Learning with parameterized quantum-thermodynamic circuits

Abstract

Systems and methods of preparing a mixed quantum state using a thermodynamic - quantum hybrid scheme are provided. A system comprises a thermodynamic processor, a quantum processor, and a controller operatively coupled to the thermodynamic processor and the quantum processor. A plurality of samples from a data state is received. A plurality of configuration parameters of a thermodynamic processor is initialized to initial values, the thermodynamic processor having a plurality of visible nodes and a plurality of hidden nodes. A plurality of configuration parameters of a quantum processor is initialized to initial values. The hidden nodes of the thermodynamic processor are sampled. Based on samples of the visible nodes and sample of the hidden nodes and on the data state, the configuration parameters of the thermodynamic processor are updated. Based on samples of the visible nodes and samples of the hidden nodes, the configuration parameters of the quantum processor are updated.

Description

EXI-00125 LEARNING WITH PARAMETERIZED QUANTUM-THERMODYNAMIC CIRCUITS RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Provisional Application No. 63 / 623,482, filed on January 22, 2024, and U.S. Provisional Application No. 63 / 677,749, filed on July 31, 2024. The entire teachings of the above applications are incorporated herein by reference. BACKGROUND

[0002] Embodiments of the present disclosure relate to thermodynamic-quantum hybrid schemes, and more specifically, to thermodynamic-quantum hybrid schemes for preparing a mixed quantum state. BRIEF SUMMARY

[0003] According to embodiments of the present disclosure, systems and methods of preparing a mixed quantum state using a thermodynamic-quantum hybrid scheme are provided.

[0004] In various embodiments, a system is provided comprising: a thermodynamic processor, having a plurality of configuration parameters, a plurality of visible nodes, and a a quantum processor, having a plurality of configuration a controller operatively coupled to the thermodynamic processor and the quantum processor. In various embodiments, the controller is configured to receive a plurality of samples from a data state, initialize the plurality of configuration parameters of the thermodynamic processor to initial values, initialize the plurality of configuration parameters of the quantum processor to initial values, sample from the hidden nodes of the thermodynamic processor, update the configuration parameters of the thermodynamic processor based on samples of the visible nodes and sample of the hidden nodes and on the data state, and update the configuration parameters of the quantum processor based on samples of the visible nodes and samples of the hidden nodes.

[0005] In some embodiment, updating the configuration parameters of the thermodynamic processor comprises: clamping the visible nodes according to the data state, therebybeginning a first clamped phase sampling the hidden nodes during the firstunclamping the visible nodes, thereby ending the first clamped phase and beginning anunclamped phase and sampling the visible and hidden nodes during the unclamped phase,EXI-00125 wherein the configuration parameters are based on a comparison between the sampling during the clamped phase and the sampling during the unclamped phase. In some embodiments, updating the configuration parameters of the quantum processor comprises: clamping the visible nodes according to the sampled visiblenodes, thereby beginning a second clamped phase sampling the hidden nodes during thesecond and preparing an ansatz on the quantum processor based on the sampling during the clamped phase and the sampling during the unclamped phase, wherein the configuration parameters are based on values measured from the quantum processor.

[0006] In some embodiments, the controller is further configured to prepare the quantum processor in an initial state defined by samples of the visible nodes. In some embodiments, the quantum processor is configured to execute a quantum circuit defined by its updated parameters on the initial state. In some embodiments, the controller is further configured to measure from the quantum processor.

[0007] In some embodiments, the initial values of the configuration parameters of the thermodynamic processor are chosen randomly.

[0008] In some embodiments, the initial values of the configuration parameters of the quantum processor are chosen randomly.

[0009] In some embodiments, the plurality of samples of the data state are measured from a quantum processor.

[0010] In some embodiments, the plurality of samples of the data state are measured from a quantum sensor.

[0011] In some embodiments, each of the plurality of visible nodes and each of the plurality of hidden nodes comprises a superconducting flux qubit.

[0012] In some embodiments, updating the configuration parameters of the thermodynamic processor comprises gradient descent.

[0013] In some embodiments, updating the configuration parameters of the quantum processor comprises gradient descent.

[0014] In some embodiments, each of the plurality of visible nodes and each of the plurality of hidden nodes comprises a flux qubit.

[0015] In various embodiments, a method of preparing a mixed quantum state using a thermodynamic-quantum hybrid scheme is provided. A plurality of samples from a data state is received. A plurality of configuration parameters of a thermodynamic processor is initialized to initial values, the thermodynamic processor having a plurality of visible nodes and a plurality of hidden nodes. A plurality of configuration parameters of a quantumEXI-00125 processor is initialized to initial values. The hidden nodes of the thermodynamic processor are sampled. Based on samples of the visible nodes and sample of the hidden nodes and on the data state, the configuration parameters of the thermodynamic processor are updated. Based on samples of the visible nodes and samples of the hidden nodes, the configuration parameters of the quantum processor are updated.

[0016] In some embodiments, the configuration parameters of the thermodynamic processor are updated. The visible nodes are clamped according to the data state, thereby beginning a first clamped phase. The hidden nodes are sampled during the first clamped phase. The visible nodes are unclamping, thereby ending the first clamped phase and beginning an unclamped phase. The visible and hidden nodes are sampled during the unclamped phase, wherein the configuration parameters are based on a comparison between the sampling during the clamped phase and the sampling during the unclamped phase.

[0017] In some embodiments, the configuration parameters of the quantum processor are updated. The visible nodes are sampled. The visible nodes are clamped according to the sampled visible nodes, thereby beginning a second clamped phase. The hidden nodes are sampled during the second clamped phase. An ansatz is prepared on the quantum processor based on the sampling during the clamped phase and the sampling during the unclamped phase, wherein the configuration parameters are based on values measured from the quantum processor.

[0018] In some embodiments, said method further comprises preparing the quantum processor in an initial state defined by samples of the visible nodes, wherein the quantum processor is configured to execute a quantum circuit defined by its updated parameters on the initial state. In some embodiments, said method further comprises measuring from the quantum processor.

[0019] In some embodiments, the initial values of the configuration parameters of the thermodynamic processor are chosen randomly.

[0020] In some embodiments, the initial values of the configuration parameters of the quantum processor are chosen randomly.

[0021] In some embodiments, the plurality of samples of the data state are measured from a quantum processor.

