X®, Apple® macOS®, UNIX®, and UNIX-like operating systems such as GNU / Linux®. In some cases, operating systems are provided via cloud computing. Non-limiting examples of suitable smartphone operating systems include Nokia® Symbian® OS, Apple® iOS®, Research In Motion® BlackBerry OS®, Google® Android®, Microsoft® Windows Phone® OS, Microsoft® Windows Mobile® OS, Linux®, and Palm® webOS®. Non-limiting examples of suitable media streaming device operating systems include Apple TV®, Roku®, Boxee®, Google TV®, Google Chromecast®, Amazon Fire®, and Samsung® HomeSync®. Suitable video game console operating systems include, by way of non-limiting example, Sony® PS3®, Sony® PS4®, Sony® PS5®, Microsoft® Xbox 360®, Microsoft® Xbox One®, Nintendo® Wii®, Nintendo® Wii U®, and Ouya®.

[0079] In some cases, a classical computer includes storage and / or memory devices. In some cases, storage and / or memory devices are one or more physical devices used to temporarily or permanently store data or programs. In some cases, storage and / or memory devices may include one or more additional data storage units located on a remote server separate from the classical computer and communicating with the classical computer via an intranet or the Internet. In some cases, the devices are volatile memory, requiring power to retain information. In some cases, the devices are nonvolatile memory, retaining stored information even when the classical computer is powered down. In some cases, nonvolatile memory includes flash memory. In some cases, nonvolatile memory includes dynamic random access memory (DRAM). In some cases, nonvolatile memory includes ferroelectric random access memory (FRAM). In some cases, nonvolatile memory includes phase change random access memory (PRAM). In some cases, nonvolatile memory includes resistive random access memory (RRAM). In some cases, the device is a storage device, such as, by way of non-limiting example, a CD-ROM, a DVD, a flash memory device, a magnetic disk drive, a magnetic tape drive, an optical disk drive, cloud-based storage, etc. In some cases, the storage device and / or memory device is a combination of devices as disclosed herein.

[0080] In some cases, a classical computer includes a display for outputting visual information to a user. In some cases, the display is a cathode ray tube (CRT). In some cases, the display is a liquid crystal display (LCD). In some cases, the display is a thin film transistor liquid crystal display (TFT-LCD). In some cases, the display is an organic light emitting diode (OLED) display. In some cases, the OLED display is a passive matrix OLED (PMOLED) or an active matrix OLED (AMOLED) display. In some cases, the display is a plasma display. In some cases, the display is a video projector. In some cases, the display is a combination of devices as disclosed herein.

[0081] In some cases, a classical computer includes an input device for receiving information from a user. In some cases, the input device is a keyboard. In some cases, the input device is a pointing device, including, by way of non-limiting example, a mouse, trackball, trackpad, joystick, game controller, stylus, etc. In some cases, the input device is a touchscreen or multi-touchscreen. In some cases, the input device is a microphone capable of capturing voice or other acoustic input. In some cases, the input device is a video camera or other sensor capable of capturing motion or visual input. In some cases, the input device is a Kinect® or Leap Motion®. In some cases, the input device is a combination of devices as disclosed herein. An optoelectronic system for simulating the solution of SDEs.

[0082] Referring to FIG. 1, a diagram of an optoelectronic system for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process is shown, where the stochasticity of the Lévy process is expressed by the photon statistics of at least one optical mode. A stochastic differential equation (SDE) may be a system of differential equations in which one or more terms are stochastic processes. A stochastic process is a mathematical object defined as a family of random variables. These differential equations may describe the dynamic behavior of the system variables in a single execution of the system.

[0083] As disclosed herein, the stochasticity of a Lévy process can be represented by the photon statistics of at least one optical mode. An optical mode is a quantum or classical state of an electromagnetic field with a wavelength range from infrared to the edge of visible light. An optical mode is represented by an optical pulse that propagates through an optical waveguide, and the amplitude of the optical pulse may represent a variable of the underlying problem or model. A quantum optical mode is an optical mode that has quantum properties.

[0084] Examples of quantum optical modes include squeezed states and photon number (Fock) states. A squeezed state is an optical coherent state in which the dispersion in the position or momentum quadruplet components is squeezed or antisqueezed. A squeezed quantum optical mode has a state in which one quadruplet component is below the minimum quantum noise level and the conjugate quadruplet component is above the minimum quantum noise level, preserving the uncertainty principle.

[0085] As shown in FIG. 1, the optoelectronic system comprises at least one input port (104, 106, ..., 108) for receiving at least one optical mode, a computational and recursive component 100, and output ports (110, 112, ..., 114) for readout.

[0086] In some cases, a stochastic differential equation (SDE) driven by a Lévy process may be of the following form: dx(t) i=f(x(t) i ,t)dt+g(x(t) i ,t)dZ(t) i where Z(t) is a Lévy process. A Lévy process may be a Wiener process, Poisson process, Gamma process, Pascal process, Cauchy process, or Meixner process. A Lévy process may be any stochastic process that satisfies the mathematical definition of a Lévy process.

[0087] In some cases, the stochastic differential equation may be of the form: dx i =f(x i ,t)dt+g(x i ,t)dW i where dW is the derivative of the Wiener stochastic process. This type of stochastic differential equation is also called a forward stochastic differential equation. The first term, also called the drift coefficient, gives the drift force to the evolution of the stochastic process x(t). The diffusion coefficient g(x,t) gives the randomness. The combined selection of the drift coefficient and the diffusion coefficient is called the "stochastic differential equation" in this embodiment.

[0088] In some cases, the stochastic differential equation represents the Langevin dynamics involving gradient descent and noise terms, and can be written as follows: DC i =-∂ i f(c)dt+σdW i , ∀i∈{1,···,N} where the first term represents the gradient of the objective function, driving the variables towards a minimum of the objective function, while the second term makes the process stochastic, increasing the chances of finding a global minimum. Here, c is a vector of variables, σ is a constant, dW is the derivative of the Wiener stochastic process, N is the number of variables, and ∂ denotes the partial derivative of c with respect to its components.

[0089] The SDEs described elsewhere herein that describe the underlying processes of the optical devices and systems described herein assume that the system is at zero temperature. While this may not be physically feasible, it closely approximates real system conditions, such as the optoelectronic systems described herein with respect to Figures 3A, 3B, 3C, and 1. At nonzero temperatures, the underlying processes may include noise due to the effect of quantum measurements the environment makes on the system, which may be reflected in the corresponding SDEs.

[0090] The signals representing the variables and one or more parameters of the SDE may be optical or electronic. In embodiments where the signals are electronic, current or voltage patterns may be used to encode information. The patterns may include variations in the amplitude, duration, phase, and shape of the current or voltage signals. The electronic signals representing the variables and parameters of the SDE may be generated by an electronic signal generator, such as a voltage or current source, or by photoelectric conversion of an optical signal. Photoelectric conversion may be achieved using a photodetector. When the signal is optical, the amplitude and phase of a series of optical pulses may be used to encode information and represent variables or parameters of the SDE. The optical pulses are generated using a continuous-wave laser source modulated with a periodic microwave signal to form an optical pulse train. The laser generates time-separated optical pulses of the same wavelength as the laser source by cycling on and off at a specific rate.

[0091] At least one input port 104, 106, 108 is for receiving at least one optical mode. The at least one optical mode may be of various types. In some cases, the at least one optical mode is a quantum optical mode. In some cases, the at least one optical mode includes a squeezed state. In some cases, the at least one optical mode includes a vacuum state. In some cases, the at least one optical mode includes a coherent state. The at least one optical mode may be configured to be converted from a vacuum state to a squeezed state using computational and recursive components 100. The vacuum state may be injected by leaving at least one input port open, as described elsewhere herein. By leaving at least one input port open, an optical vacuum state is injected into the system. The squeezed vacuum state may be generated externally using a squeezing device that receives intense coherent light (e.g., from a laser light source) to generate the squeezed vacuum state. The output of the squeezing device is then fed to the input port of the optoelectronic system. A squeezed vacuum state has a large uncertainty in one of the four-part components of an optical field (e.g., the real part of the optical field) and a small uncertainty in the conjugate four-part component. This is a purely quantum effect, where the uncertainty in one of the four-part components is below the minimum uncertainty limit. Compared to the vacuum state, a squeezed vacuum state has a large quantum noise, allowing for greater control over the stochasticity of the Lévy process compared to vacuum state injection. Coherent states of light can also be injected into the input port of an optoelectronic system. A coherent state is a classical state of light with a minimum quantum noise limit, and stochasticity can be generated using photon statistics in at least one optical mode. The coherent state is generated by an external laser source and fed into at least one input port of the optoelectronic system. The degree of squeezing of the squeezed state can control the stochasticity (i.e., noise) of the quantum optical mode. By continuously injecting controllable squeezed states into an optoelectronic system, randomness in stochastic differential equations (SDEs) can be controlled. The more squeezing, the more randomness is added to the pulse.This injection may occur continuously for each round trip through the optoelectronic system, or once every few iterations. In some cases, the optical modes are programmable. At least one optical mode may be quantum, meaning there is quantum entanglement between the optical modes.

[0092] In some cases, multiple quantum optical modes may be injected as squeezed vacuum states using a squeezer or as coherent states using an external laser source. These laser sources may operate at the same wavelength as the rest of the optoelectronic system. Pulses of different amplitudes may be generated using a single laser source by dynamically controlling the laser's output or by sequentially using intensity and phase modulators to generate the desired initial amplitude and phase of the optical pulses. A train of pulses generated by a laser is synchronized using a delay line network to ensure that all pulses arrive at the optoelectronic system on multiple different optical lines. The delay line system may comprise a set of optical waveguides, intensity modulators, phase modulators, and beam splitters. This system may be used to form coupling between pairs of input optical pulses by delaying the optical pulses and adjusting their amplitude and phase. The optical pulses generated by the laser are all in the same optical mode but have different parameters. Each optical pulse is in an independent optical mode from the other optical pulses, but they all share the properties of the same optical mode generated using the same laser source or the same squeezer.

[0093] The computational and recursive component 100 may be configured to perform one or more operations on one or more signals originating from at least one optical mode to simulate at least one solution of the stochastic differential equation. The one or more signals originating from at least one optical mode may be of various types. The one or more signals originating from at least one optical mode may be optical pulses, optoelectronic signals, electronic signals, digital signals, or analog signals, as described elsewhere herein.

[0094] The computational and recursive component 100 may include any one or more of the following: digital electronic components, analog electronic components, optoelectronic components, and optical components. These components may be fabricated through a single integrated circuit manufacturing process and interconnected to form a single application-specific integrated circuit (ASIC). These components may also be fabricated through a heterogeneous manufacturing process, where individual components or groups of components are fabricated separately using different manufacturing technologies and then assembled. In a heterogeneous manufacturing approach, components or groups of components may be obtained from a commercial catalog or designed into an integrated circuit to meet specific requirements. The digital and analog electronic components described above are often constructed using CMOS technology, particularly for integrated circuit implementations. However, components such as diodes and transistors can also be fabricated using other monolithic manufacturing processes and circuit designs. This includes, but is not limited to, bipolar junction transistor (BJT), silicon-on-insulator (SOI), bipolar CMOS (BiCMOS) technology, and other manufacturing technologies for analog, digital, or mixed-signal (combining analog and digital circuitry) applications.

[0095] The analog electronic components may include resistive random access memory (RRAM) devices. The digital electronic components may include field programmable gate arrays (FPGAs). In some cases, the computational and recursive component 100 includes one or more analog electronic components, one or more of which may be used to amplify or shape signals to compensate for losses occurring in the system.

[0096] The optical mode can be converted to a digital signal using an optical analog-to-digital converter (ADC) and then converted back to an optical mode using an optical digital-to-analog converter (DAC). The analog electronic signal can be converted to a digital signal using an electronic ADC and then converted back to an analog electronic signal using an electronic DAC. These converters can be implemented on-chip with an FPGA or used externally using separate DAC or ADC devices. The analog optical mode can be converted to an analog electronic signal using optoelectronic components, including optoelectronic materials such as lithium niobate, and vice versa, where the electronic signal can change the optical properties of the optoelectronic material.

[0097] In some cases, the computational and recursive component 100 includes one or more optical components, at least one of which may include a coupler. The at least one coupler may receive the optical mode through at least one input port of the one or more optical components. The at least one optical component may be of various types, such as a beam splitter, a directional coupler, a phase shifter, an amplifier, a squeezer, or a nonlinear optical component. A beam splitter may extract a portion of the optical pulse for measurement or other manipulation. It may also be used to implement unitary operations on quantum optical modes. A phase shifter may be used to adjust the phase of the optical pulse to maintain coherence during combining of the optical pulses or as part of implementing a unitary operation. An amplifier may be used to compensate for losses occurring in the system by coherently pumping the optical pulse. A squeezer may be used to adjust and control the fundamental noise level of the optical pulse to tune the performance of the system. In some cases, the optical component includes at least one nonlinear optical component. These components may incorporate second- or third-order optical nonlinearities to achieve non-Gaussian optical interactions. Such interactions may be necessary to implement nonlinear (e.g., second-, third-, or fourth-order) mathematical functions. Such mathematical functions may be required to properly generate drift terms in stochastic differential equations using optical elements.

[0098] The at least one coupler may be of various types. In some cases, the at least one coupler is a tunable coupler. The tunable coupler may have a variable transmission coefficient, i.e., the amount of optical field transmitted to a particular output of the coupler may be variable and controllable. In some cases, the at least one coupler is a tunable coupler including a Mach-Zehnder interferometer (MZI). The MZI is composed of two 50:50 beam splitters and two phase shifters, and tunable coupling is achieved by varying the phase shift of the phase shifters. The computation and recursion component may include an MZI mesh comprising a network of MZIs. The MZI mesh is a network of MZIs that performs a matrix-vector product operation by forming coupling between input optical modes using the network of connected MZIs. The MZI mesh can have any configuration, such as a triangle or a rectangle. Such an MZI mesh is reconfigurable by adjusting the two phase shifters included in each MZI in the MZI mesh. The parameters of the MZI mesh may be adjusted continuously or sequentially over time. A triangular configuration of the MZI mesh may be implemented similarly to the approach presented in the paper "Experimental realization of any discrete unitary operator" (Reck et al., Physical Review Letters 73.1 (1994):58), which is incorporated herein by reference for all purposes. A rectangular configuration may be implemented similarly to the approach presented in the paper "Optimal design for universal multiport interferometers" (Clements et al., Optica 3.12 (2016):1460-1465), which is incorporated herein by reference for all purposes. Variable couplers may also be realized by other techniques using tabletop optical components.

[0099] In some cases, at least one coupler is a fixed coupler, such as a beam splitter. A fixed coupler has a fixed ratio of amplitude between the two outputs of the coupler. This ratio may be 50:50, meaning that equal amounts are transmitted to the two outputs. This ratio may be 20:80, meaning that the transmission ratio between the two outputs of the coupler is 20% and 80%. This ratio may also be any other fixed value.

[0100] In some cases, the computational and recursive component 100 includes a free-space analog optical matrix-vector product (MVM) unit. Such an MVM unit may consist of an array of intensity and phase modulators, where the amplitude and phase of optical pulses propagating through free space are manipulated using the modulators. The modulators may be electro-optical components, where applying and varying electrical signals can change the intensity and phase of the optical pulses passing through them. Each of N rows of optical pulses representing the underlying problem variables is split into N columns of optical pulses with identical amplitudes and phases, generating an array of N x N optical pulses. These pulses are then fed into an N x N modulator array. The output from the modulators is then summed into N columns by adding the optical pulses in each column of the N x N optical output, generating the MVM output as a vector of N optical pulses. This result is then fed back to other parts of the optoelectronic system via a coupler. The parameters of the modulators may be determined by the elements of the matrix used in the MVM operation.

[0101] In some cases, the computational and recursive component 100 includes nonlinear optical or electronic components. These nonlinear components may be capable of performing nonlinear mathematical functions on input optical or electronic signals. The nonlinearity of these components may be of second-order or quadratic type. The nonlinearity of these components may also be of any other nonlinear type. In some cases, optical nonlinearity may be created using a nonlinear optical material (e.g., lithium niobate) with strong second- or third-order optical nonlinearity. In some cases, optical nonlinearity may be created using a photodetector. The photodetector may generate an electronic signal proportional to the intensity experienced by the optical mode, which is in turn proportional to the square of the amplitude of the optical field. In this way, an electronic signal that is a square function of the amplitude of the optical mode may be generated. The electronic signal may then be processed using analog electronic components and converted to a digital electronic signal, or converted to an optical mode using an optical intensity modulator. Optical nonlinearity may also be generated in other ways. In some cases, the nonlinearity may be generated using an analog electronic circuit composed of at least one of an analog multiplier, an amplifier, a variable resistor, or a function generator. The electronic components that make up such an analog electronic circuit may include diodes, transistors, differential amplifiers such as operational amplifiers, and passive components such as resistors, inductors, and capacitors. Mathematical operations that compute a nonlinear function of an input signal using such an analog electronic circuit may include addition, integration, inversion, multiplication, exponential operations, logarithmic operations, and division. Nonlinear functions may also be generated by a crossbar array of memristor devices that performs a matrix-vector product between a vector of the input signal and a matrix of appropriate values ​​that map to the conductance states of the memristor devices (or other RRAM devices). The conductance states of these memristor devices may be programmed with an electronic signal. The nonlinear output may be computed using analog components such as a differential amplifier. The analog electronic signal may be converted to optical mode or vice versa using optoelectronic components.Using optoelectronic materials, these components can modulate optical modes with electronic signals, and they can also generate analog electronic signals from input optical modes using similar methods.

