Analog photonic computing system and method

Abstract

A system includes an encoding device configured to encode a continuous variable, a coefficient device including an addressable modulation element configured to modulate the encoded continuous variable, and an addressable detector configured to detect the encoded and modulated continuous variable.

Description

Docket No.1248-151WO01 / 230045 ANALOG PHOTONIC COMPUTING SYSTEM AND METHOD

[0001] This application claims the benefit of U.S. Provisional Patent Application 63 / 539,979, filed September 22, 2023, the entire content of which is incorporated herein by reference. GOVERNMENT RIGHTS

[0002] This invention was made with Government support under contract number FA8750- 23-C-1001 awarded by Air Force Research Laboratory (AFRL). The Government has certain rights in the invention. TECHNICAL FIELD

[0003] This disclosure generally relates to photonic computing. BACKGROUND

[0004] Computationally hard problems, such as NP-hard problems, are problems that cannot be solved quickly even with fast computers and also scale poorly as the number of problem variables increases. Examples include supply chain optimization, decryption, routing in communication networks, routing in public transportation, and the like. Traditional computers with central processing units (CPUs) evaluate solutions sequentially at a cadence governed by the clock cycle of the CPU and are ill-suited for many NP-hard problems. Many NP-hard problems may be reformulated as Ising problems, which are inherently nonlinear and highly coupled through many-body interactions. NP-hard problems, formulated as Ising problems can be factored into a binary representation and solved via optimization algorithms such as quadratic unconstrained binary optimization (QUBO). QUBO is isomorphic, or similar in form, to an Ising model, for example, including an array of interacting binary quantum states. Co-processors for solving Ising problems, such as Quantum Annealing (QA), overcome the disadvantages of CPUs when solving NP-hard problems through massive parallelism enabled by quantum entanglement and using simulated annealing. Quantum Annealers solve NP-hard problems by mapping them onto a QUBO model and then exploiting the symmetry of the QUBO model with the Ising model. Solutions are found by allowing the Ising spins to evolve naturally to a ground state. However, Quantum Annealers suffer from complex hardware, short decoherence times of quantum states, and poor hardware scalability.Docket No.1248-151WO01 / 230045 SUMMARY

[0005] In general, systems and techniques are described for computing solutions to NP-hard problems via a QUBO model, using an analog encoding of light used as a continuous variable that is representative of QUBO spin values. For example, an analog computing system to implement such a QUBO model may include a photonic integrated circuit (PIC) and a spatial light modulator (SLM). The PIC includes a resonator configured to modulate at least one of an amplitude, phase, or polarization of light to encode the light with an analog encoding of the light based on a differentiable function, and the SLM is configured to perform optical vector matrix multiplication (OVMM) on the encoded light by modulating the amplitude, phase, or polarization of encoded light according to an NP-hard problem mapped to an Ising problem.

[0006] In one example, this disclosure describes a system including: an encoding device configured to encode a continuous variable; a coefficient device comprising an addressable modulation element configured to modulate the encoded continuous variable; and an addressable detector configured to detect the encoded and modulated continuous variable.

[0007] In another example, this disclosure describes a method including encoding, by an encoding device, a continuous variable; modulating, by an addressable modulation element of a coefficient device, the encoded continuous variable; and detecting, by an addressable detector, the encoded and modulated continuous variable.

[0008] In another example, this disclosure describes a system for iteratively solving a quadratic unconstrained optimization (QUO) problem using gradient descent and optical vector matrix multiplication (OVMM), the system including: a photonic integrated circuit (PIC) configured to encode continuous variables represented as analog voltages onto an array of optical signals; lenses configured to perform spatial addressing of the optical signals exiting the PIC; matrices representing QUO coefficients programmed into a spatial light modulator (SLM); a detector array configured to compute an output vector from the OVMM, the output vector representing gradients of the continuous variables; and analog circuitry configured to scale the gradients of the continuous variables and add noise.

[0009] The details of one or more examples of the disclosure are set forth in the accompanying drawings, and in the description below. Other features, objects, and advantages of the disclosure will be apparent from the description and drawings, and from the claims.Docket No.1248-151WO01 / 230045 BRIEF DESCRIPTION OF DRAWINGS

[0010] FIG.1 is a block diagram illustrating an example analog photonic computing system, in accordance with the techniques of the disclosure.

[0011] FIG.2A is a block diagram illustrating an example analog photonic computing system, in accordance with the techniques of the disclosure.

[0012] FIG.2B is a block diagram illustrating an example function flow of an analog photonic computing system, in accordance with the techniques of the disclosure.

[0013] FIG.3 is a schematic diagram illustrating a perspective view of another example analog photonic computing system, in accordance with the techniques of the disclosure.

[0014] FIG.4 is a schematic diagram illustrating a portion of the example analog photonic computing system of FIG.3, in accordance with the techniques of the disclosure.

[0015] FIG.5 is a flowchart of an example method of analog photonic computing, in accordance with the techniques of the disclosure.

[0016] Like reference characters denote like elements throughout the figures and description. DETAILED DESCRIPTION

[0017] In general, one or more aspects of this disclosure describe systems and methods for solving computationally difficult problems, such as NP-hard problems. For example, a computationally efficient QUBO-like solver uses a new approach, e.g., quadratic unconstrained smooth optimization (QUSO) based on optical vector-matrix multiplication and analog encoding of variables, e.g., analog versions of binary QUBO spins. The QUSO solver, implemented in an analog photonic computing system, may be configured to encode analog spins (as opposed to binary spins) through analog optical variables and optimizes energy through a hardware implementation of gradient descent.

[0018] In some examples, systems and methods disclosed herein include fast all-to-all spin coupling of a plurality of spin variables, e.g., on the order of microseconds or less. In some examples, a system includes an encoding device, which may include a hybrid photonic integrated circuit (PIC) that includes resonant structures such as microdisk modulators configured to encode analog spin amplitudes in light coupled to the PIC, and a coefficient device, which may include a spatial light modulator (SLM) configured to encode QUSO spins. In some examples, the analog photonic computing system includes a PIC configured to encode analog variables (spins), an SLM to encode optical interaction weights (e.g., QUBO weights representative of coefficients representing the computationally hard problemDocket No.1248-151WO01 / 230045 that may be programmed into the SLM), and a plurality of sensors operating in a high-speed loop to tune parameter values and read out solutions.