[0022] In some embodiments, the plurality of samples of the data state are measured from a quantum sensor.

[0023] In some embodiments, each of the plurality of visible nodes and each of the plurality of hidden nodes comprises a superconducting flux qubit.EXI-00125

[0024] In some embodiments, updating the configuration parameters of the thermodynamic processor comprises gradient descent.

[0025] In some embodiments, updating the configuration parameters of the quantum processor comprises gradient descent.

[0026] In some embodiments, each of the plurality of visible nodes and each of the plurality of hidden nodes comprises a flux qubit. BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0027] Fig. 1 shows an illustration of an architecture to perform training and inference of a thermodynamic quantum hybrid ansatz according to an embodiment of the present disclosure.

[0028] Fig. 2 shows a preparation of a thermal quantum state according to an embodiment of the present disclosure.

[0029] Fig. 3 shows a sampling from a quantum probability distribution according to an embodiment of the present disclosure.

[0030] Fig. 4 shows an algorithm for computing a quantum probability distribution according to an embodiment of the present disclosure.

[0031] Fig. 5 shows an algorithm for generating samples from a quantum probability distribution according to an embodiment of the present disclosure. DETAILED DESCRIPTION

[0032] A variational procedure is introduced for preparing a mixed quantum state via a thermodynamic-quantum hybrid scheme using a thermodynamic computer, capable of sampling and energy estimation, and a quantum computer or quantum simulator. The procedure assumes access to multiple copies of the state of interest, as well as to a thermodynamic chip capable of producing rapid samples from a statistical ensemble of discrete variables. An overview is provided of how the thermodynamic-quantum hybrid scheme can be used for quantum modular Hamiltonian learning (QHML) effectively approximating a target mixed quantum state via learning its modular Hamiltonian.

[0033] Quantum machine learning is a generalization of classical machine learning enabled by the more general set of computing operations that can be executed on quantum processors. In variational quantum machine learning, parameterized quantum circuits can be used to learn and sample from distributions that arise from quantum experiments on quantum mechanical systems.EXI-00125

[0034] Thermodynamic computing is an emerging paradigm that harnesses the principles of thermodynamics to perform computational tasks. Unlike traditional computing, which relies on deterministic logic gates and binary representations, thermodynamic computing exploits the stochastic behavior and energy landscapes of physical systems to solve complex problems efficiently. At the core of the thermodynamic computing provided herein lies the concept of energy-based models, which represent the system’s state and its associated energy. By carefully designing the energy landscape, one can encode the desired computational problem into the physical system. The system then naturally evolves towards the equilibrium distribution that can be sampled efficiently. Thermodynamic computing offers several advantages, such as leveraging the inherent parallelism and energy efficiency of physical systems, and the ability to efficiently sample from complex probability distributions and find optimal solutions in high-dimensional spaces. The physical realization of thermodynamic computers can take various forms, such as superconducting flux qubits, which can be engineered to exhibit desired energy landscapes and coupled together to form networks capable of performing computational tasks. To harness the power of thermodynamic computing, a hybrid architecture is often employed, combining a classical digital computer for storing and updating model parameters with a thermodynamic co-processor dedicated to efficient sampling and inference. Thermodynamic computing has the potential to revolutionize various domains, including machine learning, optimization, and simulation, by offering a novel approach to tackling complex computational challenges. In the following, the acronym TC is used to refer to a thermodynamic computer.

[0035] Thermodynamic computing is a classical probabilistic computing paradigm that leverages thermal noise and dissipative non-equilibrium thermodynamic processes to execute computations on classical statistical distributions. Thermodynamic computing programs can be executed by digital simulations on traditional floating point processors, or on processors that directly implement the desired non-equilibrium physics for faster computations that sidestep the overhead of the classical digital simulation.

[0036] Hybrid quantum thermodynamic computing corresponds to computing technologies that integrate both quantum computing and thermodynamic computing methods and technologies. It is possible to mesh both paradigms in a natural framework that enables novel and improved computing capabilities. Notably, machine learning generalizes to quantum thermodynamic learning and inference to leverage both the extended representation power of quantum processors and the efficiency of thermodynamic devices to learn and infer from all possible distributions that can arise in the physical world. Establishing the foundations ofEXI-00125 machine learning on the first principles of physics will enable new capabilities such as artificial intelligence that can reason about the physical world in a sound and consistent fashion from the ground up.

[0037] The present disclosure describes an integration of quantum modular Hamiltonian learning with thermodynamic computing. This integration can enable the next generation of multimodal machine learning models. The fast-sampling capabilities of the thermodynamic devices presented in this disclosure were combined with either quantum devices or simulators.

[0038] In quantum modular Hamiltonian learning (QMHL), one is given a mixed quantum state and tasked with finding the effective modular Hamiltonian from which it is generated. This effectively corresponds to finding the logarithm of the density matrix.

[0039] QMHL can be used in various contexts. The mixed state may come from one or several networked quantum or classical sensors where QMHL is used to construct a generative model which mimics the response of the sensors. Such a subroutine can be used in an active inference setting to train the generative model of the agent. The mixed state may come from a quantum simulation on a quantum processor, in which case QMHL simply recovers the Hamiltonian whose thermal state is closest to the input distribution.

[0040] In both QMHL, processing the classical part of the distributions on a thermodynamic processor simplifies and accelerates the learning and inference processes. The complete procedures to execute those tasks on computing systems that integrate classical digital, thermodynamic and quantum computing devices are provided.

[0041] The present disclosure provides methods to train and infer from energy-based models (EBMs) with latent variables as implemented on hybrid quantum-thermodynamic computing hardware. Sampling from these models directly via thermodynamic hardware and executing quantum circuits on initial states prepared from the samples can produce an ensemble of quantum states, a mixed state, whose variational parameters can be optimized via gradients consisting of physically accessible quantities on the thermodynamic chip and the QPU.

[0042] The formalism corresponding to the quantum-thermodynamic hybrid hardware interface, on which the procedure mentioned above is executed, is introduced.