[0102] In some cases, photonic integrated circuits, electronic integrated circuits, and memristor chips with crossbar arrays of memristor devices (or other RRAM devices) may be assembled and interconnected on silicon interposers or advanced organic substrates to form multi-chip computational and recursive components 100.

[0103] In some cases, the computation and recursion component 100 includes a measurement module for performing a homodyne measurement on at least one optical mode. The measurement module may include a homodyne detection device for performing the homodyne measurement on the optical mode. By measuring the intensity of the optical field and comparing its phase to a reference phase, the results of the homodyne measurement can be used to estimate the results of a quantum measurement in the position or momentum basis. The homodyne detection device may include a beam splitter, with the optical mode of interest fed to one input and an optical field from a local oscillator (e.g., a laser source) fed to the other input. The outputs of the beam splitter are connected to separate photodetectors, which measure the intensity of the input optical field. The output electronic signals from these photodetectors are subtracted using electronic devices to estimate the amplitude and phase of the optical mode. The measurement module may include a measurement processing unit. The measurement processing unit may receive the electronic signals output from the homodyne detector and perform some processing on the signals. This processing may be of various types. This processing may include converting an analog electronic signal to an analog optical signal. This processing may be a conversion from an analog electronic signal to a digital electronic signal. The measurement processing unit may perform an identity function on the measurement results, i.e., use the analog electronic signal unchanged. In some cases, the measurement module may include a memory for storing the measurement results. This memory may be implemented electronically by a digital or analog processing unit within the measurement module. In some cases, the measurement module may include other optoelectronic devices for converting optical energy into electronic signals for digital or analog processing. Such devices may perform quantum measurements on quadruplet components of quantum optical modes, functioning similarly to homodyne measurements.

[0104] Quantum measurements can take any form, including photon counting, homodyne, heterodyne, and photon-number-resolved (PNR) measurements. Photon-counting measurements can be performed using a photodetector that generates an electrical photocurrent proportional to the number of photons in the optical mode. Homodyne measurements can be performed by combining the optical field of the signal and the optical field of a local oscillator (e.g., a laser source) using a beam splitter and performing optical detection on one or both of the outputs. When both outputs are measured, subtracting the photocurrents from the two photodetectors yields a measurement of a specific quadruplet component of the optical field. Heterodyne measurements are similar to homodyne measurements and can be performed by simultaneously measuring two conjugated quadruplet components of the optical field using two homodyne measurements that are 90 degrees out of phase with each other, or by measuring with a detuned local oscillator field. Photon-number-resolved measurements are performed similarly to photon counting measurements, but the detector is designed to provide greater precision with respect to the number of photons reaching the detector. The output of these quantum measurements may be an analog electronic signal that can be used as is, converted to an optical signal, or converted to a digital electronic signal for further manipulation and processing. Quantum measurements may also be made weakly by the environment in the low-photon regime, where the coupler represents a photon loss to the reservoir.

[0105] In some cases, stochasticity is controlled by quantum measurements. In some cases, the quantum measurements change the photon statistics of one or more of the at least one optical mode. In some cases, a quantum measurement of one of the at least one optical mode generates an optical mode with photon statistics different from those of the measured optical mode. This is done by internally or externally generating another optical mode whose photon statistics are inherently different from those of the measured optical mode, but whose parameters are determined based on the measured properties of the measured optical mode. In such cases, the newly generated optical mode has different photon statistics from the original measured optical mode, but its properties depend on the measurement results.

[0106] In some cases, the computational and recursive component 100 includes a feedback loop. In some cases, the feedback loop includes electronic components. In some cases, the feedback loop includes optical components. In some cases, the optoelectronic system includes an optical waveguide. The optical waveguide can be of various types. In some cases, the optical waveguide includes an optical fiber.

[0107] The measurement-feedback system may be a system including different components such as a homodyne measurement system, a digital-to-analog converter, an analog-to-digital converter, and a classical computing device, which may be used to generate a coupling between input optical pulses by measuring the amplitude of the input optical pulses and generating an output optical pulse that is a function of the other input optical pulses.

[0108] In some cases, the computation and recursion component 100 includes a signal aggregator for summing signals. In some cases, the signal aggregator includes a circuit node. A circuit node operates based on Kirchhoff's law of electrical circuits, where the total current output from the node is equal to the total current flowing into the node. In some cases, the signal aggregator includes a summing amplifier circuit. This summing amplifier circuit may generate a voltage output proportional to the sum of voltage inputs provided to one or more of its input ports. In some cases, the signal adder includes one or more optical components, including at least one coupler. An optical signal adder operates by providing input optical pulses to two input ports of an optical coupler, and the output of the coupler may be proportional to the addition or subtraction of the two input pulses. To add or subtract more than two optical pulses, two or more optical couplers may be used, the number of which is proportional to the number of optical pulses to be added or subtracted.

[0109] In some cases, the computational and recursive component 100 includes a processing network. The output of the processing network may at least partially represent a drift term of a stochastic differential equation (SDE). In some cases, the processing network generates at least one programmable optical mode. In some cases, the generated at least one programmable optical mode at least partially represents a drift term of the stochastic differential equation. In some cases, the generated at least one programmable optical mode may be a coherent state amplitude-modulated with a value representing the drift term. This coherent state may be generated using an intensity modulator (IM) that modulates an optical field from a local oscillator with a value determined by the processing network 34. In some cases, the generated at least one programmable optical mode may be a squeezed state amplitude-modulated with a value representing the drift term. This programmable squeezed state may be generated by combining the modulated coherent state with a squeezed vacuum state using a beam splitter or coupler. The squeezed vacuum state may be generated externally using an optical squeezer. In some cases, the generated at least one programmable optical mode is quantum, i.e., quantum entanglement exists between the variables and the optical modes representing one or more parameters of the model. The generated at least one programmable optical mode is controlled by a measured optical mode amplitude that is a random variable. Such a term can be expressed as g(xa i ,dW i , t), where the coefficient g depends on the Wiener process.

[0110] The output of the processing network may at least partially represent the probabilities (i.e., chances) in a stochastic differential equation (SDE), which may be generated using the results of a homodyne measurement of at least one optical mode.

[0111] In some cases, the computational and recursive component 100 includes a function approximator that includes one or more parameters and receives one or more variables as input. In some cases, when the computational and recursive component 100 includes a processing network, the processing network includes the function approximator. The one or more parameters may be analytically determined and set prior to processing. The one or more parameters may be continuously or continuously determined by training the processing network. The one or more input variables may be received as input signals. They may be received as digital signals for a function approximator constructed using a digital processor (e.g., an FPGA). The one or more input variables may be received as electronic signals for a function approximator constructed using optical components. They may be received as optical modes for a function approximator using optical components. The one or more input variables of the function approximator may be received as optical modes. The one or more input variables of the function approximator may be received as electronic signals received from the output of a measurement system. The function approximator may be trained to approximate a score function of a machine learning model. The function approximator may be capable of performing an exact mathematical form of the score function or a polynomial approximation thereof. θ is the gradient of the log-likelihood (with respect to the random variable, i.e., the data) with respect to the random variable, i.e., the data, TIFF2025536240000003.tif749. Because it's a gradient, it provides a measure of how the log-likelihood changes when the data changes slightly. Therefore, if we can accurately approximate the score function, we can steer the stochastic process toward maximizing the log-likelihood and producing the most likely samples, achieving the goal of generative machine learning.

[0112] Machine learning (ML) may be a combination of an architecture (e.g., a model), numerical parameters, and a method for adjusting (i.e., training or learning) the parameter values ​​so that the model performs a specified task. In some cases, a machine learning model may be a generative machine learning model. Generative machine learning is a branch of machine learning that is concerned with learning the probability distribution from which a training dataset comes, or learning to sample from that probability distribution.

[0113] Generative machine learning models are sometimes classified as implicit density models and explicit density models. Implicit density models, such as the generative adversarial networks (GANs) presented in Goodfellow et al., "Generative Adversarial Networks," arXiv:1406.2661, which is incorporated herein by reference in its entirety, are not interested in learning the underlying probability density function of a dataset, but rather in learning a procedure that can generate samples from that probability density function.Explicit density models include denoising diffusion probabilistic models (DDPMs) ("Deep Unsupervised Learning using Nonequilibrium Thermodynamics", Sohl-Dickstein et al., Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:2256-2265, 2015, incorporated herein by reference in its entirety), variational autoencoders ("Auto-encoding Variational Bayes", Kingma & Welling, arXiv:1312.6114, incorporated herein by reference in its entirety), and Boltzmann machines ("Restricted Boltzmann machines for collaborative filtering", Salakhutdinov et al., Proceedings of the 24th International conference on Machine learning, ICML '07:791-798, 2015). 2007, incorporated herein by reference in its entirety), and PixelCNN (presented in the following paper, “Conditional image generation with PixelCNN decoders”, van den Oord et al., arXiv:1606.05328, incorporated herein by reference in its entirety), which aim to explicitly learn representations of probability density functions. Having access to the probability density function itself is useful in situations where analyzing the probability density function is more meaningful than analyzing individual samples (e.g., wave functions in quantum mechanics), and in applications where an interpretation of likelihood is useful, such as anomaly detection.

[0114] A latent variable may be an intermediate transformation of input data into another latent space. The dimensionality of the latent space may differ from the input data, for example, the low-dimensional compressed space in an autoencoder. The latent space is usually continuous, whereas the input data is not continuous, such as images. Latent representations are sometimes used in machine learning techniques.

[0115] Training can be the process of improving the performance of a machine learning (ML) model by providing training data and adjusting tunable parameters of the ML model, such as tunable parameters of a neural network, e.g., weights and biases. The tunable parameters can be one or more of the learnable / trainable parameters in the ML model, such as a neural network model. Training data is data used to improve the ML model, where the ML model, e.g., a neural network model, uses data that is different from the training data.

[0116] In some cases, the function approximator is trained to approximate a function that represents the logarithm of the probability distribution in the data for the machine learning model. In such an embodiment, the function approximator itself does not represent the score function itself, and an additional step for evaluating the score function (e.g., differencing) is used to calculate the gradient and therefore the score function. This embodiment has the advantage of having direct access to the likelihood function, which may be desirable for modeling purposes.

[0117] In some cases, the function approximator is trained to approximate the gradient of a function that represents the logarithm of the probability distribution in the data of the machine learning model. In such embodiments, the output of the function approximator can be considered the score function itself without the additional step required to evaluate the gradient.

[0118] In some cases, the function approximator includes a neural network. In some cases, the neural network may approximate and be used to generate drift terms in stochastic differential equations. There are various types of neural networks. In some cases, the neural network is an optical neural network implemented using optical components. The optical neural network may include any one or a combination of optical elements, such as phase shifters, beam splitters, phase-sensitive amplifiers, squeezers, and other linear or nonlinear optical elements. In some cases, the neural network is an analog optoelectronic system implemented using a combination of linear optical components, such as beam splitters, phase shifters, and squeezers, and nonlinear optical components, such as phase-sensitive amplifiers, nonlinear optical crystals, or other nonlinear optical components using optoelectronic materials. The parameters of these components may be externally controlled by controlling electronic signals that control the optical properties of these components. In some cases, neural networks are analog electronic systems that combine crossbar arrays of resistive memory (or other RRAM devices) with analog electronic components such as transistors, operational amplifiers, transimpedance amplifiers, analog switched-capacitor circuits, analog passive delay lines, and other resistor and capacitor elements. In some cases, neural networks are analog electronic systems. Analog electronic systems that implement neural networks may include circuits that provide nonlinearities. These nonlinearities may be generated by analog electronic circuits, as disclosed elsewhere herein. In some cases, neural networks are implemented with a combination of analog electronic and analog optical components. The analog components may be any combination of optical and electronic components disclosed elsewhere herein. Optical modes may be converted to electronic signals, and vice versa, by converters implemented using optoelectronic materials.

[0119] In some cases, the neural network is digital and implemented using a digital computing device (e.g., a digital computer or a field programmable gate array (FPGA)). The digital computer can be of various types. The digital computer can be any digital computer disclosed elsewhere herein. The parameters of the neural network can be pre-programmed (e.g., during inference) or dynamically controlled using external parameters such as the results of a measurement system (e.g., during training).

[0120] In some cases, neural networks are analog electronic systems. Analog electronic systems implementing neural networks may include circuits that provide nonlinearities. These nonlinearities may be generated by analog electronic circuits as disclosed elsewhere herein.

[0121] In some cases, measurement results from measurements of an optoelectronic system are interpreted as a latent encoding of input data for use in a machine learning method. In such cases, the optoelectronic system is initialized to the input data. It then evolves according to a stochastic differential equation that describes the dynamics of the optical device. The state of the optical device is then measured. Thus, the measurement is a probabilistic transformation of the original input data, and the transformation is performed by the optoelectronic system. This transformed input data may be saved and used as input to a machine learning method. For example, it may be used to train a score function approximator in a diffusion model of generative machine learning.

[0122] In some cases, where computation and recursion component 100 includes a processing network, the processing network is used to perform an arbitrary function. If the function is a linear function of input variables, the function is performed using linear operations. If the function is nonlinear, the nonlinear function is performed exactly or approximately using a combination of linear and nonlinear operations as disclosed herein with respect to processing network 34 of FIG. 3B. The arbitrary function performed may be the gradient of an objective function representing the optimization problem. The arbitrary function performed may be any function related to the gradient of the objective function. The arbitrary function may be a function based on variables in a previous step of Euler's method.

[0123] The Euler algorithm is an integral method for solving stochastic differential equations (SDEs), where the variables of the SDE are initially set to zero and the variables are updated using the differential changes obtained by the SDE. This process can be repeated any number of times until the variables reach their values. Coherent optical networks can approximately implement the Euler algorithm by repeatedly updating the amplitude of optical pulses with the differential changes described by the SDE any number of times.

[0124] In some cases, the processing network includes at least one optical cavity for holding the optical pulses and using them in the calculation of any implemented function, where the optical cavity can hold the optical pulses for as many round trips as necessary before being used in the next Euler method step.

[0125] In some cases, the drift term of the stochastic differential equation includes the drift term of the inverse stochastic differential equation that represents inverse diffusion. In some cases, the drift term represents an optimization problem. In such cases, the drift term may represent an objective function. In some cases, the drift term may represent the gradient of the objective function. In some cases, the drift term may represent a function of the gradient of the objective function. In some cases, the drift term may represent a time-dependent function of the gradient of the objective function that depends on past time. The drift term may also depend on other quantities or the output of other processes.

[0126] In some cases, the optimization problem is a box-constrained quadratic programming problem. In some cases, natural properties of optical devices are used to implement the box constraints of the problem. In some cases, processing networks are used to implement the box constraints. A box-constrained quadratic programming problem is a class of non-convex continuous optimization problems where the continuous variables are constrained by lower and upper bounds. These problems are formulated as follows: Minimize TIFF2025536240000004.tif1973 constraints l i ≦x i ≦u i , (for all i) The goal is to find a continuous x value under the box constraint. i The objective function f(x) is to minimize the matrix element Q ik and vector element V i , and the lower bound l i and upper bound u i The goal is to determine one instance of a box-constrained quadratic programming problem by

[0127] In some cases, the optimization problem is the maximum-weighted independent set problem (MWIS), which is implemented using continuous variable representations. In some cases, natural properties of optical devices are used to implement the constraints of the MWIS problem. In some cases, a processing network is used to implement the constraints of the MWIS problem. Maximum-weighted independent set (MWIS) is a non-convex optimization problem in graph theory. The goal is to find the independent set with the largest weight among all possible independent sets in a given graph. It is a non-convex optimization problem that finds the set with the largest sum of weights among independent sets with no edges between nodes. An independent set in a graph is a set of nodes in the graph with no edges between the nodes in the given set. The weight of a set is equal to the sum of the weights of the nodes in that set.

[0128] In some cases, the optimization problem is a Quadratic Assignment Problem (QAP). In some cases, natural properties of optical devices are used to implement the QAP constraints. In some cases, a processing network is used to implement the QAP constraints. The Quadratic Assignment Problem (QAP) is a class of non-convex optimization problems for assigning a set of n facilities to a set of n locations. A weight (or flow) is given for each pair of facilities, and a distance is given for each pair of locations. The objective is to assign all facilities to different locations, minimizing the sum of the products of the weights (or flows) and distances.