[0019] The sensors and techniques described herein may be used to solve NP-hard and / or APX hard problems such as satisfiability (SAT), MAX-SAT, and MAX-CUT, and may provide computation and / or solutions with reduced total energy consumption and improved speed, e.g., equal to or less than 17 picoJoules (pJ) per matrix element at equal to or greater than 1 megahertz (MHz) readout rate for each matrix multiplication. For example, the systems and methods disclosed may avoid the use of digital logic, such as field programmable gate arrays (FPGAs) to compute the interaction energy between all spin states, and is instead based on analog elements which draw significantly less energy, e.g., light propagating through a PIC including resonant modulators and an SLM configured to adjust, encode, and / or modulate the amplitude, phase, or polarization of light from the photonic channels in free space and coherent optical readout via light sensors In some examples, the systems and methods provide scalable all-to-all coupling of all analog variables (e.g., spins). The all-to-all coupling may provide improved accuracy of solvers for computationally hard problems modeled as QUBO type problems. In some examples, the systems and methods provide improved computational efficiency, e.g., an improved energy or power used to output a solution. This quantity may include the number of highspeed loop iterations multiplied by the energy consumed per loop.

[0020] FIG.1 is a block diagram illustrating an example analog photonic computing system 100, in accordance with the techniques of the disclosure. In the example shown, system 100 includes light source 102, encoding device 104, decomposition 106, coefficient device 108, addressable detectors 110, processing module 112, and analog-to-digital converter (ADC) and sensor readout 114. System 100 may be configured to solve computationally hard problems modeled as QUBO type problems.

[0021] A QUBO models shares a similar form to an Ising model. An Ising model is a mathematical model of ferromagnetism in statistical mechanics, consisting of discrete variables that represent magnetic dipole moments of atomic “spins” that can be in one of two states (+1 or −1). The Ising model energy function is given by Equation (1):Docket No.1248-151WO01 / 230045

[0022] where Jij is an interaction and hi is an externally applied magnetic field. The first term is the energy contribution from the interaction between all spins in the system, and the second term is each spin’s self-energy due to the externally applied magnetic field hi. The QUBO energy function is expressed in a matrix Q where each element Qij represents the coupling between binary variables qi and qj. The canonical form of the QUBO energy equation is Equation (2): ^^^^^^ ൌ ∑^^   ^^^^^^^^^^ (2)

[0023] The form of the QUBO energy Equation (2) is mathematically similar to the Ising model, but it is non-analytic and non-differentiable. If, instead, the QUBO spin variable q is represented by an analytical and differentiable encoding function f(q∗), where the * indicates q may take on any value between 0 and 1, e.g., f(q∗) is an analog function of continuous variable q∗, the analytical form of the energy function QUBO may be a differentiable quadratic unconstrained smooth optimization (QUSO) having energy Equation (3): ^^^^ୗ^ ൌ ∑^^   ^^^^^^^^^^∗^^^൫^^^∗൯ (3)

[0024] In gradient descent, variable optimization follows from Equation (4):where t is an iteration variable. Equation (4) may be simplified to Equation (5) by differentiating the QUSO energy equation as dEQUSO / dqi.

[0025] In this simplified form of Equation (5), the quadratic dependence on the spin becomes linear, and the spin encoding f(q) and its derivative 2f’(q) are computed directly in each iteration t (e.g., each time step). Each continuous variable (e.g., analog of QUBO spin value in Equation 5) updates by computing the difference between the prior continuous variable and the α-scaled gradient of the QUSO energy, which is proportional to the Vector MatrixDocket No.1248-151WO01 / 230045 Multiplication of the prior continuous variable and the coupling matrix Q. In Equations (4) and (5) it is implied that the variable q is a real-valued continuous variable on the interval [0,1] (e.g., the * notation is omitted hereafter).

[0026] The final form of the energy gradient of Equation (5) is proportional to the Vector Matrix Multiplication (VMM) of the encoded input spins and the coupling matrix. System 100 may be used to encode input spins and perform VMM of the encoded spins and the coupling matrix, e.g., using optical components.

[0027] In the example shown, light source 102 is configured to output light to encoding device 104. Light source 102 may be a laser, light emitting diode, or any suitable light source. Light source 102 may output ultraviolet, visible, infrared, or any suitable wavelength and / or frequency of light. Light source 102 may be a broadband light source configured to output light comprising a plurality of wavelengths / frequencies, or to output monochromatic or substantially monochromatic light, e.g., laser light. In some examples, light source 102 may output light with random polarization (e.g., unpolarized light), partially polarized light, or polarized light, e.g., linear, circular, or elliptically polarized light.

[0028] Encoding device 104 is configured to receive light 120 from light source 102. For example, encoding device 104 may include a PIC that includes a waveguide, and light source 102 and the PIC may be arranged such that light 120 from light source 102 is coupled to the waveguide of PIC.

[0029] Encoding device 104 may be configured to encode a continuous variable. For example, a property of the light 120 coupled to encoding device 104 may represent a continuous variable. That is, a measurable property of the light 120, e.g., an amplitude, a phase, and / or a polarization of the light 120 may represent a continuous variable. A continuous variable may have any value, or an infinite number of possible values within a range (e.g., from 0 to 1), as opposed to, for example, a binary variable having only two values or a digital or discrete variable having a finite number of possible values within the range. Encoding device 104 may be configured to encode the continuous variable, e.g., the property of light 120. For example, encoding device 104 may be configured to modulate the property of light 120, e.g., according to an analog and differentiable function. For example, in Equations (3)–(5), the amplitude, phase, and / or polarization of light 120 may be represented by q, and encoding device 104 may be configured to modulate the amplitude, phase, and / or polarization of the light 120 according to f(q), where f() is a differentiable function. In some examples, encoding device 104 is configured to encode the continuous variable, e.g., the amplitude, phase, and / or polarization of light 120, using an analog encoding. For example,Docket No.1248-151WO01 / 230045 encoding device 104 may include an opto-electronic component such as a resonator that is continuously tunable via an analog voltage and / or current and may be configured to proportionally change a property of light 120. In some examples, the resonator may comprise a microdisk modulator. At least a portion of the light 120 coupled to the waveguide of encoding device 104 may be coupled to the resonator, and the resonator may be tuned and / or changed to modulate the amplitude, phase, and / or polarization of the portion of the light 120 according to the analog and differentiable function, e.g., proportionally change the property of the portion of the light by a first amount based on the analog and differentiable function. For example, the modulation of the amplitude, phase, and / or polarization of the light 120 (the encoding of the continuous variable) by the resonator may also be continuous and result in one of an infinite number of values within a range of the differentiable function corresponding to the range of the continuous variable. (As used herein, the phrase “A and / or B” means “at least one of A or B.”)