[0043] QHBMs are a class of generative quantum-probabilistic models that learn mixed quantum states by approximating their modular Hamiltonians. The model stateis defined as:EXI-00125 Equation 1 where represents a variational latent state, and denotes a parameterized unitary transformation applied to . This transformation explores the space of quantum state configurations, leading to the output state , which is engineered to possess desired properties or to approximate a target quantum state. The latent state is defined as the thermal state of the latent modular Hamiltonian :Equation 2 is parameterized by , and is the partition function. The latent space can have a factorized or general classical structure. Here, the modularity of the Hamiltonian refers to the fact that the density matrix is fully encoded in . They have the same eigenvectors, and their eigenvalues are simply related.

[0044] In order to incorporate thermodynamic processors into the QHBM framework, the following ansatz is used:Equation 3

[0045] Here, represents a thermodynamic probabilistic model parameterized by , which assigns probabilities to different visible states . The parametrized quantum circuit acts on the visible states, creating a quantum superposition of the classical probabilistic model. This ansatz allows for the seamless integration of thermodynamic probabilistic models with parameterized quantum circuits, enabling the exploration of complex quantum state spaces.

[0046] One of the key tasks of Quantum Modular Hamiltonian Learning (QMHL) is to learn a mixed state by minimizing the quantum relative entropy. The modular Hamiltonian in this case is learned implicitly and can be accessed via is parameterized as an energy based model (EBM), this simplifies

[0047] The parameterization of the modular Hamiltonian in QHBMs can facilitate not only the simulation of quantum states but can also provide a framework for the learning ofEXI-00125 quantum dynamics. The flexibility in manipulating and allows for the optimization of the state to achieve high fidelity with respect to the target state, which can be useful for tasks ranging from quantum simulation to complex quantum system analysis.

[0048] Gradients are computed for the parameterized modular Hamiltonian and unitary. Applications include quantum simulations, state reconstructions and tomography, quantum thermodynamic predictions and machine learning. More concretely, QMHL allows to learn unknown states , which is also referred to as the quantum data. This data can come from a state in a laboratory or another quantum computer.

[0049] An overview of the thermodynamic chips implemented in the training and inference protocols involved in hybrid quantum-thermodynamic computing is provided.

[0050] Thermodynamic computing is a novel paradigm that leverages the principles of thermodynamics to perform computational tasks. Unlike conventional computing, which relies on deterministic logic gates and binary representations, thermodynamic computing harnesses the stochastic behavior and energy landscapes of physical systems to solve complex problems efficiently.

[0051] A thermodynamic processor is a physical computing device whose average state evolution is engineered to equilibrate to a steady state through thermal dissipation. A thermodynamic neuromorphic chip (herein "thermodynamic chip", "thermo chip") is a thermodynamic processor whose computing units consists in an ensemble of coupled artificial neurons designed to physically sample from and / or learn directly the parameters of energy based models. For instance, the couplings between the neurons can be chosen to represent Hopfield networks of Boltzmann machines and their variants such as restricted Boltzmann machines and deep Boltzmann machines.

[0052] The neurons in an energy-based model can be labeled as visible when they represent observable data and hidden when used to represent latent information about the data. The state of those neurons are dynamical variables which can fluctuate during the inference process and the training steps. The chip models a distribution of the hidden and visible neurons as . To perform QMHL, the thermodynamic neuromorphic chip can be operated in two phases. The first phase is called the clamped phase where a strong clamping potential is turned on to fix the value of the visible variables, with the hidden neurons evolving freely following Langevin dynamics. In this phase, one can obtain states of the hidden neurons from the conditional distribution . The second phase is called the unclamped phase where allEXI-00125 variables evolve freely following dissipative dynamics such as Langevin dynamics. In this phase, one can obtain states of the hidden and visible neurons from the joint distribution . In both cases, the state of the visible and hidden variables can be initialized to some values and at different steps of the training and inference processes. The state of the visible and hidden variables, whose position degrees of freedom are encoded in and , can also be measured at any given time or integrated over a period of time. The parameters of the model can be static variables representing tunable physical parameters of the chip and they can correspond to the initial state of a subset of hidden neurons .

[0053] The thermodynamic chip which are used can formally be thought of as an energy- based model (EBM) with visible units and hidden units . The energy of each state of the thermodynamic chip is . The probability distribution of the state variables of the EBM representing the thermodynamic chip is given by, Equation 4 where are the model's parameters, is the energy function of the thermodynamic chip and its partition function is given by. Equation 5

[0054] One can sample the visible units from the distribution of the visible variables on the thermodynamic chip, obtained formally by marginalizing over the hidden variables. Equation 6

[0055] Since EBMs are properly defined over pairs of conjugate variables, in some cases the momentum of the variable may be measurable at any time or over periods of time as well. The variables of the EBM defined by the circuit parameters of a thermodynamic neuromorphic chip are said to evolve under free evolution when no external operations areEXI-00125 applied on the controls of the chip. The direction of the free evolution of the variables of the chip can be represented by a Hamiltonian and a dissipative process.

[0056] Both in the clamped and unclamped modes, it is possible to measure the quantities that are used to update the parameters of the model. These quantities can be sampled and computed from the state of the variables at specific times or integrated over periods of time.

[0057] QMHL is agnostic to the substrate of the TD platform and the quantum computing platform. The main ideas of the superconducting flux qubit platform of the TD chip are outlined.

[0058] As an example, the total potential energy of the thermodynamic processor implemented on a superconducting circuit can be written asEquation 7 where the 's position degrees of freedom, the 's are the corresponding momenta and and are constants. The energy function is defined on a graph with vertices and edges . Following the previous definition of EBMs, these vertices are split into visible and . The nodes are the visible nodes used for the data encoding and are the hidden nodes for auxiliary information. In contrast to classical neural networks, the weights and biases on a TD chip are implemented via additional nodes for the weights and for the biases that are coupled to the nodes acting as neurons. The weight and bias nodes are also part of the hidden nodes of the system, i.e., for . The subscript s in stands for synapse, referring to the connection between neurons because acts as the coupling between the neurons and . Visible and hidden nodes are the same type of subcomponent on the TD hardware, only their use how they are coupled distinguishes their function. The energy functionrepresents the energy of the visible and hidden nodes, represents the energy of the hidden nodes that correspond to the coupling strength and is the energy of the individual the biases . The term corresponds to the coupling of the nodes and via the coupling strength induced by . And the term corresponds to the bias terms of the visible nodes induced by theEXI-00125 bias term . The sums in the above equation are either over the edges or the visible nodes of the graph of the TD model.