[0129] The readout output ports 110, 112,..., 114 can take a variety of forms. In some cases, the output ports 110, 112,..., 114 include optical waveguides, such as optical fibers or on-chip photonic waveguides, or free-space optical propagation, where the output of the computational and recursive component 100 is coupled into free space. In some cases, the output ports 110, 112,..., 114 include electronic circuit components, such as wires, electronic waveguides, resistors, or filter circuits. The output ports 110, 112,..., 114 can be analog electronic outputs using electronic waveguides or electrical connections. The output ports 110, 112,..., 114 can be digital electronic outputs using electrical connections or electronic waveguides. The readout can take a variety of forms. In some cases, the readout is converted into an image and sent to a display device. In some cases, the readout is stored in a database. In some cases, the output of the optoelectronic system is an optical field. In some cases, these optical fields directly generate image pixels without further processing. In some cases, these optical fields are converted to analog electronic signals and either retained for further processing or stored in an analog memory unit for further processing. In some cases, these optical fields are converted to digital electronic signals and stored in a database or for further processing. In some cases, the output of the optoelectronic system is an analog electronic signal. In some cases, the analog signals of the optoelectronic system are converted to an optical mode and used to generate an image or for further processing. In some cases, these analog electronic signals are retained for further processing or stored in an analog memory unit for further processing. In some cases, these analog electronic signals are converted to digital electronic signals and stored in a database or for further processing.

[0130] Continuing with reference to FIG. 2, a flowchart of a method for simulating at least one solution of a stochastic differential equation (SDE) driven by a Levy process is shown, where the stochasticity of the Levy process is expressed by the photon statistics of at least one optical mode.

[0131] In some cases, the stochastic differential equation (SDE) driven by the Lévy process takes the form: dx(t) i =f(x(t) i ,t)dt+g(x(t) i ,t)dZ(t i ) where Z(t) is a Lévy process. A Lévy process may be a Wiener process, Poisson process, gamma process, Pascal process, Cauchy process, or Meissner process. A Lévy process is any stochastic process that satisfies the mathematical definition of a Lévy process.

[0132] In some cases, the stochastic differential equation takes the form: dx i =f(x i ,t)dt+g(x i ,t)dW i where dW is the differential of the Wiener stochastic process. This type of stochastic differential equation is sometimes called a "forward stochastic differential equation." The first term is called the "drift coefficient" and effectively imposes a net drift force on the evolution of the stochastic process x(t). The diffusion coefficient g(x,t) adds randomness. In this embodiment, the choice of the drift and diffusion coefficients is called a "stochastic differential equation."

[0133] In some cases, the stochastic differential equation represents the Langevin dynamics, including gradient descent and noise terms, and can be written as: TIFF2025536240000005.tif11100Here, the first term represents the gradient of the objective function and drives the variables towards a minimum of the objective function, while the second term makes the process stochastic, increasing the chances of finding a global minimum.Here, c is a vector of variables, σ is a constant, dW is the derivative of the Wiener stochastic process, N is the number of variables, and ∂ denotes the partial derivative with respect to the members of c.

[0134] Continuing with reference to FIG. 2, in process step 202, an optoelectronic system is used to generate signals representing variables and initial conditions of one or more parameters of a stochastic differential equation (SDE). The optoelectronic system includes at least one input port accepting at least one optical mode, an arithmetic and recursive component, and an output port for readout. The optoelectronic system may be of various types, such as any of the optoelectronic systems disclosed herein with respect to any of FIGS. 3A, 3B, 3C, and 1. The signals representing the variables and one or more parameters of the SDE may be optical or electronic. In embodiments where the signals are electronic, a current or voltage pattern is used to encode information. This pattern may include variations in amplitude, duration, phase, or shape of the current or voltage signal. The one or more parameters of the SDE are generated by an electronic signal generator, such as a voltage source or current source, or from photoelectric conversion of an optical signal. The photoelectric conversion may be performed by a photodetector. If the signal is optical, the amplitude and phase of a train of optical pulses are used to encode information and represent one or more parameters of the SDE. These optical pulses can be generated using a continuous-wave laser source modulated with a periodic microwave signal to generate the optical pulse train. The laser is turned on and off at a specific frequency, producing optical pulses separated by a fixed time interval, but whose wavelength is identical to that of the laser source.

[0135] Still referring to FIG. 2 , in process step 204, at least one input port is used to inject a plurality of optical modes having photon statistics that at least partially represent the stochasticity of the Lévy process driving the stochastic differential equation (SDE). These optical modes can take a variety of forms. In some cases, the optical modes are quantum optical modes. In some cases, the optical modes include squeezed states. In some cases, the optical modes include vacuum states. In some cases, the optical modes include coherent states. The optical modes may also include vacuum states converted to squeezed states. The vacuum states can be injected by leaving at least one input port open, as described elsewhere herein. By leaving the input port open, an optical vacuum state is injected into the system. The squeezed vacuum state can be generated externally using a squeezing device that receives intense coherent light (e.g., from a laser light source) and generates a squeezed vacuum state. The output of the squeezing device is then provided to at least one input port of the optoelectronic system. In a squeezed vacuum state, the uncertainty in one quartet component of the optical field (e.g., the real component of the optical field) is large, while the uncertainty in the conjugate quartet component is small. This is a purely quantum effect, and the uncertainty in one quartet component can be below the minimum uncertainty. Compared to the vacuum state, the squeezed vacuum state has more quantum noise, which allows for greater control over the stochasticity of the Lévy process compared to the vacuum state input. Coherent states of light can also be injected into at least one input port of an optoelectronic system. Coherent states are classical optical states with minimal quantum noise limits, and stochasticity can be generated by utilizing the photon statistics of the optical mode. Coherent states can be generated using an external laser source and fed into at least one input port of an optoelectronic system. The amount of squeezing in the squeezed state can control the stochasticity (i.e., noise) of the quantum optical mode.By continuously injecting controllable squeezed states into an optoelectronic system, it is possible to control the randomness in the stochastic differential function (SDE). The greater the amount of squeezing, the greater the dispersion of the randomness added to the pulse. This injection may be done continuously for each round trip of the pulse through the optoelectronic system, or every few iterations. In some cases, multiple optical modes are programmable. The optical modes may be quantum, meaning that quantum entanglement exists between them.

[0136] In some cases, multiple quantum optical modes are injected as squeezed vacuum states using a squeezer or as coherent states using an external laser source. These laser sources operate at the same wavelength as the rest of the optoelectronic system. Using a single laser source, pulses with different amplitudes can be generated by dynamically controlling the laser's output or continuously using intensity and phase modulators to generate optical pulses of the desired initial amplitude and phase. The pulse trains generated by the laser are synchronized using a delay line network, and all pulses arrive at the optoelectronic system on different optical lines, such as those disclosed with respect to Figures 3A, 3B, 3C, and 1. These optical lines can be connected to the optoelectronic system using optical cavities similar to those disclosed with respect to Figures 3A, 3B, 3C, and 1. All optical pulses generated by the laser belong to the same optical mode, but each has different parameters. Each optical pulse is an independent optical mode from the other optical pulses, but all share the characteristics of the same mode created by the same laser source or squeezer.

[0137] Continuing with reference to FIG. 2 , according to processing operation 206, one or more operations are performed on signals representing variables and one or more parameters of a stochastic differential equation (SDE) and a plurality of optical modes using a computational and recursive component, such as computational and recursive component 100 disclosed herein in connection with FIG. 1 . The one or more operations may be of various types. In some cases, the operations are analog optical operations. In some cases, the operations are digital electronic operations. In some cases, the one or more operations are analog optical operations. In some cases, the one or more operations are digital electronic operations. In some cases, the one or more operations are analog electronic operations. In some cases, the one or more operations include any combination of analog optical operations, digital electronic operations, and analog electronic operations. The one or more operations may include converting a digital electronic signal to an optical mode. The one or more operations may include converting an analog electronic signal to an optical mode. The one or more operations may include converting an analog electronic signal to a digital electronic signal. The one or more operations may include converting a digital electronic signal to an analog electronic signal. The one or more operations may include converting an optical mode to a digital electronic signal. The one or more operations may include converting an optical mode to an analog electronic signal. Such operations may be performed using an optical MZI mesh, as described elsewhere herein. MVM operations may be performed using a free-space light modulator, as described elsewhere herein. MVM operations may be performed using a crossbar array of memristor devices (or other RRAM devices), as described elsewhere herein. The one or more operations may be optically generated nonlinear operations using second- or third-order optical nonlinearities, as described elsewhere herein. The one or more operations may be electronically generated nonlinear operations, as described elsewhere herein. The one or more operations may be amplitude modulation of an electronic signal using an amplitude modulator circuit. The one or more operations may be signal duplication or distribution to circuit branches using a current mirror or splitter circuit.One or more of the operations may be signal subtraction or addition using a current subtraction circuit or a differential voltage amplifier.

[0138] In some cases, the one or more operations may be quantum measurements. Quantum measurements may be of any type, including photon counting, homodyne, heterodyne, and photon-number-resolved (PNR) measurements. Photon counting measurements can be performed using a photodetector that generates an electrical photocurrent proportional to the number of photons in the optical mode. By measuring the intensity of the optical field and comparing its phase to a reference phase, the results of the homodyne measurement can be used to infer quantum measurements in the position or momentum basis. Homodyne measurements can be performed by combining the signal optical field and the local oscillator (e.g., laser source) optical field using a beam splitter and performing photodetection on one or both outputs. When both outputs are measured, the difference between the photocurrents from the two photodetectors provides a measurement of a specific quartet component of the optical field. Heterodyne measurements can be performed similarly to homodyne measurements by simultaneously performing two homodyne measurements on two conjugated quartet components of the optical field with a 90-degree phase difference, or by shifting the frequency of the local oscillator field. Photon-number-resolved measurements are performed similarly to photon-counting measurements, but the detector is designed to measure the number of arriving photons with greater precision. The output of these quantum measurements is an analog electronic signal that can be used as is, converted to an optical signal, or converted to a digital electronic signal for further processing. Quantum measurements can also be performed as weak measurements in the low-photon regime, where a coupler acts as a proxy for photon loss to the reservoir.

[0139] In some cases, the stochasticity of a stochastic differential equation (SDE) is controlled using quantum measurements. In some cases, the quantum measurements change the photon statistics of one or more of a plurality of optical modes. In some cases, a quantum measurement on one or more of a plurality of optical modes results in the creation of a new optical mode with different photon statistics than the measured optical mode. This is achieved by externally or internally generating another optical mode that is inherently different from the photon statistics of the measured optical mode but has parameters that depend on the measured properties of the measured optical mode. In such cases, the newly generated optical mode has different photon statistics from the original optical mode being measured, but has photon statistics that depend on the measured properties of the measured optical mode.

[0140] Continuing with reference to FIG. 2 , if a stopping condition is met in decision step 208, a readout of the state of the optoelectronic system is performed according to process step 210. The stopping condition can take a variety of forms. In some cases, the stopping condition is when a maximum number of iterations is reached. In some cases, the maximum number of iterations is specified by a user. In some cases, the maximum number of iterations is determined by another method using the method. In some cases, the other method is a machine learning method or optimization method, as described elsewhere herein. The readout is performed in a variety of ways. In some cases, the readout is performed by homodyne measurement of the optical modes. In some cases, the readout is converted into an image and sent to a display. In some cases, the readout is stored in a database. In some cases, the output of the optoelectronic system includes optical fields. In some cases, these optical fields generate the pixels of an image without further processing. In some cases, the optical fields are converted into analog electronic signals and kept as is or for further processing, or stored in an analog memory unit and for further processing. In some cases, the optical fields are converted into digital electronic signals and stored in a database or for further processing. In some cases, the output of the optoelectronic system is an analog electronic signal. In some cases, the analog electronic signal is converted to an optical mode and used for image generation or processing. In some cases, these analog electronic signals are retained for further processing or stored in an analog memory unit for further processing. In some cases, these analog electronic signals are converted to digital electronic signals and stored in a database or used for further processing. If the stopping condition is not met, processing operations 204 and 206 are repeated using at least one input port and arithmetic and recursive components of the optoelectronic system, respectively. In embodiments where the signals representing the variables and one or more parameters of the stochastic differential equation are optical pulses, this repetition is performed optically, using at least one optical cavity.Repeated propagation through the optical cavity and optoelectronic system recursively performs the same operations on the optical pulse, ultimately causing the amplitude and phase of the optical pulse to converge to values ​​that represent a solution to a stochastic differential equation (SDE) driven by a Lévy process. The solution to the SDE may be converted to a solution to an optimization problem or data points from a specific distribution in a machine learning model. In some cases, at each iteration, the optical pulse in the optical cavity is amplified using a phase-sensitive amplifier to compensate for amplitude degradation due to photon loss within the optical cavity. The optical cavity may be connected to other parts of the optoelectronic system using at least one optical coupler. In embodiments where the signals representing the variables and one or more parameters of the stochastic differential function are electronic signals, the iterations are performed by sampling and storing the output signal using a sample-and-hold circuit and feeding it back to the input of a computational and recursive component, such as computational and recursive component 100 disclosed with respect to FIG. 1 . The sample-and-hold circuit may also perform signal normalization to map the output signal to an appropriate range of input values. The same or another circuit may be used to amplify or shape the signal to compensate for signal losses within the system. The circuit may include capacitors, transistors, and be controlled by a clock. The clock may be triggered by a digital processing unit, such as an FPGA or digital microcontroller circuit. The clock may also be triggered by an analog ring oscillator circuit or a signal generated within an optoelectronic system. The ring oscillator circuit may include multiple stages and may include a current-limited inverter circuit. The repetition may also be performed by a switched-capacitor circuit, which stores the output signal in a capacitor and, after a predetermined delay, returns the signal to the input of a computational and recursive component, such as computational and recursive component 100 disclosed with respect to FIG. 1.

[0141] Referring to FIG. 3A, a diagram of an optoelectronic system for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process is shown, where the stochasticity of the Lévy process is represented by photon statistics of at least one optical mode, and the optoelectronic system comprises optical components.

[0142] The optoelectronic system includes at least one coupler 326, 328,..., 330 and a measurement module 310. The at least one coupler 326, 328,..., 330 comprises one or more optical components. The at least one coupler 326, 328,..., 330 can receive quantum optical modes through at least one input port 314, 316,..., 318 of the one or more optical components. The optical components can be of various types, such as beam splitters, directional couplers, phase shifters, amplifiers, squeezers, and nonlinear optical components. The at least one coupler 326, 328,..., 330 can be of various types. A variable coupler can have a variable transmission coefficient, meaning that the amount of optical field transmitted to a particular output of the coupler is variable and controllable. In some cases, at least one coupler 326, 328,..., 330 is a variable coupler, such as a Mach-Zehnder interferometer (MZI). The MZI consists of two 50:50 beam splitters and two phase shifters, and variable coupling is achieved by changing the phase shift on the phase shifters. Variable couplers may also be implemented in other ways, such as incorporating tabletop optical components. In some cases, at least one coupler 326, 328,..., 330 may be a fixed coupler, such as a beam splitter. A fixed coupler has a fixed ratio between the amplitudes of the coupler's two outputs. This ratio may be 50:50, meaning that equal amounts are transmitted to the two outputs. This ratio may be 20:80, meaning that the transmission rates between the coupler's two outputs are 20% and 80%, respectively. This ratio may also be any other fixed value.

[0143] The measurement module 310 includes homodyne detectors 320, 322,..., 324 that perform homodyne measurements of quantum optical modes. By measuring the intensity of the optical field and comparing its phase with a reference phase, the results of the homodyne measurements can be used to estimate the results of the quantum measurement in the position or momentum basis. The homodyne detectors may include a beam splitter, with the optical mode to be measured fed into one input and the optical field from a local oscillator (e.g., a laser source) fed into the other input. The outputs of the beam splitter are connected to two separate photodetectors, each measuring the intensity of the optical field. The output electronic signals from these photodetectors are subtracted by electronic devices to estimate the amplitude and phase of the optical mode. The measurement module 310 includes a measurement processing unit 312. The measurement processing unit 312 receives the electronic signals output from the homodyne detectors and can perform some processing on these signals. This processing can be of various types, including conversion from analog electronic signals to analog optical signals. It may be a conversion of an analog electronic signal to a digital electronic signal. The processing unit may also implement an identity function on the measurement result, i.e. the analog electronic signal is used as is without modification.

[0144] In some cases, the measurement module 310 includes a memory for storing measurement results. This memory may be electronically implemented in a digital or analog processing unit within the measurement module 310. The measurement module 310 may be operatively optically connected to at least one coupler 326, 328, . . . , 330 using one or more optical waveguides 332. The one or more optical waveguides 332 may be of various types. In some cases, the one or more optical waveguides 332 are comprised of optical fiber. In some cases, the one or more optical waveguides 332 are comprised of free space. In some cases, the one or more optical waveguides 332 are on-chip photonic waveguides.

[0145] In some cases, measurement module 310 may include other optoelectronic devices that convert optical energy into electronic signals for digital or analog processing. These devices can perform quantum measurements on the quartet components of quantum optical modes, similar to homodyne measurements.

[0146] In some cases, one or more optical waveguides 332 are arranged as a closed optical loop connecting at least one coupler 326, 328, ..., 330 and the measurement module 310. The output of the homodyne quantum measurement for the quantum optical mode may be fed back to at least one coupler 326, 328, ..., 330. One or more optical waveguides 332 may be connected directly to the output of the measurement module 310 instead of a closed optical loop. The output of the measurement module 310 may be connected to the input ports 314, 316, ..., 318 of the at least one coupler 326, 328, ..., 330.

[0147] In some cases, the optoelectronic system includes one or more optical cavities, including one or more optical waveguides 332 in a closed-loop configuration connecting at least one coupler 326, 328, ..., 330 to the measurement module 310. The output of the homodyne quantum measurement for the quantum optical mode may be fed back to one or more optical cavities. This is achieved by connecting the output of the homodyne measurement system to the optical cavity using another set of couplers arranged in a configuration similar to at least one coupler 326, 328, ..., 330 at the output of the measurement module. These couplers may have one input connected to the optical cavity and the other input connected to the output of the measurement module via optical waveguides. The outputs of these couplers may be connected to the optical cavity.