[0030] In some examples, encoding device 104 may include a plurality of waveguides and / or a waveguide that splits into a plurality of waveguides, e.g., N waveguides. Each waveguide of the plurality of waveguides may correspond to an output channel of encoding device 104, e.g., encoding device 104 is configured to output a plurality of channels of light to coefficient device 108. Encoding device 104 may include a plurality of modulators (e.g., resonators) corresponding to the plurality of channels, for example, encoding device 104 may include a different modulator optically coupled to the waveguide of each channel. Encoding device 104 may be configured such that each modulator is independently addressable and / or controllable, e.g., to change the resonance and / or modulation of the light in that channel independent of all of the other channels. Encoding device 104 may be configured such that each channel represents an independent continuous variable and is configured to independently encode each of the plurality of independent variables of each channel, e.g., as described above.

[0031] The continuous variables, e.g., the amplitude, phase, and / or polarization of the light in each channel, may be representative of a parameter (e.g., a spin value) of a quadratic unconstrained model (QUO), such as a QUSO model. Decomposition 106 is configured to map a computationally hard problem to the QUO (e.g., QUBO / QUSO) matrix Q. Decomposition 106 may represent any system or process for decomposing the computationally hard problem to matrix Q, e.g., populating Q with values representative of the computationally hard problem. The values representative of the computationally hard problem may be the coefficients of the matrix Q.Docket No.1248-151WO01 / 230045

[0032] Coefficient device 108 may be configured to perform vector matrix multiplication on the encoded continuous variable, e.g., encoded (or modulated) light 122 from encoding device 104. For example, coefficient device 108 include one or more addressable modulation elements configured to modulate one or more encoded continuous variables, e.g., to modulate a property of one or more continuous variables in an analog domain. In some examples, the one or more modulation elements of coefficient device 108 may be representative of a coefficient of the QUO model, e.g., the coefficients of Q. Coefficient device 108 may be configured to modulate the encoded continuous variable by modulating the encoded property of light, by proportionally changing, by a second amount, the property of the light based on one or more coefficients of the QUO model.

[0033] In some examples, coefficient device 108 may include a spatial light modulator (SLM) configured to modulate the amplitude, phase, and / or polarization of encoded light 122 incident on, and propagating through, the SLM. For example, encoded light 122 exiting each channel of encoding device 104 may be converted to a vector, e.g., by spatially spreading out the encoded light 122 in a direction and focusing the encoded light 122 across a plurality of modulation elements, e.g., M modulation elements, of the SLM in the direction. The SLM of coefficient device 108 is then configured to perform element-wise multiplication of the vector by modulating the amplitude, phase, and / or polarization of the encoded light 122 by an amount corresponding to its matrix Q value. For example, each modulation element of the SLM may correspond to an element of the matrix Q and may be configured to have a transmission, e.g., from 0% to 100%, that is proportional to the value of the corresponding matrix element. The output light 124 from the plurality of modulation elements of the SLM may be focused onto addressable detectors 110, and addressable detectors 110 may be configured to detect, sense, and / or measure the amplitude, phase, and / or polarization of the now encoded and vector-matrix multiplied output light 124, e.g., detect, sense, and / or measure an encoded and modulated continuous variable. In some examples, the focusing of the output light 124 from the plurality of SLM modulation elements and detecting, sensing, and / or measuring of the focused amplitude, phase, and / or polarization of the output light 124 provides summing of the vector-matrix multiplication, e.g., according to Equations (3) and (5) above.

[0034] In the example shown, system 100 may include processing module 112, which facilitates a feedback loop. For example, the addressable detectors 110 may detect the vector-matrix multiplied output light 124 from the N channels of encoding device 104 and coefficient device 108 at a first time tn, or iteration tn, and processing module 112 may adjustDocket No.1248-151WO01 / 230045 the encoding of the continuous variable, e.g., the light, at a second time and / or iteration tn+1, e.g., per Equation (5) above, based on the sensed output light 124. In some examples, processing module 112 may be configured to receive one or more signals 126 from addressable detectors 110 indicative of values representative of the summation portion of Equation (5). For example, addressable detectors 110 may be configured to output signals 126 as an analog electrical signal that is proportional to the mathematical product of the encoded continuous variables and coefficients of the QUO model to processing module 112 and / or analog-to-digital converter (ADC) and readout 114. Processing module 112 may be further configured to output signals 128 representative of the value of the next iteration of the continuous variable, qi(tn+1), of Equation (5). Processing module 112 may output the signals 128 as voltage and / or current signals, for example, and encoding device 104 may be configured to receive the signals 128, e.g., at the modulators of each of the N channels. The signals 128 may cause the resonances of the modulators of encoding device 104 to change in a manner corresponding to the new (subsequent second time / iteration) value of the continuous variable, e.g., to encode the new value of the continuous variable. As described above, this encoding may involve modulating the light 120 according to the new resonance.

[0035] System 100 outputs one or more signals indicative of the values of the continuous variables q, which may indicate a partial, per-iteration, and / or converged solution. In the example shown, ADC and readout 114 may be configured to convert the sensed / measured amplitude, phase, and / or polarization of the light 124 to digital values 130 representative of the values of the continuous variables q of the computationally hard problem. In some examples, ACD and readout 114 may be configured to output digital values 130 of each of the N continuous variables (channels) q at each iteration and / or upon convergence of the gradient descent feedback loop of system 100 to a solution, e.g., a minimized QUSO energy of Equation (3), at which the q values may stabilize according to convergence criteria. In some examples, signals output by sensors 112 may be output from system 100. In some examples, system 100 may output digital values 140 and / or signals 128 to an external processing module. In some examples, processing module 112 may include ADC and readout 114, or may be configured to convert the sensed / measured amplitude, phase, and / or polarization of the light 124 to digital values 130 representative of the values of the continuous variables q of the computationally hard problem.