[0059] Each neuron is an oscillator which can have non-linearities to shape the structure of the energy model. For example, the energy functions of the neurons can have the formEquation 8

[0060] When built with superconducting circuits, the mass term is a capacitance, the harmonic potential energy term is an inductance, the non-linear potential term is a Josephson energy and bothare external fluxes that allow to adjust the parameters of the thermodynamic chip. In the case of superconducting circuits, the position terms are in units of flux and the momentum terms are in units of charge. The label is meant the represent either the visible neurons, the weights neurons or the biases neurons . The placeholder stands for the corresponding indices.

[0061] Referring to Fig. 1, an architecture used to perform training and inference of the thermodynamic-quantum hybrid ansatz is shown including a dilution refrigerator 101, a quantum processing unit (QPU) 102. a thermo chip 103, and a controller 104 (e.g., FPGA or ASIC controller). This ensures optimal communication between the two processors. Combining the thermodynamic model expressed above with a quantum circuitparameterized by allows for fast sampling and high latency interaction of the thermal and the quantum device. As depicted in Fig. 1, one can exploit the fast sampling capabilities of the thermodynamic hardware best when the QPU 102 and the thermo chip 103 share the same controller 104.

[0062] During inference, data is sampled from the visible neurons of the thermodynamic chip and the corresponding state is prepared on the quantum computer.

[0063] Referring to Fig. 2, a preparation of the thermal quantum state is shown. The procedure involves: 1) sampling the thermal processor 201 and sending the samples to the controller 202, 2) preparing the state on the quantum processor 203 and applying the unitary . As shown in Fig. 2, the quantum circuitis executed on the quantum processor 203 and the distribution of the model can be sampled by performing measurement on the quantum processor 203. In practice, the quantum processor can be aEXI-00125 noisy quantum processor, a fault-tolerant quantum computer or a classical computer simulating a quantum computer.

[0064] As an example, a hybrid scheme of a thermodynamic chip with superconducting flux qubits in tandem with a superconducting QPU to learn a target density matrix is presented. As shown in Fig. 1 the thermodynamic chip can be placed in the same dilution refrigerator as the superconducting QPU and it can be controlled with the same FPGA or ASIC controller for faster communication between the two processors. The scheme is not limited to a specific quantum or thermodynamic platform. Alternative quantum hardware platforms that can be used in tandem with the thermodynamic chip are, trapped ions, neutral atom, photonic, silicon quantum dots, diamond vacancy centers and topological quantum devices. Also, the thermodynamic platform is not limited to superconducting flux qubits. Other approaches include and are not limited to semiconductors and mechanical resonators.

[0065] The parametrized quantum circuit architecture is ideally informed by the target quantum state that will be learned. This is problem dependent and there are many good strategies how to chose such an ansatz. One can also choose a less problem specific but more hardware efficient ansatz (HEA) or a more general quantum graph neural network.

[0066] In this example a general Hamiltonian variational ansatz is chosen. This ansatz can be efficiently implemented on superconducting QPUs for local spin Hamiltonians. In this case,the ansatz consists of k-local parameterized 2-qubit gates of the formwithrepresent the standard Pauli matrices acting on qubits and respectively. The parameters of the quantum circuit represent the angles of the rotations in this example.

[0067] The distribution is represented by a thermodynamic device. The thermodynamic framework allows a lot of freedom in the choice of the parameterization of . A relatively simple choice would be a deep Boltzmann machine, that has an energy function of the formwhere all the interactions are quadratic. The parameterization of the thermodynamic device hence represents the couplings between the variables.

[0068] During the training of the hybrid ansatz the parameters and of both, the TD and the QC are adjusted such that the KL divergenceis minimal. After training one can use the trained TD and QPU for various tasks, such as calculating expectation values of arbitrary operators and measuring entanglement.EXI-00125

[0069] In QMHL, the task is to learn a representation of the mixed state as an approximate parameterized density matrix . This objective can be achieved by minimizing the quantum cross-entropy between the data stateand the model , which is represented as the loss function. Equation 9

[0070] Minimizing yields the optimal parametersEquation 10

[0071] This is done by gradient descent. The general training proceeds by performing the following tasks: 1. Initializing the thermodynamic chip parameters and the quantum neural network parameters .2. Selecting a training schedule, namely an ordering of steps for updating the parameters and until a set of criteria such as convergence and runtime are met. 3. For the training steps of the thermodynamic chip, the stochastic update rule Equation 12 is used. 4. For the training steps of the quantum neural network, the stochastic update rule Equation 16 is used.

[0072] In the following, it is shown how to obtain the gradient update rule for a quantum- thermodynamic model, where the thermodynamic model has latent variables.

[0073] The quantum probability distribution is first defined. Equation 11 which can be sampled following the preparation shown in Fig. 3. Referring to Fig. 3, a preparation of the pulled-back quantum data 301 and sampling from the quantum probability distribution 302 are shown. Building on a stochastic learning methodEXI-00125 provided, it is shown that the update rule for the parameters of the thermodynamic chip at a given step is given byEquation 12 where is one of the samples from the quantum probability distribution in Equation 11. The noise is sampled from the normal distribution and the step sizes satisfy the conditionsand to maintain good exploration and convergence properties. The functional form satisfies these properties in general for positive hyperparameters and and .

[0074] The first part of Equation 12, also called the positive phase termis computed by using the clamped phase of the thermodynamic chip. First, samples are sampled from the quantum probability distribution prepared on a quantum processor according to the quantum circuit shown in Fig. 3. The samples can also be drawn from a quantum simulator on a classical computer. For each sample , one clamps the thermodynamic chip and measures a corresponding sample of the hidden units from . This yields a batch of samples . The positive phase term is then computed by averaging over these samplesEquation 13

[0075] The second term of Equation 12, also called the negative phase term is computed in the unclamped phase of the thermodynamic chip. The first step is to obtain samples for by physically sampling the visible nodes of the thermodynamic chip. The negative phase terms are computed by averaging over these samplesEquation 14EXI-00125

[0076] Combining the expressions from Equation 13 and Equation 14 into Equation 12, the final update rule is given byEquation 15

[0077] The expression for the update rule of the parameterized quantum circuit at step with step size is given byobservable which can be measured on a quantum processor using a Hadamard interferometer or the parameter shift rule. To compute the update rule, for each sample which is measured on the quantum processor along with its corresponding measurement result , the visible units of the thermodynamic chip are clamped to and a set of samples can be used to approximate . In this case, the noise is also sampled from the normal distribution and the step sizes satisfy the conditions and . The functional form can be used with the same conditions on the hyperparameters as for the training of the thermodynamic chip parameters.