[0148] In some cases, the optical modes in one or more optical cavities may not be coherent. This is the case when squeezed states are injected into the cavity or when nonlinear elements are present in the system. In such cases, the extraction of a portion of the optical pulse in the optical cavity during the measurement process can introduce "kickback noise" into the remaining optical pulse. The larger the fraction of the optical pulse that is extracted, the greater the noise introduced.

[0149] In some cases, one or more optical cavities can be used as optical memories. Optical memories consist of optical cavities of appropriate length that can store optical pulses for a time proportional to the length of the optical pulses. These optical cavities may also include a phase-sensitive amplifier (PSA) to coherently amplify the optical pulses held within the cavity, thus compensating for optical losses during storage while keeping the pulses within the cavity.

[0150] In some cases, stochastic differential equations (SDEs) driven by Lévy processes have the following form: Dx(t) i =f(x(t) i ,t)dt+g(x(t) i ,t)dZ(t) i where Z(t) is a Lévy process. A Lévy process may be a Wiener process, Poisson process, Gamma process, Pascal process, Cauchy process, or Meissner process. A Lévy process may be any stochastic process that satisfies the mathematical definition of a Lévy process.

[0151] In some cases, the stochastic differential equation can be expressed in the following form: dx i =f(x i ,t)dt+g(x i ,t)dW i Here, dWdW is the differential of the Wiener random process. This type of stochastic differential equation is also called a forward stochastic differential equation. The first term, also called the drift coefficient, provides the drift force for the evolution of the stochastic process x(t). The diffusion coefficient g(x,t) provides randomness. The selection of the drift coefficient and diffusion coefficient constitutes the stochastic differential equation in this embodiment.

[0152] In some cases, the stochastic differential equation represents the Langevin dynamics consisting of gradient descent and a noise term, and can be written as follows: TIFF2025536240000006.tif998Here, the first term represents the gradient of the objective function and drives the variables towards a minimum of the objective function, while the second term makes the process stochastic, increasing the likelihood of finding a global minimum.Here, c is a vector of variables, σ is a constant, dW is the derivative of the Wiener stochastic process, N is the number of variables, and ∂ is the partial derivative with respect to the components of c.

[0153] In some cases, controllable noise in the quantum optical mode can be generated using quantum measurement of the quantum optical mode. A measurement system including measurement module 310 can introduce controllable noise into the pulse in the cavity by discarding the output optical pulse (e.g., for a noise-addition process) or can measure the output amplitude using homodyne detection devices 320, 322,..., 324. Measurement module 310 can perform quantum measurement on the quantum optical mode by extracting a portion of the optical pulse from the optical cavity using at least one coupler 326, 328,..., 330 and performing homodyne measurement on it (e.g., using homodyne detection devices 320, 322,..., 324). Assuming the pulse in the optical cavity is not coherent, this measurement process introduces backaction noise to the pulse remaining in the cavity. This controllable noise represents at least a portion of the randomness (i.e., stochasticity) in the stochastic differential equation. The pulses measured by the homodyne measurement system (e.g., homodyne detectors 320, 322, . . . , 324) have the following values: TIFF2025536240000007.tif1753, these values ​​provide the noise associated with each pulse, where: TIFF2025536240000008.tif87 is the measured pulse amplitude, μ i is the amplitude of the pulse inside the cavity, j is the measured intensity, dW i / dt is the randomness associated with each pulse. TIFF2025536240000009.tif87 or a function thereof is again injected into the input of at least one coupler 326, 328, . . . , 330, The randomness in the values ​​of TIFF2025536240000010.tif87 introduces randomness (i.e., probability) into the stochastic differential equations of the system. If the states in an optoelectronic system are coherent, the SDE can be written as: TIFF2025536240000011.tif1566Here, the argument of the function f is Since TIFF2025536240000012.tif87 is a random variable as described above, stochasticity is introduced through the second term.

[0154] In some cases, the quantum optical mode is in a squeezed state. The amount of squeezing in the squeezed state can control the controllable noise (i.e., stochasticity) of the quantum optical mode. Continuously injecting controllable squeezed states into an optoelectronic system can control the randomness in the stochastic differential equation. The greater the amount of squeezing, the greater the variance of the randomness added to the pulse. The squeezed state may be injected through at least one coupler 326, 328, ..., 330. The squeezed state may be generated externally using an optical squeezer and fed into at least one coupler 326, 328, ..., 330 via input ports 314, 316, ..., 318. This injection may occur continuously during each round trip of the pulse circulating within the optical cavity, or every few iterations.

[0155] In some cases, the measurement module 310 models the environment: quantum measurements may be weakly performed by the environment in the low photon count region, and in such cases, at least one coupler 326, 328, . . . , 330 represents photon loss to the reservoir.

[0156] In some cases, the measurement module 310 includes a digital processing unit to convert the measurement results into digital signals for processing, which may include a field programmable gate array (FPGA).

[0157] In some cases, the measurement module 310 includes an analog electronic processing unit that converts the measurement results into analog electronic signals for processing, which may include resistive random access memory (RRAM) devices.

[0158] Referring to FIG. 3B, a diagram of an optoelectronic system with a processing network is shown for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process, where the photon statistics of at least one optical mode represent the stochasticity of the Lévy process.

[0159] The optoelectronic system includes at least one coupler 48, 50,..., 52, the measurement module 30, and the processing network 34. The at least one coupler 48, 50,..., 52 includes one or more optical elements. The at least one coupler can receive the quantum optical mode through at least one input port 36, 38,..., 40 of the one or more optical elements. The optical elements can be of various types, such as beam splitters, directional couplers, phase shifters, amplifiers, squeezers, and nonlinear optical elements. The at least one coupler 48, 50,..., 52 can also be of various types. A variable coupler can have various transmission coefficients, such that the amount of optical field sent to a particular output of the coupler is variable and controllable. In some cases, the at least one coupler 48, 50,..., 52 is a variable coupler, such as a Mach-Zehnder interferometer (MZI). The MZI consists of two 50:50 beam splitters and two phase shifters, and tunable coupling can be achieved by changing the phase shift in the phase shifters. Variable couplers can also be implemented using other methods of combining tabletop optical elements. In some cases, at least one of the couplers 48, 50, ..., 52 is a fixed coupler, such as a beam splitter. A fixed coupler has a fixed ratio between the amplitudes of the coupler's two outputs. This ratio can be 50:50, which splits the power equally between the two outputs. This ratio can also be 20:80, such as a 20% / 80% transmission ratio between the two outputs. This ratio can be any fixed value.

[0160] The measurement module 30 includes homodyne detectors 42, 44,..., 46 that perform homodyne measurements on quantum optical modes. By measuring the intensity of the optical field and comparing its phase with a reference phase, the homodyne measurement results can be used to estimate the quantum measurement results in the position or momentum basis. The homodyne detector consists of a beam splitter that provides the optical mode of interest at one input and the optical field from a local oscillator (e.g., a laser source) at the other input. The beam splitter outputs are connected to two photodetectors, each of which measures the intensity of the input optical field. The output electronic signals of the photodetectors are subtracted by electronic devices to estimate the amplitude and phase of the optical mode. The measurement module 30 also includes a measurement processing unit 32. The measurement processing unit 32 receives the electronic signals output by the homodyne detectors and performs some processing on them. This processing can take various forms, including converting analog electronic signals to analog optical signals or converting analog electronic signals to digital electronic signals. The processing unit may perform an identity function on the measurement results, and the analog electronic signal is used as is without any processing.

[0161] In some cases, the measurement module 30 includes a memory for storing measurement results. The memory may be electronically implemented in a digital or analog processing unit within the measurement module 30. The measurement module 30 may be operatively optically connected to at least one coupler 48, 50, . . . , 52 using one or more optical waveguides 62. The one or more optical waveguides 62 may be of various types. In some cases, the one or more optical waveguides 62 are comprised of optical fiber. The one or more optical waveguides 62 may be free space. In some cases, the one or more optical waveguides 62 may be on-chip photonic waveguides.

[0162] In some cases, measurement module 30 includes other optoelectronic devices that convert optical energy into electronic signals for digital or analog processing. Such devices can perform quantum measurements on the number of quadrature components of quantum optical modes, as well as homodyne measurements.

[0163] In some cases, one or more optical waveguides 62 are wired in a closed-loop configuration, connecting at least one coupler 48, 50, ..., 52 to the measurement module 30. The output of the homodyne quantum measurement for the quantum optical mode may be fed back to at least one coupler 48, 50, ..., 52. One or more optical waveguides 62 may also be connected directly to the output of the measurement module, instead of creating a closed loop. Additionally, the output of the measurement module may be connected to one of the input ports 36, 38, ..., 40 of at least one coupler 48, 50, ..., 52.

[0164] In some cases, the optoelectronic system includes one or more optical cavities, including one or more optical waveguides 62 in a closed-loop configuration connecting at least one coupler 48, 50, ..., 52 to the measurement module 30. The output of the homodyne quantum measurement for the quantum optical mode may be fed back to these one or more optical cavities. This is done by connecting the output of the homodyne measurement system to the optical cavity through another set of couplers arranged similarly to the at least one coupler 48, 50, ..., 52 at the output of the measurement module. One input of these couplers is connected to the optical cavity, and the other input is connected to the output of the measurement module via an optical waveguide. The outputs of these couplers are connected to the optical cavity.

[0165] In some cases, measurement module 30 does not perform a quadrature measurement on the optical pulse. In such cases, the pulses extracted by at least one coupler 48, 50, ..., 52 may be fed directly to processing network 34, bypassing homodyne detectors 42, 44, ..., 46. In this case, the input pulses to processing network 34 are analog optical pulses.

[0166] In some cases, measurement module 30 does not transmit the results of the quartet component measurements to processing network 34. In such embodiments, homodyne detectors 42, 44, ..., 46 are used to measure and determine the values ​​of the variables of interest at the end of the process, while simultaneously feeding a portion of the optical pulse extracted from the optical cavity directly to processing network 34 via connection 60. By implementing this system configuration, all processing is performed optically, and homodyne detectors 42, 44, ..., 46 are used to evaluate the values ​​of the optical pulses at the end of the process.

[0167] In some cases, measurement module 30 is not connected to processing network 34. In such cases, connection 60 is broken. These embodiments may represent cases where measurement module 30 is used to evaluate the values ​​of the underlying problem variables at the end of the process, and the parameters of processing network 34 are either preset or evaluated at each step without using the value of the light pulse at each step.

[0168] In some cases, the optical modes within one or more optical cavities are not coherent. This is the case when squeezed states are injected into the cavity or when nonlinear elements are present in the system. In these cases, extracting a portion of the optical pulse from the optical cavity during the measurement process may introduce kickback noise into the portion of the optical pulse that remains in the cavity. The larger the fraction of the optical pulse that is extracted, the greater the noise introduced.

[0169] In some cases, one or more optical cavities may be used as optical memories. The optical memories may consist of optical cavities of appropriate length to hold optical pulses for a time proportional to the length of the optical cavity. These optical cavities may also include a phase-sensitive amplifier (PSA) that coherently amplifies the held optical pulses in the cavity to compensate for losses of the optical pulses while they are held in the cavity.

[0170] In some cases, the stochastic differential equation (SDE) driven by the Lévy process is of the form: dx(t) i =f(x(t) i ,t)de+g(x(t) i ,t)dZ(t) i where Z(t) is a Lévy process. Lévy processes may include Wiener, Poisson, Gamma, Pascal, Cauchy, or Meissner processes. A Lévy process may be any stochastic process that satisfies the mathematical definition of a Lévy process.

[0171] In some cases, the stochastic differential equation is of the form: dx i =f(x i ,t)dt+g(x i ,t)dW i where dW is the differential of the Wiener stochastic process. This type of stochastic differential equation is also called a forward stochastic differential equation. The first term, also called the drift coefficient, effectively provides a drift force for the evolution of the stochastic process x(t). The diffusion coefficient g(x,t) provides randomness. The choice of the drift coefficient and diffusion coefficient together is called the stochastic differential equation in this embodiment.

[0172] In some cases, the stochastic differential equation describes the Langevin dynamics involving gradient descent and a noise term, as follows: TIFF2025536240000013.tif797Here, the first term represents the gradient of the objective function, driving the variables towards a minimum of the objective function, while the second term makes the process stochastic, increasing the chances of finding a global minimum. Here, c is a vector of variables, σ is a constant, dW is the derivative of the Wiener stochastic process, N is the number of variables, and ∂ denotes the partial derivative of c with respect to its components.

[0173] In some cases, quantum measurements on quantum optical modes may be used to generate controllable noise for the quantum optical modes. A measurement system including measurement module 30 may introduce controllable noise into the pulses within the optical cavity by discarding the extracted optical pulses (e.g., for noise addition processing) or by measuring the extracted amplitude using homodyne detectors 42, 44,..., 46. Measurement module 30 may perform quantum measurements on the quantum optical modes by extracting a portion of the optical pulses from the optical cavity using at least one coupler 48, 50,..., 52 and performing homodyne detection on them (e.g., using homodyne detectors 42, 44,..., 46). Assuming the pulses within the cavity are not coherent, this measurement process introduces back-action noise into the pulses remaining within the cavity. This controllable noise represents at least a portion of the randomness (i.e., stochasticity) of the stochastic differential equation. The pulse measured by the homodyne measurement system (e.g., homodyne detectors 42, 44, . . . , 46, etc.) provides the following values: TIFF2025536240000014.tif1346 which provides the noise associated with each pulse. TIFF2025536240000015.tif77 is the measured pulse amplitude, μ i is the pulse amplitude inside the cavity, j is the measured intensity, and dWi / dt is the randomness associated with each pulse. When TIFF2025536240000016.tif77 or any function thereof is re-injected into the input of at least one coupler 48, 50, ···, 52, The inherent randomness in the values ​​of TIFF2025536240000017.tif77 introduces randomness (i.e., probability) into the system's stochastic differential equation. If the states in an optoelectronic system are coherent, the SDE can be expressed as: TIFF2025536240000018.tif1464Here, the argument of the function f is TIFF2025536240000019.tif77, so probability is introduced through the second term, TIFF2025536240000020.tif77 is a random variable as shown in the formula disclosed herein.

[0174] In some cases, the quantum optical mode is in a squeezed state. The amount of squeezing in the squeezed state can control the controllable noise (i.e., stochasticity) of the quantum optical mode. Continuously injecting controllable squeezed states into an optoelectronic system can control the randomness in the stochastic differential equation. The greater the amount of squeezing, the greater the variance of the randomness added to the pulse. The squeezed state may be injected using at least one coupler 48, 50, ..., 52. The squeezed state may be generated externally using an optical squeezer and fed into at least one coupler 48, 50, ..., 52 through input ports 36, 38, ..., 40. This injection may occur continuously every time the pulse circulates around the optical cavity, or every few iterations.

[0175] In some cases, the measurement module 30 models the environment: quantum measurements may be weakly performed by the environment in the low photon regime, and at least one coupler 48, 50, ..., 52 represents photon loss to the reservoir.

[0176] The SDEs described elsewhere that represent the processes underlying the optical devices and systems disclosed herein assume that the system is at zero temperature. While this may be physically difficult to achieve, it represents a good approximation of the conditions in real systems, such as the optoelectronic systems described herein with respect to Figures 3A, 3B, 3C, and 1. At non-zero temperatures, the underlying processes may contain noise due to the effects of quantum measurements made on the system by the environment, which is reflected in the corresponding SDEs.

[0177] In some cases, measurement module 30 includes a digital processing unit that converts and processes the measurement results into one or more digital signals, which may include a field-programmable gate array (FPGA).

[0178] In some cases, measurement module 30 includes an analog electronic processing unit that converts and processes the measurement results into one or more analog electronic signals, which may include a resistive random access memory (RRAM) device.

[0179] The optoelectronic system includes a processing network 34 operably connected to at least one coupler 48, 50, . . . , 52. The processing network 34 may generate at least one programmable optical mode. In some cases, the at least one generated programmable optical mode may represent at least a portion of a drift term of a stochastic differential equation. In some cases, the at least one generated programmable optical mode may be a coherent state, the amplitude of which may be modulated with a value representing the drift term. The coherent state may be generated using an intensity modulator (IM) that modulates an optical field from a local oscillator (LOO) with a value determined by the processing network 34. In some cases, the at least one generated programmable optical mode may be a squeezed state, the amplitude of which may be modulated with a value representing the drift term. The programmable squeezed state may be generated by combining the modulated coherent state with a squeezed vacuum state via a beam splitter or coupler. The squeezed vacuum state may be generated externally using an optical squeezer. In some cases, at least one generated programmable optical mode may be quantum, meaning that there is quantum entanglement between the optical modes representing one or more parameters or variables of the model. At least one generated programmable optical mode may be entangled with a measured optical mode amplitude that is a random variable. TIFF2025536240000021.tif77. In this case, these terms represent a nonlinear form of randomness (i.e., stochasticity) in the stochastic differential equation. They may be represented by g(xi, dW i, t) in the stochastic differential equation disclosed in connection with FIG. 3A.