[0036] Processing module 112 may be implemented as software, but may in some examples include any combination of hardware, firmware, and software. Processing module 112 may include processing circuitry that executes instructions and memory that stores instructionsDocket No.1248-151WO01 / 230045 and / or data. The combination of processing circuitry and memory may retrieve, store, and / or execute the instructions and / or data of one or more applications, modules, or software. The processing circuitry and memory may also be operably coupled to one or more other software and / or hardware components, including, but not limited to, one or more of the components illustrated in FIG.1. Processing module 112 may be implemented within one or more processors, including one or more microprocessors, digital signal processors (DSPs), application specific integrated circuits (ASICs), field programmable gate arrays (FPGAs), or any other equivalent integrated or discrete logic circuitry, as well as any combinations of such components. The term “processor” or “processing circuitry” may generally refer to any of the foregoing logic circuitry, alone or in combination with other logic circuitry, or any other equivalent circuitry.

[0037] FIG.2A is a block diagram illustrating an example analog photonic computing system, and FIG.2B is a block diagram illustrating an example function flow of the computing system of FIG.2A. FIGS.2A and 2B are described together below. System 200 may be an example of system 100 described above.

[0038] In the example shown in FIG.2A, spin encoding (e.g., encoding of a continuous variable) may be represented by the amplitude of light. Laser 202 is configured to output light to splitter 204. Splitter 204 is configured to split the input power of the light over N channels. For example, encoding device 104 may include splitter 204 configured to split the light from light source 202 across N waveguides of encoding device 104, each channel including a different variable optical attenuator (VOA) of VOAs 206. VOAs 206 may be examples of modulators of encoding device 104, and VOAs 206 may be configured to initialize the amplitude of the light representative of a continuous spin value of each channel. In some examples, VOAs 206 comprise fiber-based microelectromechanical (MEMS) devices, or a resonator such as a microdisk resonator, or any suitable means for electrically modulating the transmission of the light in each of the N channels. Referring to FIG.2B, VOAs 206 may be configured to encode the continuous spin values as optical fields at (252), where the encoding is the modulation of the amplitude of the light by VOAs 206.

[0039] In the examples shown, each channel may scalar initialize a spin vector V by subdividing the light output from each channel of VOAs 206 (e.g., output from encoding device 104) into M equal sub-elements VM. For example, the N channels of VOAs 206 may be arranged along a first direction, and a cylindrical lens may focus the light exiting each of the N channels of VOAs 206 along a plurality of modulation elements of SLM 208 in a second direction substantially perpendicular to the first direction.Docket No.1248-151WO01 / 230045

[0040] In some examples, the light exiting VOAs 206 may be linearly polarized, e.g., in the transverse electric (TE) plane, and an NP-hard problem may be mapped to a QUSO (or QUBO) form and normalized to values between 0 and 1, e.g., by encoding coupling coefficients at (254) as QUSO matrix Q as grayscale transmission values of SLM 208 and analyzer 210 between pass (e.g., “1” or substantially 100% transmission) and block (e.g., “0” or substantially 0% transmission).

[0041] In some examples, the coupling matrix Q may be encoded by shifting the relative phase of the TE and transverse magnetic (TM) modes of the light. The resulting phase shift transforms the initially linear polarization into an elliptical polarization. In some examples, the SLM 208 may perform the encoding with a relative phase delay of up to 10 bits of precision across 1200 x 1920 arrays of individual modulation elements. In some examples, the vector VM of the light exiting VOAs 206 may have a linear polarization that is rotated halfway between the x and y axes (e.g., relative to the arrangement of the modulation elements of SLM 208) for improved sensitivity to phase shifts prior to incidence on the coupling surface of SLM 208. SLM 208 may be configured to impart an electrically tunable phase delay to the light propagating through each modulation element of SLM 208, thereby rotating the polarization of the light, which then propagates through analyzer 210, which may be a linear polarizer.

[0042] For example, SLM 208 may be a liquid crystal module (LCM) comprising a two- dimensional array of individually electrically addressable liquid crystal modulation elements between two crossed linear polarizers. In other examples, SLM 208 may be a microelectromechanical system (MEMS) or device comprising a two-dimensional array of individually electrically addressable modulation elements configured to modulate, change, and / or modify an amplitude, phase, and / or polarization of light. In other examples, SLM 208 may be any device configured to modulate, change, and / or modify an amplitude, phase, and / or polarization of light over a spatial area. The N channels corresponding to VOAs 206 may be arranged as a row along a first direction, and optics (e.g., including a cylindrical lens) may spread out the light along a second direction substantially perpendicular to the first direction and focus the light along M modulation elements of a column of SLM 208. SLM 208 and analyzer 210 may multiply the encoded spins and coefficients at (256) by performing vector (VM) matrix (Q) multiplication for each of the channels by modulating the amplitude of the light via the modulation elements of SLM 208 and analyzer 210, e.g., corresponding to Qijf(qi(tn)) of Equation 5 above. The vector-matrix multiplication for each channel is summed at (258) by focusing the light exiting SLM 208 and analyzer 210 to sensors 212, whichDocket No.1248-151WO01 / 230045 collect and sense the light propagating through all M modulation elements of SLM 208 to a sensor of sensors 212, which add up or sum the vector multiplied by the matrix by sensing / measuring the light focused onto the sensor. For example, there may be N sensors 212, with each sensor sensing and / or measuring the light from each of the N channels after propagation through SLM 208 and analyzer 210 and focusing to each of the N sensors. The N measurements may then be summed 214, e.g., by processing circuitry configured to receive measurements from sensors 212. In some examples, the final readout is performed by projecting the element-wise result of the polarization through a polarization analyzer 210 and summing over the N channels a total of M times to convert the input N x 1 set of vectors into a 1 x M set of output gradients.

[0043] In some examples, calibration may be necessary because the signal readout is not linear in polarization. For example, the values of the coupling coefficients of the Q matrix of SLM 208 may be pre-compensated to produce the desired output amplitude modulation from SLM 208 and analyzer 210. In some examples, other encodings (e.g., more complex encodings) may be performed by projecting copies of the input spins (e.g., VMvectors) onto different SLMs 208. For example, coupling may be performed by breaking the QUSO / QUBO matrix Q into a plurality of sparse positive definite matrices and summing the resulting outputs of each.

[0044] In some examples, processing circuitry may be configured to add the self-energy term of Equation (1), and to add noise to the resulting measured values at (260). In some examples, adding noise to the resulting measurements from sensors 212 may provide an improved convergence to a global minimum of the QUSO energy function, e.g., the exit criteria for the feedback loop and indication of a solution to the computationally hard problem. For example, noise may be required to exit and local minimum of the QUSO energy function in order to progress to a global minimum of the QUSO energy function.