[0079] In following, the full training sequence is provided.

[0080] Architecture: An exemplary thermo sampling chip is locally controlled by an FPGA (or ASIC). Superconducting QPU located in the same dilution fridge as the thermo chip is also locally controlled by an FPGA. Both the thermo and quantum chips are programmed and controlled by a CPU that inputs instructions directly to and reads outputs from their respective FPGAs.

[0081] Input: The data state , which is a state that can be accessed and measured but do not know how to prepare. It could take a variety of forms (e.g., a physical system in the lab). One can stipulate that it exists on an auxiliary set of qubits within the same dilution fridge as the coupled thermo-quantum chip. Thus, one can call the QPU on which the parameterized unitary circuit runs "QPU 1", and the QPU which can generate the training data stateEXI-00125 "QPU 2". These are both of the same architecture (e.g., superconducting) and width (number of qubits).

[0082] It is also assumed that a model of exists on the CPU, allowing one to directly computegiven samples of and .

[0083] In addition, a step size schedule for the updates of the thermo chip parameters , as well as a step size schedule for the updates of the quantum parameters are needed.

[0084] This is done by performing the following steps: 1. Randomly initializing , the thermo chip parameters, and , the quantum circuit parameters and storing the parameters digitally. 2. Setting the parameter degrees of freedom of the thermo chip to the appropriate values of via the locally coupled FPGA or ASIC. Followed by allowing the neuron degrees of freedom (i.e., the visible and hidden nodes) to thermalize according to the distribution created by fixing . 3. Generating a set of data samples by sampling from. Equation 17 The parameterized unitary operations can be performed on QPU 2, and the Metropolis update steps can be done via a combination of locally coupled FPGA / ASIC and CPU control. Followed by storing these samples . 4. Using the local digital controller to clamp the visible nodes of the thermo chip using physical mechanisms. For each of the data state samples taken in step 3, conditionally sampling the hidden nodes of the thermo chipEquation 18 where one hidden sample per training data sample is considered, but more can be considered when needed for convergence. Followed by storing the samples alongside . This is the so-called clamped phase of training, since the visible units are clamped to data values. 5. Reinitializing the thermo chip parameters and allowing the nodes to re-thermalize. Followed by obtaining samples from the joint distribution of the unclampled thermo chipEXI-00125Equation 19 for and storing these samples digitally. This is the unclamped training phase. 6. Using the clamped and unclamped samples now stored on the digital computer to estimate the positive and negative phase terms, respectively, of the gradient for . Then, combining with the update step schedule and one can approximateEquation 20 by computing using an approximation of the functional form of on CPU. The resulting quantity can be stored on the control CPU. 7. Using the stored values of and on the CPU to update the thermo chip parametersEquation 21 and storing the new parameters on CPU. 8. On QPU 2, executing the operationEquation 22 which can be done by using a parameter shift rule to compute the gradients of the unitary operators. Then, sampling the visible units from the thermo chipEquation 23 for , for some sufficiently large . For each of these visible samples, conditionally samplingthe hidden nodes with the visible nodes clampedEquation 24EXI-00125 for , for some fixed number of conditional samples . With these samples, estimating the update of the QNN parameters:Equation 25 from Equation 16. Then, storing the value of . 9. Using the stored values of and to update the QNN parameters:Equation 26 10. Updating the time step and returning to step 2 11. Repeating steps 2 through 10 times until a desired level of convergence is reached. The resulting parameters and provides a QHBM that approximates the data state

[0085] With the resulting trained QHBM state, one can, for example, estimate local operator expectation values for the data state.

[0086] In QMHL, inference consists of preparing and measuring the trained QHBM stateEquation 27

[0087] The exact process can vary depending on the exact task at hand. Here, a simple case is considered involving estimating the expectation value of an operator , i.e., computingIt is stipulated that is sufficiently simple (e.g., a product of Pauli operators) that a unitary is known such that for all qubits. The inputs for inference are the same as for training, except one does not need a second QPU capable of generating the data state as it is assumed that one has already trained a QHBM to approximate this state.

[0088] The process is as follows: 1. Initializing the thermo chip parameters via the local FPGA or ASIC digital controller. 2. Initializing the visible and hidden neurons randomly, either uniformly or according to a chosen prior probability distribution.EXI-00125 3. Drawing a sample from the visible neurons on the thermo chip. 4. Storing on the control system for the QPU. 5. Preparing a state on the QPU corresponding to the sampled bit string state in the computational basis. 6. Preparing the stateby executing the pre-determined quantum circuit ansatzwith the input state 7. Transforming to the measurement basis by subsequently executing the circuit corresponding to the operator to obtain the state 8. Measuring in the computational basis to obtain a state . 9. Repeating preparation and measurement of the state for shots to obtain an estimate for 10. Storing 11. Repeating steps 1 through 9, sampling for for samples from the thermo chip distribution, and computing and storing12. ComputingEquation 28

[0089] A derivation of Equation 3 representing the quantum modelfrom the complete microscopic description of the hybrid quantum thermodynamic chip setup is provided. The state of the thermodynamic chip is assumed to be described by an EBM. The state of the visible neurons is labeled as and the state of the unclamped hidden neurons is labeled as . The state of the visible neurons can also be prepared on a quantum computer on which a continuous group of unitary transformations can be applied. The parameters of this continuous group of transformationare labeled . The set of all possible visible states isdenoted as and the size of this set is written as | |. Depending on the specific energybased model being implemented on the device, in some cases the parameters of the model during inference and training can a subset of the hidden neurons which are clamped and arelabeled as . In this static case, the set of all possible hidden states therefore does notEXI-00125 include the clamped parameters . The static representation of the hybrid quantum thermodynamic model is given by the density matrixEquation 29 where the hidden classical degrees of freedom of the thermodynamic chip can be traced over. Note that in some other cases, the parameters of the model are the initial state of a subset of hidden neurons and are allowed to fluctuate during training and inference. In those dynamical cases, all physical expectation values are also integrated over the subsequent fluctuation of those parameters.