[0180] In some cases, the processing network 34 may be operatively connected to the measurement module 30 using connections 60. These connections may be optical waveguides through which the output of the measurement processing unit 32 is converted into one or more optical signals. The connections 60 may be analog electronic connections using electronic waveguides or electrical connections. The connections 60 may be digital electronic connections using electrical connections or electronic waveguides. The at least one generated programmable optical mode may represent at least a portion of the stochasticity (i.e., randomness) of the stochastic differential equation. The at least one generated programmable optical mode may represent at least a portion of the stochasticity (i.e., randomness) of the stochastic differential equation. It may also represent a term that is a function of TIFF2025536240000022.tif77, TIFF2025536240000023.tif77 contains noise as shown in the equations disclosed herein in connection with FIG. 3A . Thus, when the at least one generated programmable optical mode is re-injected into one or more optical cavities, it may at least partially account for the stochasticity (i.e., randomness) of the stochastic differential equation. The re-injection into the one or more optical cavities may be performed via at least one coupler 54, 56, ..., 58. The at least one generated programmable optical mode may exhibit a drift term that is an identity function, meaning that the re-injection of the optical mode into one or more optical cavities does not change the optical mode within the one or more optical cavities. In some cases, the at least one generated programmable optical mode may be generated using the results of a homodyne measurement of a quantum optical mode. The value of the optical mode measured using the homodyne measurement is: The at least one generated programmable optical mode is represented by TIFF2025536240000024.tif77. The measured amplitude The at least one generated programmable optical mode may be any function of the measured amplitude. It may be a linear function of the measured amplitude. It may be a quadratic function of the measured amplitude. It may be a higher-order nonlinear function of the measured amplitude. The at least one generated programmable optical mode may be fed back to the system. In some cases, the at least one generated programmable optical mode may be fed back using at least one coupler 54, 56, ..., 58, where the at least one generated programmable optical mode is provided to one input port of the at least one coupler 54, 56, ..., 58, and the other input port of the at least one coupler 54, 56, ..., 58 is connected to one or more optical cavities. One output port of the at least one coupler 54, 56, ..., 58 is connected to one or more optical cavities. The at least one coupler 54, 56, . . . , 58 may be tunable so that the amount of the at least one generated programmable optical mode that is fed back into the optical cavity may be dynamically controlled.

[0181] The processing network 34 may include any combination of digital electronic components, analog electronic components, optoelectronic components, and optical components. The optical mode may be converted to a digital signal using an optical analog-to-digital converter (ADC) and then converted back to an optical mode using an optical digital-to-analog converter (DAC). The analog electronic signal may be converted to a digital signal using an electronic ADC and then converted back to an analog electronic signal using an electronic DAC. These converters may be implemented on-chip with the FPGA, or separate DAC or ADC devices may be used externally. The analog optical mode may be converted to an analog electronic signal and back again using optoelectronic components incorporating optoelectronic materials, such as lithium niobate, that change the optical properties of the optoelectronic material.

[0182] The analog electronic components may include resistive random access memory (RRAM) devices. The digital electronic components may include field programmable gate arrays (FPGAs). In some cases, processing network 34 includes one or more analog electronic components, one or more of which may be used to amplify or shape signals to compensate for losses occurring within the system.

[0183] In some cases, the processing network 34 may include nonlinear optical or electronic elements. These nonlinear elements can implement a nonlinear function on the input optical or electronic signal. The nonlinearity of these elements may be of the binary quadratic (square-power) type. The nonlinearity of these elements may also be of any other nonlinear type. In some cases, optical nonlinearity may be created using a nonlinear optical material (e.g., lithium niobate), which has strong second- or third-order nonlinearity. In some cases, optical nonlinearity may be created using a photodetector. The photodetector generates an electronic signal proportional to the intensity received by the optical mode, which in turn is proportional to the square of the optical field amplitude. In this way, an electronic signal that is a quadratic function of the optical mode amplitude is obtained. This electronic signal may be further processed by analog electronic components, converted to a digital electronic signal, or converted to an optical mode using an optical intensity modulator. Optical nonlinearity can also be created in other ways. In some cases, the nonlinearity may be created using an analog electronic circuit consisting of at least one of an analog multiplier, an amplifier, a potentiometer, and a function generator. The electronic components that make up such analog electronic circuits may include diodes, transistors, differential amplifiers such as operational amplifiers, and passive components such as resistors, inductors, and capacitors. Mathematical operations that such analog electronic circuits can perform to manipulate nonlinear functions of an input signal include addition, integration, inversion, multiplication, exponentiation, logarithm, and division. Nonlinear functions can also be generated using a crossbar array of memristor devices (or other RRAM devices) that performs matrix-vector multiplication between an input signal vector and a matrix of appropriate values ​​that are mapped to the conductivity states of the memristor devices (or other RRAM devices). The conductivity states of the memristor devices are programmable by electronic signals. Nonlinear outputs may be calculated using analog components such as differential amplifiers. Analog electronic signals can be converted to optical modes and vice versa using optoelectronic components. Using optoelectronic materials, these components enable the optical modes to be modulated by electronic signals.They can also generate analog electronic signals using the input optical modulus by the same process.

[0184] In some cases, the processing network 34 may include an MZI mesh comprising a network of MZIs. An MZI mesh is a network of MZIs that performs matrix-vector multiplication (MVM) by forming couplings between input optical modes using a network of connected MZIs. The MZI mesh can have any configuration, such as a triangular or rectangular configuration. Such an MZI mesh is reconfigurable by adjusting two phase shifters included in each MZI in the MZI mesh. The parameters of the MZI mesh may be adjusted continuously or sequentially over time. The triangular configuration of the MZI mesh can be implemented similarly to the approach shown in the paper "Experimental realization of any discrete unitary operator," Reck et al., Physical Review Letters 73.1 (1994):58, which is incorporated herein by reference for all purposes, and the rectangular configuration can be implemented similarly to the approach shown in the paper "Optimal design for universal multiport interferometers," Clements et al., Optica 3.12 (2016):1460-1465, which is incorporated herein by reference for all purposes.

[0185] In some cases, the processing network 34 may include a free-space analog optical matrix-vector multiplication (MVM) unit. Such an MVM unit may be generated from an array of intensity and phase modulators, where the amplitude and phase of free-space optical pulses are manipulated. The modulators may be electro-optical devices that can change the intensity and phase of the optical pulses passing through them by applying and varying electrical signals. N rows of optical pulses representing the underlying problem variables are split into N columns with identical amplitudes and phases of the optical pulses, forming an array of N × N optical pulses. These pulses are fed into an N × N modulator array. The outputs of the modulators may be collected into N columns by adding the optical pulses to each column of the N × N optical outputs to generate the MVM output as a vector of N optical pulses. The results may be fed back to other parts of the optoelectronic system via couplers. The modulator parameters may be determined by the elements of the matrix used in the MVM operation.

[0186] In some cases, processing network 34 may be used to generate squeezed states of quantum optical modes. The amount of squeezing of the squeezed states controls the controllable noise of the quantum optical modes. Squeezed states generated using processing network 34 may have their mean field amplitude modulated with a value representing a drift term. In some cases, the controllable noise of squeezed states generated using processing network 34 may be controlled by the amount of squeezing of the optical modes. Programmable squeezed states are generated by combining modulated coherent states and squeezed vacuum states using a beam splitter or coupler. Squeezed vacuum states may be generated externally using an optical squeezer. The combined squeezed vacuum state and modulated coherent state may be fed into at least one optical cavity using couplers 54, 56, . . . , 58.

[0187] In some cases, processing network 34 may be used to implement an arbitrary function. If the function is a linear function of the input variables, the function may be implemented using linear operations. If the function is nonlinear, the nonlinear function may be implemented exactly or approximately using a combination of linear and nonlinear operations as disclosed herein for processing network 34. The arbitrary function implemented may be the gradient of an objective function representing the optimization problem. The arbitrary function implemented may be any function of the gradient of the objective function. The arbitrary function may be a function of variables in a previous step of Euler's method.

[0188] In some cases, processing network 34 may include at least one optical cavity to store the optical pulses and use them to compute the implemented arbitrary function, with the optical cavity storing the optical pulses for as many round trips as necessary before being used in the next step of the Euler method.

[0189] In some cases, the drift term of the stochastic differential equation may include a drift term of an inverse stochastic differential equation that represents inverse diffusion. In some cases, the drift term represents an optimization problem. In such cases, the drift term may represent an objective function. In some cases, the drift term may represent the gradient of the objective function. In some cases, the drift term may represent a function of the gradient of the objective function. In some cases, the drift term may represent a time-dependent function of the gradient of the objective function that depends on past time. The drift term may depend on other quantities or the output of other processes.

[0190] In some cases, the optimization problem is a quadratic programming problem with a box constraint. In some cases, the box constraint of the problem is implemented using natural properties of optical devices. In some cases, the box constraint is implemented using a processing network 34.

[0191] In some cases, the optimization problem is a maximum-weighted independent set (MWIS) problem and is implemented using a continuous variable representation. In some cases, natural properties of optical devices are used to implement the box constraints of the MWIS problem. In some cases, processing network 34 is used to implement the box constraints.

[0192] In some cases, the optimization problem is a quadratic assignment problem (QAP). In some cases, natural properties of the optical device are used to implement the QAP constraints. In some cases, processing network 34 is used to implement the QAP constraints.

[0193] In some cases, the processing network 34 includes a function approximator having one or more parameters and receiving one or more variables as input. The one or more parameters may be analytically determined and set prior to processing. The one or more parameters may be continuously and continuously determined by training the processing network 34. The one or more input variables may be received as input signals. The one or more input variables may be received as digital signals for a function approximator constructed using a digital processor (e.g., an FPGA). The one or more input variables may be received as electronic signals for a function approximator constructed using optical or analog electronic circuitry. The one or more input variables may be received as optical inputs for a function approximator constructed using optical components. The one or more input variables of the function approximator may be received as optical signals. The one or more input variables of the function approximator may also be received as electronic signals derived from the output of a measurement system. The function approximator may be trained to approximate a score function of a machine learning model. The function approximator may be capable of implementing the exact mathematical form of the score function or a polynomial approximation thereof. The score function sθ is the gradient of the log-likelihood of the probability density function (random variables, i.e., data) and is defined as: s θ (x,t)=∇x log p(x,t) Since this is a gradient, it measures the change in log-likelihood for infinitesimal changes in the data. Obtaining an accurate score function therefore allows us to drive the stochastic process towards maximizing log-likelihood, producing the most likely samples, which is the goal in generative machine learning.

[0194] In some cases, a function approximator is trained to approximate a function that represents the logarithm of the probability distribution over the data of the machine learning model. In such an embodiment, the function approximator does not represent the score function itself, but rather an additional step of evaluating the gradient (e.g., via finite differences) is used to compute the gradient, and thus the score function. This embodiment has the advantage of allowing direct access to the likelihood function, which may be desirable for modeling purposes in its own right.

[0195] In some cases, the function approximator is trained to approximate the gradient of a function that represents the logarithm of the probability distribution over the data of the machine learning model. In such embodiments, the output of the function approximator can be considered directly as the score function, without the additional processing required to evaluate the gradient.

[0196] In some cases, the function approximator includes a neural network. In some cases, the neural network is used to approximate and generate drift terms in stochastic differential equations. The neural network can take various forms. In some cases, the neural network is an optical neural network implemented using optical elements. The optical neural network may include any or a combination of linear and nonlinear optical elements, such as phase shifters, beam splitters, phase-sensitive amplifiers, squeezers, and other linear and nonlinear optical elements. In some cases, the neural network may be an analog optoelectronic system implemented using a combination of linear optical elements, such as beam splitters, phase shifters, and optical squeezers, and nonlinear optical elements, such as phase-sensitive amplifiers and other nonlinear elements using nonlinear optical crystals or optoelectronic materials. The parameters of these components can be externally adjustable by controlling electronic signals that control the optical properties of these components. In some cases, neural networks are analog electronic systems implemented using a crossbar array of resistive memory (or other RRAM devices) combined with analog electronic elements such as transistors, operational amplifiers, transimpedance amplifiers, analog switched capacitor circuits, analog passive delay lines, and other resistive and capacitive elements. In some cases, neural networks are analog electronic systems. Analog electronic systems implementing neural networks may include circuits that provide nonlinearities. These nonlinearities may be generated using analog electronic circuits as disclosed elsewhere herein. In some cases, neural networks are implemented using a combination of analog electronic elements and analog optical elements. The analog components may be any combination of optical and electronic components as disclosed elsewhere herein. Optical modes can be converted to electronic signals, and vice versa, via transducers constructed using optoelectronic materials.

[0197] In some cases, the neural network is digital, implemented using a digital computing device (e.g., a digital computer, FPGA, etc.). The digital computer can be of various types. The parameters of the neural network can be pre-programmed values ​​(e.g., during inference) or dynamically controlled using external parameters such as the results of a measurement system (e.g., during training).

[0198] In some cases, neural networks are analog electronic systems. Analog electronic systems that implement neural networks include circuits that provide nonlinearities. These nonlinearities may be generated using the analog electronic circuits described elsewhere herein.

[0199] In some cases, measurements from an optoelectronic system are interpreted as a latent encoding of input data for use in a machine learning method. In such cases, the optoelectronic system is initialized with input data. The optoelectronic system is then evolved according to a stochastic differential equation that describes the dynamic behavior of the optical device. The state of the optical device is then measured. This measurement is a probabilistic transformation of the original input data, and the transformation is performed by the optoelectronic system. This transformed input data is saved and used as input to the machine learning method. For example, it may be used to train a score function approximator in a diffusion model of generative machine learning.

[0200] In some cases, an optoelectronic system may include multiple processing networks, each including a function approximator. These processing networks may be connected using one or more optical cavities. The processing networks may be connected to the same one or more optical cavities using at least one coupler, such as at least one coupler 48, 50,..., 52. The function approximators may use measurements of the same optical mode in the same one or more optical cavities by performing homodyne measurements on the same optical mode present in the same one or more optical cavities. One or more outputs of the function approximators may be connected to the same one or more optical cavities using at least one coupler, such as at least one coupler 54, 56,..., 58. All processing networks connected to one or more optical cavities may modify the optical mode in the one or more optical cavities.

[0201] Referring to FIG. 3C, a diagram of an optoelectronic system for simulating at least one solution of a stochastic differential equation (SDE) driven by a Lévy process in which stochasticity is represented by photon statistics in at least one optical mode is shown, the optoelectronic system comprising two processing networks and two measurement modules with different functions within a machine learning model.

[0202] The first measurement module 3010 and processing network 3014 are used to train the generative diffusion model. During training, parameters of the processing network are learned to implement a function approximator for the generative diffusion model. The second measurement module 3016 and processing network 3020 are used for the generation portion of the machine learning model. Using the parameters obtained from the first processing network 3014, the second processing network 3020 is prepared to generate samples with a distribution similar to the training data.

[0203] In some cases, this means that feedback generated from first processing network 3014 is not re-injected into one or more optical cavities containing one or more optical waveguides 3070, and at least one coupler 3052, 3054, ..., 3056 may be omitted. In such cases, first processing network 3014 is trained without updating the optical pulses in one or more optical cavities.

[0204] In some cases, at least one of measurement modules 3010 and 3016 may not perform homodyne measurements, in which case homodyne detection devices 3034, 3036,..., 3038 or 3040, 3042,..., 3044 are bypassed and the optical pulses are input to processing network 3014 or 3020 directly through connection 3072 or 3074, which may represent an optical waveguide.

[0205] In some cases, the optical pulses in one or more optical cavities may not be measured by the second measurement module 3016. In such cases, the second measurement module 3016 and at least one coupler 3058, 3060,..., 3062 may be omitted, and the second processing network 3020 may generate samples and provide them to one or more optical cavities using at least one coupler 3064, 3066,..., 3068.

[0206] Referring to FIG. 4, a flowchart of a method for generating approximate samples from a distribution of training data is shown.

[0207] In process operation 402, parameter values ​​of a trained function approximator are obtained. The function approximator represents, at least in part, a machine learning model. The parameter values ​​may be obtained by training. Training may be performed classically and non-optically, or may be performed using an optoelectronic system. In some embodiments, training is performed by simulating an optical device. In some cases, training may include performing machine learning techniques such as backpropagation and gradient descent. Training may be performed optically using an additional optoelectronic system specifically designed and implemented to perform backpropagation and gradient descent optically. The type of function approximator may vary. The function approximator may be any function approximator disclosed elsewhere herein, such as the function approximators disclosed in connection with FIGS. 3B, 3C, and 1. In some cases, the trained function approximator approximates a score function of a machine learning diffusion model. The trained function approximator may approximate a function representing the logarithm of the probability distribution of the data of the machine learning model. The trained function approximator may approximate the gradient of a function that represents the logarithm of the probability distribution of data in a machine learning model. In some cases, the function approximator may include at least one of an artificial neural network, a derivative architecture of an artificial neural network, a convolutional neural network (CNN), a transformer network, or a graph neural network (GNN). In some cases, the function approximator may be an optical neural network. The function approximator may be trained using machine learning techniques. See, for example, Ho et al., “Denoising diffusion probabilistic models,” 2020, rXiv:2006.11239, which is incorporated herein by reference in its entirety, which sets the standard for diffusion models and uses a convolutional U-Net architecture that includes downsampling layers, residual connections, and upsampling layers to efficiently utilize spatial image data.