[0045] In some examples, processing circuitry may receive signals indicative of the measurements of sensors 212, add the self-energy term based on the current spin values (260), add noise (26), and update the real-valued spin parameters at (264). For example, the processing circuitry may update the encoding of the continuous spin values (e.g., light) of VOAs 206 at a time subsequent to the latest measurements by sensors 212 in an iterative feedback loop. In some examples, the processing circuitry may apply any needed nonlinearity or changes in nonlinearity at (262), e.g., compensation of SLM 208 due to its nonlinear polarization response and / or nonlinear readout of sensors 212.Docket No.1248-151WO01 / 230045

[0046] FIG.3 is a schematic diagram illustrating a perspective view of an example analog photonic computing system 300, and FIG.4 is a schematic diagram illustrating a portion of the example analog photonic computing system 300 of FIG.3. FIGS.3 and 4 are described together below. System 300 may be an example of systems 100 and / or 200 described above.

[0047] In the examples shown, system 300 includes PIC 304, SLM 308, and sensors 310A and / or 310B. PIC 304 may be representative of encoding device 104 of FIG.1 and may be configured to encode a continuous variable, e.g., a quantity representing an analog QUSO spin value, e.g., any value between 0 and 1. In some examples, a property of light, e.g., the amplitude, phase and / or a polarization of the light, may be representative of the continuous variable, and PIC 304 may be configured to encode the continuous variable by modulating the amplitude, phase and / or a polarization of the light. In the example shown, the light 302 is 1550 nanometer (nm) laser light, and PIC 304 includes a plurality of modulators 324 configured to modulate the amplitude of light 302, e.g., according to a differentiable function f, e.g., of Equation (5).

[0048] In the example shown, PIC 304 includes splitter 322. Splitter 322 may comprises a first portion 322a configured to receive light 302, and a second portion 322b configured to couple the received light to a plurality of waveguides, e.g., to split the light to a plurality of channels (e.g., N channels), where each channel includes a waveguide. Each channel of the N channels may correspond to one of N continuous variables, e.g., the N channels carry light representing N analog QUSO spin values. The plurality of modulators 324 may each be optically coupled to a respective waveguide of the N channels, and may be configured to modulate the amplitude of the light before the light exits the channel. Modulators 324 may be any optical component, or opto-electronic component, configured to modulate any of the amplitude, phase, and / or polarization of the light. In the example shown, modulators 324 each comprises a microdisk modulator or microdisk resonator. Each modulator 324 is configured to be independently electrically addressable, e.g., the resonance of each modulator 324 may be controlled by the application of a voltage or current signal via electrical contacts 326, thereby controlling the amplitude of light exiting the channel. PIC 304, using modulators 324, may be configured to encode a continuous variable, e.g., light, using an analog encoding of a differentiable function. In the example shown, the light of each channel i (where i = 1 to N) may be representative of a continuous variable qi, and each of modulators 324 may be configured to encode the continuous variable qiusing analog encoding, e.g., by adjusting the resonance of each modulator such that it outputs an amplitude of the light coupled to each modulator 324 according to f(qi), where f is a continuously differentiableDocket No.1248-151WO01 / 230045 function. In the example shown, PIC 304 is configured to output the modulated light to SLM 308, e.g., as encoded light.

[0049] Lens 332 may be configured to focus light from each channel from PIC 304 onto a plurality of modulation elements of SLM 308. For example, the outputs of the N channels of PIC 304 may be arranged in a row in the x-direction shown in FIG.3. Lens 332 may include a collimating lens and a cylindrical lens, and lens 332 may be configured to expand the light from PIC 304 in the y-direction and focus the light from PIC 304 onto a plurality of modulation elements of SLM 308. The expanded, focused encoded light from PIC 304 may represent a vector, VM, where the value of each element j of the vector is f(qi).

[0050] SLM 308 may be representative of coefficient device 108 of FIG.1. SLM 308 may comprise N x M individually addressable modulation elements arranged in an array, each modulation element configured to have a transmission between 0% and 100%. For example, each modulation element of SLM 308 is configured to modulate the amplitude, phase, and / or polarization of incident light according to an analog modulation, e.g., any grayscale transmission, phase delay, and / or retardance value, respectively. In the example shown, each modulation element of SLM 308 is configured to modulate the amplitude of the encoded light exiting PIC 304 by controlling the transmission of the modulation element. In some examples, each addressable modulation element of SLM 308 corresponds to a different coefficient of a QUO model and is configured to modulate an encoded continuous variable of the QUO model by an amount corresponding to the corresponding coefficient.

[0051] In some examples, each modulation element of SLM 308 is configured to modulate the encoded variable using optical vector matrix multiplication (OVMM) by modulating, independent of each of the other modulation elements of the plurality of modulation elements of the SLM, at least one of the amplitude, the phase, or the polarization of the encoded light exiting PIC 304. For example, the transmission values of each modulation element of SLM 308 may be representative of a coefficient representing a computationally hard problem, e.g., after decomposition of the computationally hard problem. The M x N modulation elements of SLM 308 may then be representative of the QUSO matrix Q, and each modulation element of SLM 308 may modulate the amplitude of the encoded light by attenuating a portion of the encoded light as it propagates through the modulation element, e.g., from no attenuation (pass, or 100% transmission) to full attenuation (block, or 0% transmission). For example, lens 322 may spread out encoded light of the nthchannel into M values, and each value is multiplied by each of the M modulation elements of the nthcolumn of SLM 308 by virtue of propagation through each of the M modulation elements.Docket No.1248-151WO01 / 230045

[0052] Lens 334 may be configured to focus the encoded light modulated by each modulation element of the plurality of modulation elements of SLM 308 onto the sensors 310A or 310B. For example, lens 334 may be configured to focus light exiting each modulation element of each of the N columns of SLM 308 to each corresponding N sensor of N sensors of sensors 310A or 310B.

[0053] Sensors 310A and / or 310B (collectively, “sensors 310”) may be configured to sense and / or measure the amplitude, phase, and / or polarization of the encoded light after modulation by SLM 308. For example, sensor 310A may comprise a two-dimensional CCD array of pixels, and sensor 310B may comprise a one-dimensional array of photodiodes having pixels elongated in the other direction relative to the array. In some examples, SLM 308, lens 334, and sensors 310A or 310B are configured to perform the OVMM between Q and f(q) and summation of Equation (5).