[0090] The update rule for the parameters of the thermodynamic chip has the formEquation 30

[0091] Therefore, one must compute the gradient of the loss function with respect to the parameters of the thermodynamic chip to proceed in the training of the model. The loss function in Equation 9 can be written in a more explicit form to derive the expressions for the gradient with respect to the thermodynamic chip parameters:Equation 31EXI-00125

[0092] Another useful representation of the marginal distribution written in Equation 6 is given byEquation 32 where the partition function of the marginal distribution isEquation 33 with the normalization propertyEquation 34

[0093] To get the expression of the gradient of the cost function with respect to the parameters of the thermodynamic chip as written in Equation 37, it is useful to consider the quantityEquation 35 which is involved in the gradient of the log-likelihoodEXI-00125 Equation 36

[0094] Using the relation given above, the gradient with respect to of isEquation 37

[0095] The parameter update rule Equation 12 is obtained by inserting Equation 37 into Equation 4.

[0096] The update rule for the parameters of the quantum neural network has the formEquation 38

[0097] To compute this quantity, the expression for the gradient of the cost function with respect to the quantum neural network parameters is derived. The modular Hamiltonian used to represent the variational model is where the modular HamiltonianEquation 39 where from Equation 4 is used. The derivative of the QMHL cost function isEquation 40EXI-00125

[0098] In this case, the marginalized distribution isEquation 41 and using from Equation 46Equation 42 is a probability distribution and is a function. The gradient of to isEquation 43 whereEquation 44 using the chain rule. This last expression yields the gradientEquation 45 The update rule Equation 16 is obtained by inserting Equation 45 into Equation 38.

[0100] Sampling from the quantum probability distributiona quantum simulator using a classical computer can be done by applying the following procedure: 1. Storing in an appropriate format (e.g., full density matrix or tensor network). 2. Conjugating by unitary operatorsEXI-00125 Equation 46 3. Defining the function shown in Fig. 4. Fig. 4 shows an algorithm computing4. Generating samples from (e.g., via the metropolis Hastings algorithm shown in Fig. 5 for sampling

[0101] An exemplary system combines a digital computer for storing and updating parameters with a "thermo co-processor" engineered to follow Langevin dynamics for efficient sampling. The neurons representing hidden and visible nodes are constructed from superconducting flux qubits.

[0102] In systems where synapses are introduced, these represent the couplings between the neurons which are then no longer static. Some exemplary architectures do not contain call synapses. Instead, the only degrees of freedom on the TC are the neurons which represent the variables of interest. The couplings between the neurons are static and controlled by the classical controller. To change the interaction strength between two neurons the classical control must be changed.

[0103] Thermodynamic hardware, such as systems of coupled superconducting flux qubits, can be engineered to naturally represent energy-based models (EBMs) at equilibrium. The physical system is described by a Hamiltonian , where is the kinetic energy and is the energy function of the EBM, parametrized by . At equilibrium, the system minimizes its total energy, and the position variables follow the Boltzmann distribution defined by the energy function :Equation 47 where is the inverse temperature and is the partition function that normalizes the distribution. By carefully engineering the couplings and parameters of the physical system, the energy function can be designed to represent the desired probability distribution associated with the EBM. This allows the thermodynamic hardware to naturally capture the dependencies between variables and their configurations. Sampling from the equilibrium distribution of the thermodynamic hardware allows for efficient inference and learning in energy-based models. The sampled values of the position variables x correspond to the configurations of the EBM and can be obtained directly through measurement of the physical system. This direct sampling from the hard- ware eliminates theEXI-00125 need for computationally expensive Markov Chain Monte Carlo (MCMC) methods typically used in EBMs, leading to faster and more efficient training and inference.

[0104] Thermodynamic hardware provides a natural platform for representing and sampling from energy-based models by engineering the couplings and parameters of the physical system to match the desired energy function at equilibrium. This approach offers the potential for accelerated learning and inference in EBMs, leveraging the inherent parallelism and energy efficiency of the physical hardware.

[0105] In energy-based models, the inputs to the energy function can be divided into two categories: visible units and hidden units. Visible units, denoted as , represent the observed data or the variables of interest, while hidden units, denoted as , are latent variables that are not directly observed. This split is arbitrary and serves the purpose of dedicating some variables to the data and others to capturing additional dependencies and improving the expressivity of the model. Hidden units can also be used to mitigate hardware limitations, such as missing physical connections between variables. The joint probability distribution over both visible and hidden units in an EBM is given by:Equation 48 where is the energy function that takes both visible and hidden units as inputs, and is the partition function. Clamping is a technique used in EBMs where certain variables, typically the visible units x, are forced to take on specific values, usually corresponding to the observed data. When clamping the visible units, the model samples only the hidden units z from the conditional distribution:Equation 49 where is the partition function conditioned on the clamped visible units . In superconducting hardware, clamping can be achieved by increasing the bias of the potential of a single variable.

[0106] In contrast to clamping, un-clamping means that the conditional values of an EBM are not enforced anymore and that all the variables are left to evolve freely. Leading to a sample from the joint distribution.

[0107] In systems lacking synapses, the thermodynamic hardware is trained via standard contrastive divergence. In this setup, the TC is used for sampling only. Samples are stored onEXI-00125 the classical controller and gradient calculations are done on the classical control. This requires one to know the potential energy E(x) of the hardware because one has to evaluate it on the classical controller for a given sample x.

[0108] In following, the learning strategy using the stochastic gradient optimization algorithm adapted to energy-based models is provided. The posterior distribution is defined as where is the energy function and is the partition function. The parameter update rule is derived using stochastic gradient Langevin dynamics (SGLD):Equation 50

[0109] On hardware the states can be obtained via measurement. To simulate this on classical hardware one can use the Langevin MCMC algorithm to sample from the posterior distribution:Equation 51

[0110] If there are hidden variables, the parameter update rule generalizes to:. Equation 52

[0111] In the EBM literature the two averages in Equation 50 and 52 are referred to as the positive and the negative phase of the contrastive divergence algorithm. The positive phase referring to the average over the data:Equation 53

[0112] The negative phase refers to the second average, where both the hidden and visible nodes are un-clamped and the average is taken over the model distribution, i.e., via samples from the model itself.Equation 54

[0113] An exemplary training model is as follows:EXI-00125 1. Initializing the parameters at random 2. For , :a. Sampling a batch of data { of sizeb. For all (positive phase): i. Clamping the variables and sample the conditional distribution ii. Calculating iii. Averaging c. For j in k (negative phase): i. Sampling the joint distribution ii. Calculating iii. Averaging d. Updating parameters according to the difference of the two averages:Equation 55

[0114] In the above training mode, is the maximal number of gradient updates, is the batch size, is the number of samples taken from the un-clamped phase.