[0208] Techniques such as score matching may be used to train function approximators, as disclosed, for example, in Song et al., “Score-Based Generative Modeling through Stochastic Differential Equations,” 2020, ArXiv:2011.13456, incorporated herein by reference in its entirety.

[0209] The score matching technique may be implemented as follows: θ Let (x,t) represent the output of the function approximator, which is the data distribution is known as the score of TIFF2025536240000026.tif648. Here, a simple objective function may be used to minimize the Euclidean distance between the function approximator and the score of the true data distribution, i.e., the Fisher divergence, given by: TIFF2025536240000027.tif1178

[0210] However, the true score function sdata(x,t) is unknown. Through mathematical manipulation, we obtain the following equivalent formula: TIFF2025536240000028.tif11108Using this formula does not require knowledge of the true scores of the data distribution, but it does involve the derivative of the output of the function approximator, which can be computationally expensive for high-dimensional data x.

[0211] Sliced ​​score matching, disclosed by Song et al. in "Sliced ​​Score Matching," 2019, ArXiv:1905.07088, which is incorporated herein by reference in its entirety, overcomes the problem of high computational cost by comparing random projections of scores, making it easy to compute. However, this method is an approximation of the objective function mentioned above. The objective function for sliced ​​score matching is given as follows: TIFF2025536240000029.tif12121

[0212] where v~pv is a random projection vector with the same dimensions as the data, sampled from a Rademach distribution (i.e., a distribution with equal probability of -1 and 1). In other cases, the projection vector may be sampled from other distributions, such as a uniform distribution. This objective function can be computed and used to train a core function approximator using backpropagation and gradient descent. This is a supervised learning technique.

[0213] To obtain the training data, a stochastic differential equation (also called a forward stochastic differential equation) of the following form is used: dxi=f(xi,t)dt+g(xi,t)dWi Here, the first term is called the drift coefficient and effectively imposes a drift force on the evolution of the stochastic process x. The diffusion coefficient g(x,t) imposes randomness. One may choose drift and diffusion coefficients that are easy to model mathematically, such as the following stochastic differential equation used by Song et al. in "Score-Based Generative Modeling through Stochastic Differential Equations," 2020, ArXiv:2011.13456, which is incorporated herein by reference in its entirety: TIFF2025536240000030.tif1372Alternatively, drift and diffusion coefficients may be used to describe the evolution of a physical system, such as an optoelectronic system.

[0214] During training, x is initialized to a sample from a training data set. The differential equation is then evolved to time t. In some cases, this is performed by evolving an optoelectronic system, such as the optoelectronic systems disclosed herein in connection with FIGS. 3A, 3B, 3C, and 1. The state of the optoelectronic system may be recorded, i.e., x at time t and time t. The parameter θ of the score function approximator may be updated based on an objective function (e.g., Fisher divergence) as disclosed elsewhere herein.

[0215] For every forward stochastic differential equation, there exists a backward stochastic differential equation, given by: dx=[f(x,t)-g 2 (t)s(x,t)]dt+g(t)dW

[0216] Once, score approximation Once TIFF2025536240000031.tif648 is trained, it is possible to derive the stochastic process into a training distribution by expanding this backward stochastic differential imputation formula.

[0217] 4, in process operation 404, the learned function approximator is implemented on an optoelectronic system using the obtained parameter values. The optoelectronic system may include at least one input port for receiving at least one optical mode, computational and recursive components, and an output port for readout. The optoelectronic system may take various forms. The optoelectronic system may be any of the optoelectronic systems disclosed elsewhere herein, such as those disclosed in connection with FIGS. 3A, 3B, 3C, and 1.

[0218] In some cases, the learned function approximator is implemented on a computational and recursive component, such as computational and recursive component 100 disclosed in connection with FIG. 1 . The learned function approximator may also be implemented on a processing network, such as processing network 34 disclosed herein in connection with FIG. 3B or processing network 3014 or 3020 disclosed herein in connection with FIG. 3C . The learned function approximator may be implemented on a computational and recursive component or network by mapping parameters of the learned function approximator to parameters of the computational and recursive component or processing network. For analog electronic or optical computational and recursive components or processing networks, the mapping is performed by finding parameters of the computational and recursive component or processing network such that the mathematical form of the generated output signal approximates the learned function approximator. For digital computational and recursive components or processing networks, the learned function approximator may be implemented directly in the computational and recursive component or processing network because digital computational and recursive components or processing networks can implement any mathematical function. Computational and recursive components or nonlinear components (e.g., electronic, optical, or optoelectronic) within the processing network may be deployed to more accurately implement the learned function approximator. Because the learned function approximator is generally a nonlinear function, incorporating nonlinear components within the processing network may be necessary to implement an operational function approximator.

[0219] Referring to FIG. 4 and in accordance with process operation 406, an optoelectronic system is executed to simulate a solution to at least one stochastic differential equation (SDE) driven by a Levy process that represents a distribution of training data for a machine learning (ML) model. The optoelectronic system may include at least one input port for receiving at least one optical mode, computational and recursive components, and an output port for readout. The optoelectronic system may be of various types. The optoelectronic system may be any of the optoelectronic systems disclosed elsewhere herein, such as the optoelectronic systems disclosed herein with respect to FIGS. 3A, 3B, 3C, and 1.

[0220] The optoelectronic system can be deployed to perform a method for simulating the solution of at least one stochastic differential equation (SDE) driven by a Levy process as disclosed herein with respect to FIG.

[0221] An optoelectronic system can be evolved to implement the method disclosed herein with respect to FIG. 8, which implements dynamics described by a stochastic differential equation. The system states are initialized to random noise, and an inverse stochastic differential equation is used to evolve the system. At each iteration, a function approximator provides a drift correction, which guides the evolution of the system toward a predetermined training data distribution.

[0222] The machine learning model may be a machine learning diffusion model. The stochastic differential equation representing the distribution of the training data of the machine learning model may be of various types.

[0223] In one or more embodiments, the stochastic differential equation may be an inverse stochastic differential equation, such as the stochastic differential equation disclosed herein with respect to process step 402.

[0224] The training data distribution for the machine learning diffusion model may be of various types. In one or more embodiments, the training data distribution may be a dataset of images. In some examples, the training data distribution may be a dataset of handwritten digits, a dataset of images of celebrities, or a dataset of music recordings.

[0225] A machine learning diffusion model can describe multiple components working together to provide a framework for generative machine learning. One component is a stochastic process that gradually adds noise to the distribution of training data. In some cases, the stochastic process is represented by a forward stochastic differential equation as disclosed elsewhere herein. The stochastic differential equation may be implemented using a computational model. Alternatively, the stochastic differential equation may be implemented on an optoelectronic system by utilizing physical internal functions. The optoelectronic system may take a variety of forms. The optoelectronic system may be any optoelectronic device or system disclosed elsewhere herein, such as the optoelectronic systems disclosed in connection with FIGS. 3A, 3B, 3C, and 1.

[0226] The second component of the machine learning diffusion model is a function approximator. The function approximator may be of various types. The function approximator may be any function approximator disclosed elsewhere herein, such as the function approximators disclosed in connection with FIGS. 3B, 3C, and 1 in process step 402. In some cases, the function approximator includes an artificial neural network, a derivative architecture of an artificial neural network, a convolutional neural network, a transformer neural network, a graph neural network, etc. In some cases, the function approximator is an optical neural network. In some cases, the function approximator is an analog electronic neural network. The function approximator is trained using machine learning techniques. A convolutional neural network may include any number of several types of computational layers, including, but not limited to, convolutional layers, upsampling layers, downsampling layers, activation layers, copy layers, residual layers, fully connected layers, etc. The combination of these layers and the specific selection of hyperparameters constitutes the neural network architecture.

[0227] The third component of a machine learning diffusion model is the reverse stochastic process. For the forward stochastic differential equation above, there is always a corresponding inverse stochastic differential equation (also called backward SDE), which can be written as: dx=[f(x,t)-g 2 (t)s(x,t)]dt+g(t)dW

[0228] Once, the score function approximation Once TIFF2025536240000032.tif648 is trained, the inverse stochastic differential equation can be developed to guide the process to the distribution of the training data. This inverse stochastic differential equation is implemented using a function approximator. More precisely, the term s(x,t) is provided by the function approximator disclosed elsewhere herein.

[0229] Continuing with reference to FIG. 4, in process step 408, the results are reported to generate an approximate sample from the distribution of training data. A readout is performed at the end of the system evolution to represent the approximate sample from the distribution of training data. In some cases, a homodyne measurement is performed at the end of the system evolution by measuring the amplitude of the quantum optical mode to represent the approximate sample from the distribution of training data.

[0230] In some cases, prior to process step 402, a function approximator is trained to obtain parameter values.

[0231] Referring to FIG. 5, a flow chart of a method for training a function approximator is shown.

[0232] According to process operation 502, an iteration number t ranging from 0 (zero) to a maximum iteration number T is obtained. The maximum iteration number T can be determined empirically. The maximum iteration number T may be large enough so that all information in the data sample is destroyed through the forward stochastic process and the terminal state x(T) of the system is easily a sample from the sampled prior distribution. In some cases, the prior distribution may be Gaussian. The iteration number t may be uniformly sampled between 0 and T, or may be sampled using a sampling technique such as importance sampling for performance purposes. When importance sampling is used, the sampling distribution from which t is drawn is weighted by a function proportional to the error in the score prediction.

[0233] Continuing with reference to FIG. 5 , according to process operation 504, the optoelectronic system is initialized with samples of training data for the machine learning model. More specifically, in some cases, the initial state x(0) of the system is set to a sample from a training dataset, i.e., sampled from a training distribution. The signal representing the training data sample may be an analog optical or electronic signal. In the case of an analog optical signal, an optical pulse may represent the initial state of the system. The initial optical pulse may include a vacuum state, a squeezed state, a coherent state, or other optical quantum state. In the case of an analog electronic signal, the variable representing the initial state may be encoded as a voltage or current signal having a predetermined amplitude, duration, phase, or waveform. These optical or electronic signals representing the initial state of the system may be generated within a computational and recursive component, such as computational and recursive component 100 shown herein in connection with FIG. 1 , or may be generated externally and provided to the system via an input port, as described elsewhere herein. The optoelectronic system may be of various types. The optoelectronic system may be any optoelectronic system disclosed elsewhere herein, such as the optoelectronic systems disclosed herein in connection with Figures 3A, 3B, 3C, and 1.

[0234] In some cases, multiple optical modes are used with photon statistics that at least partially represent the stochasticity of the Lévy process, which drives a stochastic differential equation (SDE). The optical modes may be of various types. In some cases, the optical modes may be quantum optical modes. In some cases, the optical modes may include squeezed states. In some cases, the optical modes may include vacuum states. In some cases, the optical modes may include coherent states. The optical modes may be vacuum states converted into squeezed states. The vacuum states may be injected by leaving at least one input port open, as described elsewhere herein. An optical vacuum state may be injected into the system by leaving at least one input port open. The squeezed vacuum state may be generated externally using a squeezing device that receives intense coherent light (e.g., from a laser light source) and generates the squeezed vacuum state. The output of the squeezing device may then be fed to at least one input port of the optoelectronic system. A squeezed vacuum state may have a larger uncertainty in one of the quadruplet components of the optical field (e.g., the real component of the optical field) and a smaller uncertainty in the conjugate quadruplet component. This is a purely quantum effect, and the uncertainty in one of the quadruplet components may be below the minimum uncertainty. Compared to the vacuum state, the squeezed vacuum state has a larger quantum noise, allowing for greater control over the stochasticity of the Lévy process compared to injecting a vacuum state. Coherent states of light may also be injected into at least one input port of an optoelectronic system. Coherent states are classical optical states with a minimal quantum noise limit, and stochasticity can be generated using photon statistics in the optical mode. Coherent states may be generated using an external laser source and fed into at least one input port of the optoelectronic system. The amount of squeezing in the squeezed state can control the stochasticity (i.e., noise) of the quantum optical mode. Continuously injecting controllable squeezed states into an optoelectronic system allows for control over the randomness in stochastic differential equations (SDEs).The greater the amount of squeezing, the greater the dispersion of randomness added to the pulse. This injection may occur continuously within the optoelectronic system, either after each round trip of the pulse or after several repetitions. In some cases, multiple optical modes are programmable. The optical modes may also be quantum, meaning that quantum entanglement exists between them.

[0235] In some cases, multiple quantum optical modes may be injected as squeezed vacuum states using a squeezer or as coherent states using an external laser source. These laser sources operate at the same wavelength as the rest of the optoelectronic system. Pulses with different amplitudes can be generated using a single laser source by dynamically controlling the power of the laser output or by sequentially using intensity and phase modulators to generate optical pulses with desired initial amplitudes and phases. The pulse trains generated by the laser may be synchronized using a delay line network so that all pulses on different optical lines arrive at an optoelectronic system such as those disclosed in connection with Figures 3A, 3B, 3C, and 1. These optical lines may be connected to the optoelectronic system using optical cavities similar to those disclosed in connection with Figures 3A, 3B, 3C, and 1. The optical pulses generated by the laser all belong to the same optical mode but with different parameters. Each optical pulse is an independent optical mode from the other optical pulses, but all share the properties of the same optical mode generated using the same laser source or the same squeezer.

[0236] 5 , in accordance with process operation 506, the optoelectronic system is operated iteratively t times to simulate a solution of at least one stochastic differential equation representing a machine learning (ML) model up to t. A forward form of the stochastic differential equation is used to evolve the system.

[0237] In some cases, the optoelectronic system is deployed to perform the method disclosed herein in connection with FIG. 2 for simulating the solution of at least one stochastic differential equation (SDE) driven by a Levy process.

[0238] In some cases, the optoelectronic system is deployed to perform the method disclosed herein in connection with FIG. 8 for implementing dynamic behavior described by stochastic differential equations.

[0239] In some cases, the amplitude and phase of the optical pulses are converted to values ​​that represent a prior distribution, such as random Gaussian noise with mean zero. A value representing a system snapshot used for training, i.e., x(t), may be obtained by measuring the amplitude of the quantum optical mode at repetition number t using homodyne measurements. A value representing a system snapshot used for training, i.e., x(t), may be obtained by performing the readout process disclosed herein in connection with FIGS. 2 and 1.

[0240] In some cases, the stochastic differential equation is given by: dx=f(x,t)dt+g(t)dW where the first term, also known as the drift coefficient, effectively imposes a drift force on the evolution of the stochastic process x(t). The diffusion coefficient g(t) provides randomness. Mathematically tractable drift and diffusion coefficients can be chosen, as described in Song et al., "Score-Based Generative Modeling through Stochastic Differential Equations," 2020, ArXiv:2011.13456, which is incorporated herein by reference for all purposes. In another embodiment, drift and diffusion coefficients that describe the evolution of a physical system, such as an optoelectronic system, may be used.

[0241] This forward stochastic differential equation is used to generate one or more training signals for the function approximator.

[0242] The training data distribution for a machine learning diffusion model may be of various types. In some cases, the training data distribution may be a dataset of images. In some cases, the training data distribution may be a dataset of handwritten digits, a dataset of images of celebrities, or a dataset of music recordings.

[0243] A machine learning diffusion model may describe multiple components that work together to provide a generative machine learning framework. One of the components is a stochastic process that gradually adds noise to the distribution of training data. In some cases, this stochastic process may be represented by a forward stochastic differential equation as disclosed elsewhere herein. The stochastic differential equation may be implemented using a computational model. Alternatively, the stochastic differential equation may be implemented on an optoelectronic system using the system's physical internal functions. The optoelectronic system may be of various types. The optoelectronic system may be any optical device or system disclosed herein, such as the optoelectronic system disclosed herein in connection with FIGS. 3A, 3B, 3C, and 1.

[0244] The second component of the machine learning diffusion model is a function approximator. The function approximator may be of various types. The function approximator may be any function approximator disclosed elsewhere herein, such as the function approximators disclosed herein in connection with processing operation 402 of FIG. 4 and FIGS. 3B, 3C, and 1. In some cases, the function approximator includes at least one of an artificial neural network, a derivative architecture of an artificial neural network, a convolutional neural network, a transformer neural network, and a graph neural network. In some cases, the function approximator is an optical neural network. In some cases, the function approximator is an analog electronic neural network. The function approximator may be trained using machine learning techniques. A convolutional neural network may include any number of several types of computational layers, including, but not limited to, convolutional layers, upsampling layers, downsampling layers, activation layers, copy layers, residual layers, and fully connected layers. The combination of these layers and the selection of hyperparameters comprise the architecture of the neural network.

[0245] The third component of the machine learning diffusion model is the inverse stochastic process. The general form of the forward stochastic differential equation disclosed herein has the inverse stochastic differential equation (also called backward stochastic differential equation), given by: dx=[f(x,t)-g 2 (t)s(x,t)]dt+g(t)dW

[0246] First, approximate the score function Once TIFF2025536240000033.tif648 is trained, this inverse stochastic differential equation can be evolved to guide the process toward the training distribution. This inverse stochastic differential equation may be implemented using a function approximator. More specifically, the s(x,t) term is provided by a function approximator disclosed elsewhere herein.