[0054] In some examples, SLM 308, lens 334, and sensors 310A or 310B are configured to perform coherent OVMM. For example, local oscillators may be added to the input vector VM and applying a pure phase modulation at SLM 308. Phase coherence may then be achieved by controlling the optical path, e.g., via optical elements such as lenses, mirrors, prisms, baffles, apertures, and the like, by limiting the output to the chief ray of the forward propagating light. In some examples, system 300 may be configured to perform OVMM by using a coherent light source (e.g., a laser) over an incoherent light source, e.g., a vertical cavity side emitting laser (VCSEL) to integrate the spin encoder (modulators 324) with PIC 304.

[0055] Sensors 310 may be representative of detectors 110 of FIG.1 and / or sensors 212 of FIG.2. In some examples, sensors 310 may be configured to output one or more signals representative of the measured amplitude, the measured phase, or the measured polarization of the modulated and encoded light. For example, sensors 310 may be configured to output signals 342 representative of the measured amplitudes. Signals 342 may be voltage and / or amplitude electrical signals, and modulators 324 may be configured to receive signals 342 and change the encoding based on the received signals. For example, modulators 324 may be configured to change the resonance of modulators 324 based on signals 342 to encode light with a second and subsequent encoding, e.g., in a feedback loop. In some examples, PIC 304 may be configured to encode a first encoding a continuous variable (e.g., encode a property of light) at a first time, and modulators 324 (e.g., an opto-electronic component) of each channel is configured to encode the property of the light with a second encoding at a second time subsequent to the first time by proportionally changing the property of the light based onDocket No.1248-151WO01 / 230045 the electrical signal 342 output by a corresponding addressable detector element of the detector array 310.

[0056] In some examples, system 300 may be configured to impart an analog electronic nonlinearity and / or noise to signals 342. For example, an exponentially decaying noise of Equation (6) may be injected into signals 342, e.g., to improve convergence accuracy. ^^^^^^ ൌ ^^^exp ^െ^^ଶ / ^^ଶ^ (6)

[0057] In some examples, sensors 310 comprise a detector array comprising plurality of addressable detector elements configured to output a plurality of analog electrical signals (e.g., signals 126 and / or 342), wherein each addressable detector element is configured to output a corresponding analog electrical signal of the plurality of analog electrical signals that is proportional to a sum of one or more encoded and modulated continuous variables.

[0058] FIG.5 is a flowchart of an example method of analog photonic computing, in accordance with the techniques of the disclosure. Although the method is described with reference to systems 100, 200, and 300 FIGS.1-3, the methods discussed herein may include and / or utilize other systems and methods in other examples.

[0059] Encoding device 104 may encode a continuous variable (502). The continuous variable may be representative of a parameter of a quadratic unconstrained optimization (QUO) model. In some examples, encoding device 104 may encode the continuous variable based on a continuously differentiable function. Encoding device 104 may encode the continuous variable by encoding a property of light, e.g., at least one of an amplitude of the light, a phase of the light, or a polarization of the light.

[0060] Coefficient device 108 may modulate the encoded continuous variable (504). For example, SLM 308 may be a coefficient device 108, and SLM 308 may comprise one or more addressable modulation elements representative of one or more coefficients of the QUO model. SLM 308 may modulate the encoded continuous variable by modulating the encoded property of light, e.g., by modulating at least one of the encoded amplitude of the light, the encoded phase of the light, or the encoded polarization of the light.

[0061] One or more detector 110 may detect the encoded and modulated continuous variable (506). For example, sensors 310 may be representative of detectors 110, and sensors 310 may detect, sense and / or measure the encoded and modulated property of light exiting SLM 308. Sensors 310 may output one or more analog electrical signals 342 that are proportional to the product of the encoded continuous variable (e.g., the encoded property of light) andDocket No.1248-151WO01 / 230045 one or more coefficients of the QUO model (e.g., one more of the addressable modulation elements of SLM 308) and output signals 342.

[0062] In some examples, PIC 304 may output encoded light from a plurality of channels, e.g., N channels, the encoded light representative of an analog encoded continuous variable, and SLM 308 may modulate at least one of an amplitude, phase, or polarization of the encoded light that is output from PIC 304. In some examples, the method includes decomposing a computationally hard problem into a set of coefficients, and the coefficients are represented by a transmission value of each element of the plurality of modulation elements of SLM 308, e.g., to modulate the amplitude of the encoded light. In some examples, SLM 308 may modulate the encoded light using OVMM by modulating, independent of each of the other modulation elements of the plurality of modulation elements of the SLM, the at least one of the amplitude, the phase, or the polarization of the encoded light exiting PIC 304.

[0063] In some examples, system 300 may operate in an iterative feedback loop configured to converge on a convergence criteria, such as a local or global minimum of the QUSO energy function of Equation (3), a stability criteria of measured amplitudes by sensors 310, or any suitable convergence criteria. For example, system 300 may comprise a system for iteratively solving a quadratic unconstrained optimization (QUO) problem using gradient descent and optical vector matrix multiplication (OVMM). System 300 may include PIC 304 configured to encode continuous variables represented as analog voltages onto an array of optical signals, lenses 332 configured to perform spatial addressing of the optical signals exiting PIC 304 (e.g., spatial addressing of the optical signals from the plurality of channels from PIC 304), and matrices (e.g., Q) representing QUO coefficients programmed into SL 308 (e.g., programmed into the modulation elements of SLM 308). System 300 may include a detector array (e.g., sensors 310) configured to compute an output vector (e.g., signals 342) from the OVMM (e.g., a vector of detected, sensed, and / or measured encoded and modulated properties of light from each of the detectors of detector array as one or more signals 342). The output vector may represent gradients of the continuous variables, e.g., of the QUO model. System 300 may include analog circuitry configured to scale the gradients of the continuous variables and add noise, and in some examples, apply and / or add nonlinearity to signals 342

[0064] The following examples may illustrate one or more aspects of the disclosure:

[0065] Example 1: A system including: an encoding device configured to encode a continuous variable; a coefficient device including an addressable modulation elementDocket No.1248-151WO01 / 230045 configured to modulate the encoded continuous variable; and an addressable detector configured to detect the encoded and modulated continuous variable.