[0115] The neurons used to encode the data and hidden states are based on a flux qubit design. The value of the neuron is described by a phase / flux degree of freedom and the design is based on the DC SQUID which contains two Josephson junctions. In systems lacking synapses, inference and sampling can be derived from the equation of motion for a system of particles undergoing Langevin dynamics.

[0116] In various systems, the TC consists of neurons and synapses. Neurons represent the variables of interest. Neurons are used to encode for example the data on the TC and they can also be used for hidden variables that increase the expressivity of the model or mitigate limited connectivity of the hardware. The synapses can be understood as the biases and weights between the neurons, i.e., representing the coupling between the neurons and their local bias value. Both, neurons and synapses, can be constructed via a flux qubit. For some exemplary hardware, the values of the couplings and the biases are not encoded as separate oscillators, rather they are set by pulses controlled by the classical control.EXI-00125

[0117] A single neuron, which in some embodiments can also be a synapse, is an oscillator that can be constructed from a flux qubit. Their potentials can have many different forms. In Equation 56, they have a double wall potential. One could also use a single well potential of the formOther types of potentials are also possible. The couplings in Equaiton 56 are of linear form, one could also couple the oscillators with a quadratic coupling which could take the form. In some embodiments, is a coupling induced by control pulses. In some embodiments, is the position degree of freedom of any synapse oscillator that is itself also free to evolve.

[0118] The whole system of neurons and synapses is best understood as a graph, where the neurons represent the vertices V of the graph. The synapses can either represent the interaction between these vertices, hence be on the edges E of the graph, or be coupled to each node separately as a bias. The system is described by the following Hamiltonian:Equation 56 which allows one to construct for example a Boltzmann machine on the hardware. Note that x represents any position value of the flux qubits. The first two terms in Equation 56 represents the individual potentials of the visible and hidden neurons and the following terms are the coupling between the neurons and the bias of the neurons.

[0119] In the context of energy-based models (EBMs), the potential energy function E(x) is derived from the Hamiltonian H by considering only the terms that depend on the position variables x, while omitting the momentum terms. This potential energy function E(x) is equivalent to the energy function used in the EBM.

[0120] When the system described by the Hamiltonian H reaches equilibrium, the position variables x can be sampled to obtain the values of interest directly through measurement. This sampling process is based on the Boltzmann distribution, which states that the probability of observing a particular configuration of the system is proportional to exp( H), where is the inverse temperature.

[0121] In the equilibrium state, the position variables x can be sampled according to theBoltzmann distribution exp( E(x)) / Z. With the inverse temperature. These sampled valuesEXI-00125 of x correspond to the desired outputs of the EBM and can be obtained directly through measurement.

[0122] The proposed architectures for inference and sampling include: neurons represented by superconducting flux qubits, and weights and biases trained on an FGPA or ASIC.

[0123] In exemplary embodiments, in addition to the neurons, the synapses are dynamical degrees of freedom on a thermodynamic processor. During their time evolution arising from Langevin dynamics, the gradients needed for learning algorithms can be obtained by performing measurements of the momentum degrees of freedom of the synapses. It is assumed to perform fast measurements of the momentum such that the evolution of the system can be discretized in small time intervals. In exemplary embodiments, the couplings and biases of the neurons are also represented by an oscillator. Furthermore, the gradient can calculated via momentum measurements of the synapse oscillators.

[0124] The system is described by the following Hamiltonian:Equation 57

[0125] Here, the technical details of the gradient calculation on hardware are described. It is assumed that the momentum degree of freedom can be measured quickly in small time intervals given by t. Repeatedly measuring the momentum in short time intervals allows one to compute gradient updates which can be used in learning algorithms. The gradient of thepotential energy U ( , x, z) with respect to the synapse parameters is given by:Equation 58

[0126] Note that in Equation 58 the partial derivative is taken with respect to and not . This is attributed to the weights and biases being represented as oscillator degrees of freedom which are often denoted as q. The momentum measurements denotes the r-thEXI-00125 measurement of the momentum for measurements at time steps t. The constant a is friction dependent.

[0127] An additional exemplary architecture consists of a three-chip architecture for a self- learning neuro-thermodynamic computer. Each of the three chips consists of a neuro- thermodynamic processor. In particular, two of the three chips have the following roles: in the first chip, the visible neurons are always clamped to the data, and in the second chip, the visible nodes are always free to evolve while being un-clamped to the data. Both of the latter two chips are coupled to a third chip via position-position and momentum-momentum type couplings. The coupling terms are carefully chosen such that the position update rules for small time increments yield both the desired positive and negative phase terms which are required for energy-based model learning algorithms. A three-dimensional layout for the three chips is possible, which may allow for a hardware implementation which uses fewer wires. With the exception of changing the training data after certain time periods, no gates are required throughout the learning process. As such, this architecture provides a framework for a full self-learning neuro-thermodynamic computer.

[0128] The present disclosure may be embodied as a system, a method, and / or a computer program product. The computer program product may include a computer readable storage medium (or media) having computer readable program instructions thereon for causing a processor to carry out aspects of the present disclosure.

[0129] The computer readable storage medium can be a tangible device that can retain and store instructions for use by an instruction execution device. The computer readable storage medium may be, for example, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. A non-exhaustive list of more specific examples of the computer readable storage medium includes the following: a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), a static random access memory (SRAM), a portable compact disc read-only memory (CD- ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanically encoded device such as punch-cards or raised structures in a groove having instructions recorded thereon, and any suitable combination of the foregoing. A computer readable storage medium, as used herein, is not to be construed as being transitory signals per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagatingEXI-00125 through a waveguide or other transmission media (e.g., light pulses passing through a fiber- optic cable), or electrical signals transmitted through a wire.