[0247] Continuing with reference to FIG. 5 , according to processing operation 508, the state of the system after t iterations (x(t), t) is output, and the parameters θ of the score function approximator are updated according to the objective function. This may be performed using backpropagation to calculate partial derivatives of the objective function with respect to each parameter θ of the function approximator. Small optimization steps based on gradient descent may then be performed to modify θ in a direction that minimizes the objective function. Gradient descent may include any update rule, such as stochastic gradient descent or Adam, as described by Kingma & Ba in “Adam: A Method for Stochastic Optimization,” 2014, arXiv:1412.6980, which is incorporated herein by reference for all purposes.

[0248] The state of the system may be observed by performing homodyne measurements on the quantum optical modes of the optoelectronic system. These measurements can provide amplitude and phase values ​​of the optical pulses by comparing the optical pulses to a reference light source, such as an external laser, that generates the optical pulses. These values ​​are interpreted as representing a random process that is used as data points in the training process and are considered training samples used to train the function approximator. The state of the system may be observed by performing the readout process described herein in connection with FIG. 2.

[0249] Referring to FIG. 6, a flowchart of a method for generating samples from a distribution of training data for a generative machine learning model is shown, where the method uses a hybrid optoelectronic system having an optical device coupled with an electronic score function approximator. The hybrid optoelectronic system may be of various types. The hybrid optoelectronic system may be any hybrid optoelectronic system disclosed elsewhere herein, such as the hybrid optoelectronic systems disclosed in connection with FIGS. 3A, 3B, 3C, and 1. The optical device may also be of various types. The optical device may be any optical device disclosed elsewhere herein, such as the optical device disclosed in connection with FIGS. 3A, 3B, 3C, and 1. The parameters of the electronic score function approximator are considered fixed and may be determined through a process called “learning,” as disclosed elsewhere herein. The electronic score function approximator may be of various types. The electronic score function approximator may be any score function approximator including electronic components disclosed elsewhere herein. In some cases, the electronic score function may include the use of a field programmable gate array (FPGA). The electronic score function approximator may be a digital neural network. In some cases, the electronic score function approximator may include the use of a crossbar array of memristor devices (or other RRAM devices). The electronic score function approximator may include an analog neural network.

[0250] All forward stochastic differential equations of the general form disclosed elsewhere in this specification have an inverse stochastic differential equation (also called backward stochastic differential equation), which can be expressed as: dx=[f(x,t)-g 2 (t)s(x,t)]dt+g(t)dW

[0251] score function Once an approximation of TIFF2025536240000034.tif648 is learned, an inverse stochastic differential equation can be developed to guide the process toward the learned distribution. The inverse stochastic differential equation may be implemented using an electronic score function approximator. More specifically, the term s(x,t) is provided by the electronic score function approximator disclosed herein.

[0252] Continuing with reference to FIG. 6 , in accordance with process operation 602, the optoelectronic system is initialized to a state representing a prior distribution. The prior distribution is determined by the terminal state of a forward stochastic process (described by a forward stochastic differential equation) used to train the electronic score function approximator. The forward stochastic process may be selected so that the prior distribution is easily sampleable. In some cases, the prior distribution may be a zero-mean Gaussian distribution. The optoelectronic system may be of various types. The optoelectronic system may be any of the optoelectronic systems disclosed elsewhere herein, such as those disclosed in connection with FIGS. 3A, 3B, 3C, and 1. The optical device is initialized with background vacuum noise upon startup of the optoelectronic system to generate zero-mean Gaussian states. These vacuum states exist within the optical waveguide of the optoelectronic system.

[0253] According to process operation 604, a maximum number of time steps N is selected. This value may be chosen to be the same as the number of time steps used to train the electronic score function approximator.

[0254] In accordance with process operation 606, a time step counter is initialized to zero.

[0255] In accordance with process operation 608, the time step counter is incremented by one.

[0256] According to process operation 610, an optoelectronic system is developed and its state is measured. To develop the optoelectronic system, a train of optical pulses may be generated, representing random variables as initial conditions for the dynamic behavior described by the stochastic differential equation. These pulses may be generated as a vacuum state, as a squeezed vacuum state using a squeezer, or as a coherent state using an external laser source. These pulses then pass through a network of optical components, such as beam splitters, directional couplers, phase shifters, amplifiers, squeezers, and other linear and nonlinear optical elements. The amplitude of the pulses, representing the random process of the stochastic differential equation, may be obtained at the end of the process using homodyne measurements.

[0257] According to processing operation 612, the measured conditions are converted into a digital representation in the form of a training data set. In some cases, the digital representation is an image. In some cases, the digital representation may include other structured data. The conversion to the digital representation may be performed using an analog-to-digital converter (ADC). The ADC may be located within a homodyne detection device.

[0258] According to decision operation 614, if the value of the time step counter exceeds the maximum number of time steps N, processing operation 616 is performed, and if the value of the time step counter is less than or equal to the maximum number of time steps N, processing operation 618 is performed.

[0259] In accordance with processing operation 616, the digital representation is output, with the process terminating thereafter. The output digital representation is considered to be a sample from the training distribution, i.e., a sample generated by the generative machine learning model.

[0260] According to processing operation 618, the digital representation is input to an electronic score function approximator. The electronic score function approximator may include a digital neural network. The digital neural network may include any number of computational layers, including, but not limited to, convolutional layers, upsampling layers, downsampling layers, activation layers, copy layers, residual layers, and fully connected layers. The output of the digital neural network may be an approximation of the score function. The digital neural network may be connected to the optoelectronic system via multiple elements, including an analog-to-digital converter (ADC), a digital-to-analog converter (DAC), an intensity modulator (IM), and a phase modulator (PM). The ADC converts the optical signal into an electronic signal that can be processed by a digital processor. The DAC converts the digital signal output from the digital processor into an optical signal at an appropriate operating wavelength for the hybrid optoelectronic system. The intensity modulator may be used to adjust or correct the amplitude of the optical signal generated using the DAC to match the amplitude level of the optical pulse before it is input to the digital processor. A phase modulator may be used to adjust the phase of the optical pulses generated by the DAC so that they are in phase with other optical pulses in the optical cavity. The IM and PM may be connected to the main laser source to receive clock timing, amplification energy, and a reference phase.

[0261] In accordance with process operation 620, the output of the electronic score function approximator is injected into an optoelectronic system. More specifically, the output of the score function approximator is optically encoded as an optical pulse and incorporated into the system. The optoelectronic system continues to evolve, and the process moves to process operation 608. Injecting the output optical signal from the function approximator into the system may be performed using a beam splitter, where the optical signal from the function approximator is combined with the signal inside the optical cavity through an input of the beam splitter and combined at an output of the beam splitter to produce an optical signal that remains within the optical cavity.

[0262] Referring to FIG. 7 , a flowchart of a method for generating samples from a distribution of training data for a generative machine learning model is shown, using an optoelectronic system with an optical score function approximator. The optoelectronic system with an optical score function approximator may be of various types. The optoelectronic system with an optical score function approximator may be an optoelectronic system with an optical score function approximator disclosed elsewhere herein, such as the optoelectronic system with an optical score function approximator disclosed herein in connection with FIGS. 3B, 3C, and 1. The optical score function approximator may also be of various types. In some cases, the optical score function approximator may include a neural network, such as the neural network disclosed herein in connection with FIGS. 3B, 3C, and 1. The optical score function approximator may represent a parameterized model, whose parameter values ​​are determined through a process called “training,” as disclosed elsewhere herein. This training process may be performed electronically, and the determined parameter values ​​may be statically encoded as parameters of the optoelectronic system. The optical score function approximator may also be optically trained through an optical optimization process.

[0263] Continuing with reference to FIG. 7 , in accordance with process operation 702, the optoelectronic system is initialized to a state representing a prior distribution. The prior distribution is determined by the terminal state of a forward stochastic process (described by a forward stochastic differential equation) used to train the optical score function approximator. The forward stochastic process may be selected so that the prior distribution is an easily sampleable distribution. In some cases, the prior distribution may be a Gaussian distribution. The optoelectronic system may be of various types. The optoelectronic system may be any of the optoelectronic systems disclosed elsewhere herein, such as those disclosed in connection with FIGS. 3A, 3B, 3C, and 1. The optoelectronic system is initialized using background vacuum noise to generate a zero-mean Gaussian state. Such a vacuum state exists within the optical waveguide of the optoelectronic system.

[0264] According to process operation 704, a maximum number of time steps N is selected, which may be chosen to be the same as the number of time steps used to train the optical score function approximator.

[0265] According to process operation 706, an optoelectronic system designed to represent the inverse stochastic process is evolved over N steps. The optical score function approximator is directly optically coupled to the optical device and may operate simultaneously without user interaction. The optoelectronic system designed to represent the inverse stochastic process has the same diffusion coefficient g(t) as the forward process. It also includes a drift term f(x,t) of the forward process and a modified drift term comprising the square of the diffusion coefficient coupled to the output of the score function approximator, which is consistent with the following inverse stochastic differential equation: dx=[f(x,t)-g 2 (t)s(x,t)]dt+g(t)dW

[0266] According to process operation 708, the state of the optoelectronic system is measured. The state of the system may be observed by performing homodyne measurements on the optical modes of the optoelectronic system. By comparing the optical pulses to a reference light source, such as an external laser, that generates the optical pulses, these measurements can provide values ​​for the amplitude and phase of the optical pulses. These values ​​may be interpreted as outputs of a generative machine learning model.

[0267] According to processing operation 710, the measured state is converted into a digital representation. This conversion may be performed using a digital-to-analog converter (DAC), which may be located within the homodyne detection device.

[0268] In accordance with processing operation 712, the digital representation is output, with the process terminating thereafter. The output digital representation is considered to be a sample from the training distribution, i.e., a sample generated by the generative machine learning model.

[0269] Referring to FIG. 8, a flowchart of a method for implementing dynamic behavior described by stochastic differential equations is shown.

[0270] In accordance with process operation 802, a plurality of quantum optical modes representing random variables that serve as initial conditions for the dynamic behavior described by the stochastic differential equations are injected into an optoelectronic system comprising at least one coupler having one or more optical components. The optoelectronic system may be of various types. The optoelectronic system may be any of the optoelectronic systems disclosed elsewhere herein, such as those disclosed in connection with FIGS. 3A, 3B, 3C, and 1.

[0271] Multiple quantum optical modes may be injected as squeezed vacuum states using a squeezer or as coherent states using an external laser source. These laser sources may operate at the same wavelength as the rest of the optoelectronic system. A single laser source can be used to dynamically control the power of the laser output or to sequentially use intensity and phase modulators to generate the desired initial amplitude and phase of the optical pulses. The pulse trains generated by the laser may be synchronized using a delay line network so that all pulses on different optical paths arrive at the optoelectronic system as disclosed in connection with Figures 3A, 3B, 3C, and 1. These optical paths may be connected to the optoelectronic system using optical cavities similar to those disclosed in connection with Figures 3A, 3B, 3C, and 1. All optical pulses generated by the laser belong to the same optical mode but have different parameters. Each optical pulse is an independent optical mode from the other optical pulses, but all share the characteristics of the same optical mode generated using the same laser source or the same squeezer.

[0272] In accordance with process operation 804, one or more quantum measurements are performed on a plurality of quantum optical modes. The one or more quantum measurements may be used to control random noise, which corresponds to a noise (also called spread) coefficient in a stochastic differential equation.

[0273] In accordance with processing operation 806, one or more optical operations are performed on the plurality of quantum optical modes using one or more optical components. The one or more optical operations may be of various types. The one or more optical operations may include analog optical operations, digital operations, or analog electronic operations. Digital operations may include converting the quantum optical modes to digital signals, performing digital processing on the digital signals, and converting the digital signals back to quantum optical modes. Analog electronic operations may include converting the quantum optical modes to analog electronic signals, performing analog electronic processing on the analog electronic signals, and then converting the analog electronic signals back to quantum optical modes.

[0274] Converting the quantum optical mode to a digital signal may include converting the quantum optical mode to an analog electronic signal and then converting the analog electronic signal to a digital signal. Converting the digital signal back to the quantum optical mode may include converting the digital signal to an analog electronic signal and then converting the analog electronic signal back to the quantum optical mode.

[0275] The one or more digital operations may be performed on the plurality of optical pulses, which may include converting one or more of the plurality of optical pulses into a digital signal, performing digital operations on the one or more optical pulses, and converting the digital signal back into an optical pulse. The one or more optical operations and / or the one or more digital operations may be used to generate a drift term in the stochastic differential equation.

[0276] Process operations 804 and 806 may be repeated one or more times to simulate the dynamic behavior described by the stochastic differential equation. By repeating process operations 804 and 806 using multiple instances of the optoelectronic system, or by synchronously cycling one or more optical pulses through the system multiple times using a feedback loop, the amplitude and phase of one or more optical pulses may converge to values ​​that represent samples from a distribution of a stochastic process corresponding to a solution to the stochastic differential equation. In some cases, the state of the optoelectronic system may be observed by performing homodyne measurements on quantum optical modes of the optoelectronic system. Values ​​of the amplitude of one or more optical pulses at each iteration may be obtained using homodyne measurements continuously during the process by measuring one or more optical pulses at each iteration. These values ​​may be obtained by measuring the amplitude of one or more optical pulses at the end of the process. By comparing the optical pulses to a reference light source, such as an external laser, that generates the optical pulses, these measurements may provide values ​​of the amplitude and phase of the optical pulses. The state of the system may be observed by performing the readout process described herein in connection with FIG. 2.

[0277] While preferred embodiments of the present invention have been shown and described herein, it will be apparent to those skilled in the art that these embodiments are provided by way of example only. It is not intended that the invention be limited to the specific examples set forth herein. While the present invention has been described based on the foregoing description, the description and illustration of these embodiments should not be construed in a limiting sense. It will be apparent to those skilled in the art that numerous changes, modifications, and substitutions are possible without departing from the invention. Moreover, all aspects of the present invention are not limited to the specific depictions, configurations, or relative proportions set forth herein, which depend upon a variety of conditions and variables. It will be understood that various alternatives to the embodiments of the present invention described herein may be employed in practicing the invention. It is therefore intended that the present invention encompass any such alternatives, modifications, variations, or equivalents. The following claims define the scope of the invention, and it is intended that methods and structures within the scope of these claims and their equivalents be covered thereby.

Claims

1. 1. An optoelectronic system for simulating at least one solution of a stochastic differential equation (SDE) driven by a Levy process, comprising: at least one input port for receiving at least one optical mode, the stochasticity of the Lévy process being represented by photon statistics of the at least one optical mode; and a computational and recursive component configured to perform one or more operations on one or more signals derived from the at least one optical mode, the computational and recursive component simulating the at least one solution of the stochastic differential equation; Optoelectronic system:

2. the computational and recursive component is further configured to implement at least a portion of a generative machine learning model based on the at least one optical mode. The optoelectronic system of claim 1 .

3. an output port configured to provide a readout, wherein the at least one solution of the stochastic differential equation is based at least in part on the readout; The optoelectronic system of claim 1 .

4. the computational and recursive components comprise one or more elements selected from the group consisting of digital electronic components, analog electronic components, optoelectronic components, and optical components; The optoelectronic system of claim 1 .

5. the computational and recursive component comprises a network of optical components; The optoelectronic system of claim 1 .

6. the analog electronic component comprises a resistive random access memory (RRAM) device; 6. The optoelectronic system of claim 5.

7. the digital electronic components include a field programmable gate array (FPGA); 6. The optoelectronic system of claim 5.

8. further comprising one or more optical waveguides, the one or more optical waveguides containing the at least one optical mode. The optoelectronic system of claim 1 .

9. the computational and recursive component includes one or more optical components, at least one of the one or more optical components includes at least one coupler, the at least one coupler configured to receive the one or more optical modes via the at least one input port; The optoelectronic system of claim 1 .

10. the computation and recursion component includes a measurement module configured to perform a homodyne measurement on the at least one optical mode; The optoelectronic system of claim 1 .

11. Each variable and one or more parameters of the SDE are represented by at least one of an optical signal or an electronic signal; The optoelectronic system of claim 1 .

12. the signal encodes information using a current pattern; 12. The optoelectronic system of claim 11.

13. the signal encodes information using a voltage pattern; 12. The optoelectronic system of claim 11.

14. the computational and recursive component comprises a feedback loop including at least one of an electronic component or an optical component; The optoelectronic system of claim 1 .

15. the computation and recursion component comprises a signal summer configured to sum signals; The optoelectronic system of claim 1 .

16. the signal summer includes one or more optical components including a circuit node, a summing amplifier circuit, or at least one coupler; 16. The optoelectronic system of claim 15.

17. the at least one optical mode comprises at least one member of the group consisting of a squeezed state, a vacuum state, and a coherent state, and further, the at least one optical mode is injected using the at least one input port. The optoelectronic system of claim 1 .

18. the optical mode comprises a vacuum state transformed into a squeezed state using the computational and recursive components; The optoelectronic system of claim 1 .

19. The stochasticity is controlled using quantum measurements. The optoelectronic system of claim 1 .

20. one of the quantum measurements changes the photon statistics of one or more of the at least one optical mode; 20. The optoelectronic system of claim 19.

21. a quantum measurement on one of the at least one optical mode produces an optical mode with different photon statistics than the measured optical mode.

20. The optoelectronic system of claim 19.

22. Operate in the low photon regime, and the quantum measurement comprises a weak quantum measurement and is performed by the environment.