[0066] Example 2: The system of example 1, wherein the continuous variable is representative of a parameter of a quadratic unconstrained optimization (QUO) model; and wherein the addressable modulation element is representative of a coefficient of the QUO model.

[0067] Example 3: The system of example 2, wherein the encoding device is configured to encode the continuous variable using an analog encoding, wherein the coefficient device is configured to modulate a property of the encoded continuous variable in an analog domain, and wherein the addressable detector is configured to output an analog electrical signal that is proportional to a mathematical product of the encoded continuous variable and the coefficient of the QUO model.

[0068] Example 4: The system of example 3, wherein a property of light is representative of the continuous variable, wherein the property of light includes at least one of an amplitude of the light, a phase of the light, or a polarization of the light, and wherein the encoding device is configured to encode the continuous variable by proportionally changing, by a first amount, the property of the light, and wherein the coefficient device is configured to modulate the encoded continuous variable by modulating the encoded property of light, by proportionally changing, by a second amount, the property of the light based on the coefficient of the QUO model.

[0069] Example 5: The system of example 4, wherein the encoding device includes a photonic integrated circuit (PIC) including an opto-electronic component configured to proportionally change the property of the light.

[0070] Example 6: The system of example 5, wherein the opto-electronic component includes a resonator including a microdisk modulator.

[0071] Example 7: The system of any one of examples 4 through 6, wherein the encoding device is configured to encode a plurality of continuous variables, wherein the coefficient device includes a plurality of addressable modulation elements, wherein each addressable modulation element of the plurality of addressable modulation elements corresponds to a different coefficient of the QUO model and is configured to modulate an encoded continuous variable of the plurality of encoded continuous variable by an amount corresponding to the corresponding coefficient, and wherein the detector includes a detector array including plurality of addressable detector elements configured to output a plurality of analog electrical signals, wherein each addressable detector element is configured to output a correspondingDocket No.1248-151WO01 / 230045 analog electrical signal of the plurality of analog electrical signals that is proportional to a sum of one or more encoded and modulated continuous variables.

[0072] Example 8: The system of example 7, wherein a property of light is representative of each continuous variable of the plurality of continuous variables, wherein the property of light includes at least one of an amplitude of the light, a phase of the light, or a polarization of the light, wherein the encoding device includes a splitter configured to couple the light into a plurality of channels of the encoding device, wherein each channel corresponds to a continuous variable of the plurality of continuous variables, wherein each channel of the plurality of channels includes an opto-electronic component including a resonator configured to encode the continuous variable by proportionally changing the property of the light within the channel.

[0073] Example 9: The system of example 8, further includes a first cylindrical lens configured to focus the light exiting the encoder device onto the plurality of addressable modulation elements of the coefficient device, wherein each addressable modulation element of the plurality of addressable modulation elements of the coefficient device is configured to modulate the encoded continuous variable by modulating, independent of each of the other addressable modulation elements of the plurality of addressable modulation elements of the coefficient device, the encoded property of light exiting the encoding device.

[0074] Example 10: The system of example 9, wherein each addressable detector element of the detector array is configured to detect the modulated and encoded property of light of a corresponding channel of the encoding device, the system further includes a second lens configured to focus the encoded and modulated property of light exiting the coefficient device onto the detector array.

[0075] Example 11: The system of example 10, wherein each addressable detector element of the detector array is configured to output an electrical signal representative of the mathematical product of a corresponding encoded continuous variable of the plurality of encoded continuous variables and at least a portion of the plurality of coefficients of the QUO model.

[0076] Example 12: The system of example 11, wherein the encoding a first encoding at a first time, wherein the opto-electronic component of each channel is configured to encode the property of the light with a second encoding at a second time subsequent to the first time by proportionally changing the property of the light based on the electrical signal output by a corresponding addressable detector element of the detector array.Docket No.1248-151WO01 / 230045

[0077] Example 13: The system of any one of examples 1 through 12, wherein the coefficient device includes a spatial light modulator (SLM).

[0078] Example 14: The system of any one of examples 1 through 13, wherein the analog encoding device is configured to encode the continuous variable based on a continuously differentiable function.

[0079] Example 15: A method includes encoding, by an encoding device, a continuous variable; modulating, by an addressable modulation element of a coefficient device, the encoded continuous variable; and detecting, by an addressable detector, the encoded and modulated continuous variable.

[0080] Example 16: The method of example 15, wherein the continuous variable is representative of a parameter of a quadratic unconstrained optimization (QUO) model, and wherein the addressable modulation element is representative of a coefficient of the QUO model.

[0081] Example 17: The method of example 16, further includes outputting, by the addressable detector, an analog electrical signal that is proportional to the product of the encoded continuous variable and the coefficient of the QUO model.

[0082] Example 18: The method of example 17, wherein encoding the continuous variable includes encoding the continuous variable based on a continuously differentiable function.

[0083] Example 19: The method of example 18, wherein encoding the continuous variable includes encoding a property of a light, wherein the property of light includes at least one of an amplitude of the light, a phase of the light, or a polarization of the light, and wherein modulating the encoded continuous variable includes modulating the encoded property of the light.

[0084] Example 20: A system for iteratively solving a quadratic unconstrained optimization (QUO) problem using gradient descent and optical vector matrix multiplication (OVMM), the system including: a photonic integrated circuit (PIC) configured to encode continuous variables represented as analog voltages onto an array of optical signals; lenses configured to perform spatial addressing of the optical signals exiting the PIC; matrices representing QUO coefficients programmed into a spatial light modulator (SLM); a detector array configured to compute an output vector from the OVMM, the output vector representing gradients of the continuous variables; and analog circuitry configured to scale the gradients of the continuous variables and add noise.

[0085] The techniques described in this disclosure may be implemented, at least in part, in hardware, software, firmware, or any combination thereof. For example, various aspects ofDocket No.1248-151WO01 / 230045 the techniques may be implemented within one or more microprocessors, DSPs, ASICs, FPGAs, or any other equivalent integrated or discrete logic QRS circuitry, as well as any combinations of such components, embodied in external devices. The terms “processor” and “processing circuitry” may generally refer to any of the foregoing logic circuitry, alone or in combination with other logic circuitry, or any other equivalent circuitry, and alone or in combination with other digital or analog circuitry.