20. The optoelectronic system of claim 19.

23. the at least one coupler is at least one of an adjustable coupler or a fixed coupler; 10. The optoelectronic system of claim 9.

24. the one or more optical components include at least one member of the group consisting of a beam splitter, a phase shifter, an amplifier, and a squeezer; 10. The optoelectronic system of claim 9.

25. the one or more optical components include at least one nonlinear optical component; 10. The optoelectronic system of claim 9.

26. the one or more optical components comprising a Mach-Zehnder interferometer (MZI) including two 50:50 beam splitters and at least one phase shifter; 10. The optoelectronic system of claim 9.

27. the computation and recursion component comprises an MZI mesh including a network of MZIs, the MZI mesh configured to implement an analog optical matrix-vector product (MVM) unit; 10. The optoelectronic system of claim 9.

28. the computation and recursion component comprises a free-space analog optical matrix-vector product (MVM) unit; 10. The optoelectronic system of claim 9.

29. the computational and recursive component includes a processing network, the output of which at least partially represents a drift term of the stochastic differential equation; The optoelectronic system of claim 1 .

30. the computational and recursive component includes a processing network, an output of the processing network representing at least in part the probabilities generated using results of the homodyne measurement of the at least one optical mode; 11. The optoelectronic system of claim 10.

31. the measurement module is operably optically connected to the at least one coupler using one or more optical waveguides, the one or more optical waveguides being arranged in a closed loop, and the output of the homodyne measurement being fed back to the at least one coupler; 11. The optoelectronic system of claim 10.

32. further comprising one or more optical cavities including the one or more optical waveguides arranged in a closed loop connecting the at least one coupler and the measurement module, wherein the output of the homodyne measurement is fed back to the one or more optical cavities.

32. The optoelectronic system of claim 31.

33. the one or more optical waveguides are optical fibers; 9. The optoelectronic system of claim 8.

34. the computation and recursion component is configured to encode a representation of a latent variable, the latent variable comprising a portion of a machine learning (ML) model; The optoelectronic system of claim 1 .

35. the computational and recursive component is configured to simulate a solution of at least one forward stochastic differential equation, and the solution of the at least one stochastic differential equation is used to train a generative machine learning (ML) model; The optoelectronic system of claim 1 .

36. the computational and recursive component comprises a function approximator that includes one or more parameters and one or more variables as input; The optoelectronic system of claim 1 .

37. the function approximator is trained to approximate a score function of a machine learning (ML) model; 37. The optoelectronic system of claim 36.

38. the drift term of the stochastic differential equation includes a drift term of an inverse stochastic differential equation representing inverse diffusion; 30. The optoelectronic system of claim 29.

39. the function approximator is trained to approximate a function representing the logarithm of a probability distribution of data in a machine learning (ML) model; 37. The optoelectronic system of claim 36.

40. the function approximator is trained to approximate the gradient of a function representing the logarithm of a probability distribution of data in a machine learning (ML) model; 37. The optoelectronic system of claim 36.

41. The stochastic differential equation represents Langevin dynamics. The optoelectronic system of claim 1 .

42. the function approximator comprises a neural network; 37. The optoelectronic system of claim 36.

43. the neural network comprises an optical neural network or an analog electronic neural network; 43. The optoelectronic system of claim 42.

44. the stochastic differential equation represents an optimization problem; The optoelectronic system of claim 1 .

45. the optimization problem is at least one of the group consisting of a box-constrained quadratic programming problem, a maximum weighted independent set optimization problem, and a two-dimensional assignment optimization problem; 45. An optoelectronic system according to claim 44.

46. 1. A method for simulating at least one solution of a stochastic differential equation (SDE) driven by a Levy process in an optoelectronic system, comprising: (a) generating signals in an optoelectronic system that represent variables and initial conditions of one or more parameters of the stochastic differential equation (SDE); (b) receiving, at the at least one input port, one or more optical modes having photon statistics that at least partially represent the stochasticity of a Lévy process driving the stochastic differential equation (SDE); and (c) performing, in a computational and recursive component operatively connected to the input port, one or more operations on the signals representing the variables and the one or more parameters of the stochastic differential equation (SDE) and one or more signals derived from the at least one optical mode to simulate the at least one solution of the stochastic differential equation; method.

47. performing a recursion by repeating steps (b) and (c) at least once in the calculation and recursion component; 47. The method of claim 46.

48. providing a readout at a readout port of the optoelectronic system; 47. The method of claim 46.

49. The stochastic differential equation (SDE) represents an optimization problem.

47. The method of claim 46.

50. the drift term of the stochastic differential equation (SDE) represents at least one of the group consisting of a gradient of an objective function, a function of the gradient of the objective function, and a time-dependent function of the gradient of the objective function that depends on past time; 47. The method of claim 46.

51. the optimization problem is at least one of the group consisting of a box-constrained quadratic programming problem, a maximum weighted independent set optimization problem, and a two-dimensional assignment optimization problem; 51. The method of claim 50.

52. The stochastic differential equation (SDE) describes the Langevin dynamics.

47. The method of claim 46.

53. the one or more operations comprise at least one of the group consisting of: (i) analog optical operations; (ii) digital operations; (iii) analog electronic operations; (iv) converting a digital signal to an optical mode; (v) converting an analog electronic signal to an optical mode; (vi) converting an analog electronic signal to a digital signal; (vii) converting a digital signal to an analog electronic signal; (viii) converting an optical mode to a digital signal; and (ix) converting an optical mode to an analog electronic signal.

47. The method of claim 46.

54. converting the optical mode to a digital signal comprises converting the optical mode to an analog electronic signal and converting the analog electronic signal to the digital signal, and further converting the digital signal to the optical mode comprises converting the digital signal to an analog electronic signal and converting the analog electronic signal to the optical mode.

54. The method of claim 53.

55. 1. A method for generating approximate samples from a distribution of training data, comprising: (a) obtaining values ​​for a plurality of parameters of a trained function approximator that at least partially represents a machine learning (ML) model; (b) implementing the learned function approximator in an optoelectronic system using at least the values ​​of the plurality of parameters, the optoelectronic system including at least one input port receiving at least one optical mode and an arithmetic and recursive component; (c) simulating, in the optoelectronic system, a solution of at least one stochastic differential equation (SDE) driven by a Levy process that represents the distribution of the training data of the machine learning (ML) model; and (d) generating approximate samples from the distribution of the training data based at least in part on the at least one solution; method.

56. before step (a), further comprising the step of training the function approximator to obtain values ​​of the parameters; 56. The method of claim 55.

57. The learning (a) obtaining an iteration number t between zero and a maximum number of iterations; (b) initializing the optoelectronic system with samples of the training data of the machine learning (ML) model; (c) running the optoelectronic system for a number of iterations t to simulate the at least one solution of the stochastic differential equation (SDE) driven by the Lévy process representing the machine learning (ML) model up to t; and (d) outputting a state of the optoelectronic system after t iterations; 57. The method of claim 56.

58. updating the values ​​of the parameters of the function approximator.

58. The method of claim 57.

59. Steps (a)-(d) and the step of updating the values ​​of the parameters of the function approximator are repeated at least once.

59. The method of claim 58.

60. 1. An optical system for simulating at least one solution of a stochastic differential equation, comprising: one or more optical components that generate controllable noise that represents the randomness of the stochastic differential equation, the controllable noise being generated using weak quantum measurements. Optical system.

61. the one or more optical components further generate a drift term in the stochastic differential equation; 61. The optical system of claim 60.

62. the weak quantum measurement is a measurement of a quantum mode; 61. The optical system of claim 60.

63. a feedback loop, wherein the output of the weak quantum measurement is used to generate a feedback signal.

63. The optical system of claim 62.

64. Operates in the low photon range, 61. The optical system of claim 60.

65. Operate in the low photon regime, and the weak quantum measurement is expressed by the environment.

65. The optical system of claim 64.

66. the feedback loop comprises one or more optical cavities; 64. The optical system of claim 63.

67. the drift term is programmable; 62. The optical system of claim 61.

68. the drift term is generated using a digital computing device; 68. The optical system of claim 67.

69. the drift term is generated using a field programmable gate array (FPGA); 69. The optical system of claim 68.

70. the drift term is generated using the optical component; 68. The optical system of claim 67.

71. the one or more optical components include at least one of the group consisting of a beam splitter, a phase shifter, an amplifier, and a squeezer; 61. The optical system of claim 60.

72. the one or more optical components include at least one nonlinear optical component; 61. The optical system of claim 60.

73. Implementing forward stochastic differential equations to train generative machine learning (ML) models; 61. The optical system of claim 60.

74. the drift term comprises a function approximator having one or more parameters and receiving one or more variables as input; 62. The optical system of claim 61.

75. the drift term of the stochastic differential equation includes a score function of a machine learning diffusion model; 62. The optical system of claim 61.

76. the function approximator is trained to approximate a score function of a machine learning diffusion model; 75. The optical system of claim 74.

77. the drift term of the stochastic differential equation includes a drift term of de-diffusion; 62. The optical system of claim 61.

78. the function approximator is trained to approximate a function that represents the logarithm of a probability distribution of data in a machine learning model; 75. The optical system of claim 74.

79. the function approximator is trained to approximate the gradient of a function representing the logarithm of a probability distribution of data in a machine learning model; 75. The optical system of claim 74.

80. the stochastic differential equation represents Langevin dynamics; 61. The optical system of claim 60.

81. the function approximator comprises a neural network; 75. The optical system of claim 74.

82. the neural network comprises an optical neural network; 82. The optical system of claim 81.

83. a plurality of function approximators having identical parameters, the function approximators being coupled using one or more optical cavities; 75. The optical system of claim 74.

84. 1. A method of simulating at least one solution of a stochastic differential equation, comprising: (a) generating, in an optical system comprising a network of optical components having one or more optical components, a plurality of optical pulses representing random variables that serve as initial conditions for dynamics described by a stochastic differential equation; (b) performing one or more quantum measurements on the plurality of optical pulses; and (c) performing one or more optical operations on the plurality of light pulses using the network of optical components; method.

85. the optical pulse comprises a vacuum state or a coherent state; 85. The method of claim 84

86. further comprising performing digital processing on the plurality of light pulses, the steps including converting one or more of the plurality of light pulses into a digital signal, performing digital processing on the one or more light pulses, and converting the one or more light pulses back into light pulses.

85. The method of claim 84

87. further comprising repeating steps (b) and (c) one or more times.

85. The method of claim 84

88. 1. A method for generating approximate samples from a distribution of training data, comprising: (a) obtaining values ​​of parameters of a trained function approximator that approximates a score function of a machine learning diffusion model; (b) implementing the learned function approximator on an optical system including one or more optical components using the obtained parameter values; (c) running the optical system to perform dynamics described by a stochastic differential equation that represents the distribution of training data for the machine learning diffusion model; and (d) reporting results to generate approximate samples from the distribution of the training data; method.

89. prior to step (a), further comprising training the function approximator to obtain values ​​of parameters; 89. The method of claim 88.

90. The learning step comprises: (a) sampling a time t between zero and a maximum time; (b) initializing the optical system with samples of the training data of the machine learning diffusion model; and (c) running the optical system for t iterations, performing dynamics described by a stochastic differential equation representing the diffusion process of the machine learning diffusion model; (d) outputting the state of the system after t iterations; 90. The method of claim 89.

91. further comprising updating parameter values ​​of the function approximator.

91. The method of claim 90.

92. 1. An optical system for simulating at least one solution described by a stochastic differential equation: (a) at least one coupler including one or more optical components, the at least one coupler receiving an optical quantum mode via at least one input port of the one or more optical components; and (b) a measurement module for performing homodyne measurements of the photon modes, the measurement module being operatively optically connected to the at least one coupler using one or more optical waveguides; the controllable noise of the light quantum mode is generated using at least a quantum measurement of the light quantum mode, the controllable noise representing at least a portion of the randomness of a stochastic differential equation; Optical system.

93. a processing network operatively connected to the at least one coupler, the processing network generating at least one programmable optical mode that represents at least a portion of the drift of the stochastic differential equation.

93. The optical system of claim 92.

94. the processing network is operably connected to the measurement module, and the generated at least one programmable optical mode represents at least a portion of the randomness of the stochastic differential equation and is generated using results of the homodyne measurement of the optical quantum modes, and the generated at least one programmable optical mode is fed back to the system.

94. The optical system of claim 93.

95. the processing network includes at least one of the group consisting of digital electronic components, analog electronic components, optoelectronic components, and optical components; 94. The optical system of claim 93.

96. the analog electronic component is a resistive random access memory (RRAM) device; 96. The optical system of claim 95.

97. the digital electronic component is a field programmable gate array (FPGA); 96. The optical system of claim 95.

98. the at least one coupler is at least one of a variable coupler and a fixed coupler; 93. The optical system of claim 92.

99. the photon mode is in a squeezed state, and the amount of squeezing of the squeezed state controls the controllable noise of the photon mode.

93. The optical system of claim 92.

100. the squeezed state is injected using the at least one coupler; 100. The optical system of claim 99.

101. the photon mode is in a squeezed state, an amount of squeezing of the squeezed state controls the controllable noise of the photon mode, and further, the squeezed state is generated using the processing network.

94. The optical system of claim 93.

102. the one or more optical waveguides are arranged in a closed loop connecting the at least one coupler and the measurement module, and an output of the homodyne quantum measurement is fed back to the at least one coupler.

93. The optical system of claim 92.

103. and one or more optical cavities including the one or more optical waveguides arranged in a closed loop connecting the at least one coupler and the measurement module, wherein the output of the homodyne quantum measurement is fed back to the one or more optical cavities.

93. The optical system of claim 92.

104. the one or more optical waveguides are optical fibers; 93. The optical system of claim 92.

105. Operates in the low photon range, 93. The optical system of claim 92.

106. Operate in the low photon regime, and the quantum measurement is a weak quantum measurement and is performed by the environment.

106. The optical system of claim 105.

107. The one or more optical components include at least one selected from the group consisting of a beam splitter, a phase shifter, an amplifier, and a squeezer.

93. The optical system of claim 92.

108. the one or more optical components include at least one nonlinear optical component; 93. The optical system of claim 92.

109. To encode latent variable representations used in machine learning methods, Use of the optical system according to claim 92.

110. For implementing forward stochastic differential equations for training generative machine learning (ML) models, Use of an optical system according to claim 92.

111. the processing network includes one or more parameters and includes a function approximator that receives one or more variables as input; 94. The optical system of claim 93.

112. the function approximator is trained to approximate a score function of a machine learning model; 112. The optical system of claim 111.

113. the drift term of the stochastic differential equation includes a drift term of de-diffusion; 94. The optical system of claim 93.

114. the function approximator is trained to approximate a function that represents the logarithm of a probability distribution of data in a machine learning model; 112. The optical system of claim 111.

115. the function approximator is trained to approximate the gradient of a function representing the logarithm of a probability distribution of data in a machine learning model; 112. The optical system of claim 111.

116. the stochastic differential equation represents Langevin dynamics; 93. The optical system of claim 92.

117. the function approximator comprises a neural network; 112. The optical system of claim 111.

118. the neural network is an optical neural network; 118. The optical system of claim 117.

119. 1. A method for implementing dynamics described by stochastic differential equations, comprising: (a) injecting, in an optical system comprising at least one coupler including one or more optical components, a plurality of optical quantum modes representing random variables that serve as initial conditions for dynamics described by a stochastic differential equation; (b) performing one or more quantum measurements on the plurality of optical quantum modes; and (c) performing one or more operations on the plurality of optical quantum modes using the one or more optical components; method.

120. The one or more operations include at least one selected from the group consisting of: (i) Analog optical operation; (ii) digitally converting one or more of the plurality of photon modes into a digital signal, performing digital processing on the digital signal, and converting the digital signal back into a photon mode; and (iii) analog electronic processing that converts one or more of the plurality of photon modes into an analog electronic signal, performs analog electronic processing on the analog electronic signal, and converts the analog electronic signal back into a photon mode; 120. The method of claim 119.

121. converting one or more of the plurality of photon modes to a digital signal comprises converting the one or more of the plurality of photon modes to an analog electronic signal and converting the analog electronic signal to a digital signal, and further wherein converting the digital signal back to a photon mode comprises converting the digital signal to an analog electronic signal and converting the analog electronic signal to a photon mode. The method of claim 120.

122. further comprising repeating steps (b) and (c) one or more times.

120. The method of claim 119.

123. 1. A method for generating approximate samples from a distribution of training data, comprising: (a) obtaining parameter values ​​of a trained function approximator representing at least a portion of a machine learning model; (b) using the obtained parameter values ​​to implement the learned function approximator on an optical system comprising one or more optical components; (c) operating the optical system to implement dynamics described by a stochastic differential equation that represents the distribution of the training data of the machine learning model; and (d) reporting results to generate approximate samples from the distribution of the training data; method.

124. prior to step (a), further comprising training the function approximator to obtain parameter values; The method of claim 123.

125. The learning: (a) obtaining t between zero and the maximum number of iterations; (b) initializing the optical system with samples of the training data of the machine learning model; (c) operating the optical system repeatedly t times to exercise the dynamics described by the stochastic differential equation representing the machine learning model up to t; and (d) outputting the state of the system after t iterations; The method of claim 124.

126. further comprising updating parameter values ​​of the function approximator. The method of claim 125.

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