[0086] For aspects implemented in software, at least some of the functionality ascribed to the systems and devices described in this disclosure may be embodied as instructions on a computer-readable storage medium such as RAM, DRAM, SRAM, magnetic discs, optical discs, flash memories, or forms of EPROM or EEPROM. The instructions may be executed to support one or more aspects of the functionality described in this disclosure.

[0087] In addition, in some respects, the functionality described herein may be provided within dedicated hardware and / or software modules. Depiction of different features as modules or units is intended to highlight different functional aspects and does not necessarily imply that such modules or units must be realized by separate hardware or software components. Rather, functionality associated with one or more modules or units may be performed by separate hardware or software components or integrated within common or separate hardware or software components. Also, the techniques may be fully implemented in one or more circuits or logic elements.

Claims

Docket No.1248-151WO01 / 230045 CLAIMS What is claimed is:

1. A system comprising: an encoding device configured to encode a continuous variable; a coefficient device comprising an addressable modulation element configured to modulate the encoded continuous variable; and an addressable detector configured to detect the encoded and modulated continuous variable.

2. The system of claim 1, wherein the continuous variable is representative of a parameter of a quadratic unconstrained optimization (QUO) model; and wherein the addressable modulation element is representative of a coefficient of the QUO model 3. The system of claim 2, wherein the encoding device is configured to encode the continuous variable using an analog encoding, wherein the coefficient device is configured to modulate a property of the encoded continuous variable in an analog domain, and wherein the addressable detector is configured to output an analog electrical signal that is proportional to a mathematical product of the encoded continuous variable and the coefficient of the QUO model.

4. The system of claim 3, wherein a property of light is representative of the continuous variable, wherein the property of light comprises at least one of an amplitude of the light, a phase of the light, or a polarization of the light, and wherein the encoding device is configured to encode the continuous variable by proportionally changing, by a first amount, the property of the light, and wherein the coefficient device is configured to modulate the encoded continuous variable by modulating the encoded property of light, by proportionally changing, by a second amount, the property of the light based on the coefficient of the QUO model.Docket No.1248-151WO01 / 230045 5. The system of claim 4, wherein the encoding device comprises a photonic integrated circuit (PIC) comprising an opto-electronic component configured to proportionally change the property of the light.

6. The system of claim 5, wherein the opto-electronic component comprises a resonator comprising a microdisk modulator.

7. The system of claim 4, wherein the encoding device is configured to encode a plurality of continuous variables, wherein the coefficient device comprises a plurality of addressable modulation elements, wherein each addressable modulation element of the plurality of addressable modulation elements corresponds to a different coefficient of the QUO model and is configured to modulate an encoded continuous variable of the plurality of encoded continuous variable by an amount corresponding to the corresponding coefficient, and wherein the detector comprises a detector array comprising plurality of addressable detector elements configured to output a plurality of analog electrical signals, wherein each addressable detector element is configured to output a corresponding analog electrical signal of the plurality of analog electrical signals that is proportional to a sum of one or more encoded and modulated continuous variables.

8. The system of claim 7, wherein a property of light is representative of each continuous variable of the plurality of continuous variables, wherein the property of light comprises at least one of an amplitude of the light, a phase of the light, or a polarization of the light, wherein the encoding device comprises a splitter configured to couple the light into a plurality of channels of the encoding device, wherein each channel corresponds to a continuous variable of the plurality of continuous variables, wherein each channel of the plurality of channels comprises an opto-electronic component comprising a resonator configured to encode the continuous variable by proportionally changing the property of the light within the channel.

9. The system of claim 8, further comprising:Docket No.1248-151WO01 / 230045 a first cylindrical lens configured to focus the light exiting the encoder device onto the plurality of addressable modulation elements of the coefficient device, wherein each addressable modulation element of the plurality of addressable modulation elements of the coefficient device is configured to modulate the encoded continuous variable by modulating, independent of each of the other addressable modulation elements of the plurality of addressable modulation elements of the coefficient device, the encoded property of light exiting the encoding device.

10. The system of claim 9, wherein each addressable detector element of the detector array is configured to detect the modulated and encoded property of light of a corresponding channel of the encoding device, the system further comprising: a second lens configured to focus the encoded and modulated property of light exiting the coefficient device onto the detector array.

11. The system of claim 10, wherein each addressable detector element of the detector array is configured to output an electrical signal representative of the mathematical product of a corresponding encoded continuous variable of the plurality of encoded continuous variables and at least a portion of the plurality of coefficients of the QUO model.

12. The system of claim 11, wherein the encoding a first encoding at a first time, wherein the opto-electronic component of each channel is configured to encode the property of the light with a second encoding at a second time subsequent to the first time by proportionally changing the property of the light based on the electrical signal output by a corresponding addressable detector element of the detector array.

13. The system of claim 1, wherein the coefficient device comprises a spatial light modulator (SLM).

14. The system of claim 1, wherein the analog encoding device is configured to encode the continuous variable based on a continuously differentiable function.

15. A method comprising: encoding, by an encoding device, a continuous variable;Docket No.1248-151WO01 / 230045 modulating, by an addressable modulation element of a coefficient device, the encoded continuous variable; and detecting, by an addressable detector, the encoded and modulated continuous variable.

16. The method of claim 15, wherein the continuous variable is representative of a parameter of a quadratic unconstrained optimization (QUO) model, and wherein the addressable modulation element is representative of a coefficient of the QUO model.

17. The method of claim 16, further comprising: outputting, by the addressable detector, an analog electrical signal that is proportional to the product of the encoded continuous variable and the coefficient of the QUO model.

18. The method of claim 17, wherein encoding the continuous variable comprises encoding the continuous variable based on a continuously differentiable function.

19. The method of claim 18, wherein encoding the continuous variable comprises encoding a property of a light, wherein the property of light comprises at least one of an amplitude of the light, a phase of the light, or a polarization of the light, and wherein modulating the encoded continuous variable comprises modulating the encoded property of the light.

20. A system for iteratively solving a quadratic unconstrained optimization (QUO) problem using gradient descent and optical vector matrix multiplication (OVMM), the system comprising: a photonic integrated circuit (PIC) configured to encode continuous variables represented as analog voltages onto an array of optical signals; lenses configured to perform spatial addressing of the optical signals exiting the PIC; matrices representing QUO coefficients programmed into a spatial light modulator (SLM); a detector array configured to compute an output vector from the OVMM, the output vector representing gradients of the continuous variables; and analog circuitry configured to scale the gradients of the continuous variables and add noise.

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