[0130] Computer readable program instructions described herein can be downloaded to respective computing / processing devices from a computer readable storage medium or to an external computer or external storage device via a network, for example, the Internet, a local area network, a wide area network and / or a wireless network. The network may comprise copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers and / or edge servers. A network adapter card or network interface in each computing / processing device receives computer readable program instructions from the network and forwards the computer readable program instructions for storage in a computer readable storage medium within the respective computing / processing device.

[0131] Computer readable program instructions for carrying out operations of the present disclosure may be assembler instructions, instruction-set-architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state-setting data, or either source code or object code written in any combination of one or more programming languages, including an object oriented programming language such as Smalltalk, C++ or the like, and conventional procedural programming languages, such as the “C” programming language or similar programming languages. The computer readable program instructions may execute entirely on the user’s computer, partly on the user’s computer, as a stand-alone software package, partly on the user’s computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user’s computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider). In some embodiments, electronic circuitry including, for example, programmable logic circuitry, field-programmable gate arrays (FPGA), or programmable logic arrays (PLA) may execute the computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry, in order to perform aspects of the present disclosure.

[0132] Aspects of the present disclosure are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the disclosure. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchartEXI-00125 illustrations and / or block diagrams, can be implemented by computer readable program instructions.

[0133] These computer readable program instructions may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks. These computer readable program instructions may also be stored in a computer readable storage medium that can direct a computer, a programmable data processing apparatus, and / or other devices to function in a particular manner, such that the computer readable storage medium having instructions stored therein comprises an article of manufacture including instructions which implement aspects of the function / act specified in the flowchart and / or block diagram block or blocks.

[0134] The computer readable program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be performed on the computer, other programmable apparatus or other device to produce a computer implemented process, such that the instructions which execute on the computer, other programmable apparatus, or other device implement the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0135] The flowchart and block diagrams in the Figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present disclosure. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of instructions, which comprises one or more executable instructions for implementing the specified logical function(s). In some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and / or flowchart illustration, and combinations of blocks in the block diagrams and / or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts or carry out combinations of special purpose hardware and computer instructions.

[0136] The descriptions of the various embodiments of the present disclosure have been presented for purposes of illustration, but are not intended to be exhaustive or limited to theEXI-00125 embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.

Claims

EXI-00125 CLAIMS What is claimed is:

1. A system comprising: a thermodynamic processor, having a plurality of configuration parameters, a plurality a controller operatively coupled to the thermodynamic processor and the quantum processor, wherein the controller is configured to receive a plurality of samples from a data state, initialize the plurality of configuration parameters of the thermodynamic processor to initial values, initialize the plurality of configuration parameters of the quantum processor to initial values, sample from the hidden nodes of the thermodynamic processor, update the configuration parameters of the thermodynamic processor based on samples of the visible nodes and sample of the hidden nodes and on the data state, and update the configuration parameters of the quantum processor based on samples of the visible nodes and samples of the hidden nodes.

2. The system of Claim 1, wherein updating the configuration parameters of the thermodynamic processor comprises: clamping the visible nodes according to the data state, thereby beginning a firstclamped phasesampling the hidden nodes during the first unclamping the visible nodes, thereby ending the first clamped phase and beginningan unclamped phase andsampling the visible and hidden nodes during the unclamped phase, wherein the configuration parameters are based on a comparison between the sampling during the clamped phase and the sampling during the unclamped phase.

3. The system of Claim 2, wherein updating the configuration parameters of the quantum processor comprises:EXI-00125 clamping the visible nodes according to the sampled visible nodes, thereby beginninga second clamped phasesampling the hidden nodes during the second and prepare an ansatz on the quantum processor based on the sampling during the clamped phase and the sampling during the unclamped phase, wherein the configuration parameters are based on values measured from the quantum processor.

4. The system of Claim 1, wherein the controller is further configured to prepare the quantum processor in an initial state defined by samples of the visible nodes, the quantum processor is configured to execute a quantum circuit defined by its updated parameters on the initial state, and the controller is further configured to measure from the quantum processor.

5. The system of Claim 1, wherein the initial values of the configuration parameters of the thermodynamic processor are chosen randomly.

6. The system of Claim 1, wherein the initial values of the configuration parameters of the quantum processor are chosen randomly.

7. The system of Claim 1, wherein the plurality of samples of the data state are measured from a quantum processor.

8. The system of Claim 1, wherein the plurality of samples of the data state are measured from a quantum sensor.

9. The system of Claim 1, wherein each of the plurality of visible nodes and each of the plurality of hidden nodes comprises a superconducting flux qubit.

10. The system of Claim 1, wherein updating the configuration parameters of the thermodynamic processor comprises gradient descent.EXI-00125 11. The system of Claim 1, wherein updating the configuration parameters of the quantum processor comprises gradient descent.

12. The system of any one of Claims 1 to 9, wherein each of the plurality of visible nodes and each of the plurality of hidden nodes comprises a flux qubit.

13. A method comprising: receiving a plurality of samples from a data stateinitializing a plurality of configuration parameters of a thermodynamic processor to initial values, the thermodynamic processor having a plurality of visible nodes and a plurality initializing a plurality of configuration parameters of a quantum processor to initial sampling from the hidden nodes of the thermodynamic processor, updating the configuration parameters of the thermodynamic processor based on samples of the visible nodes and sample of the hidden nodes and on the data state, and updating the configuration parameters of the quantum processor based on samples of the visible nodes and samples of the hidden nodes.

14. The method of Claim 1, wherein updating the configuration parameters of the thermodynamic processor comprises: clamping the visible nodes according to the data state, thereby beginning a firstclamped phasesampling the hidden nodes during the first unclamping the visible nodes, thereby ending the first clamped phase and beginningan unclamped phase andsampling the visible and hidden nodes during the unclamped phase, wherein the configuration parameters are based on a comparison between the sampling during the clamped phase and the sampling during the unclamped phase.

15. The method of Claim 14, wherein updating the configuration parameters of the quantum processor comprises:EXI-00125 clamping the visible nodes according to the sampled visible nodes, thereby beginninga second clamped phasesampling the hidden nodes during the second and prepare an ansatz on the quantum processor based on the sampling during the clamped phase and the sampling during the unclamped phase, wherein the configuration parameters are based on values measured from the quantum processor.

16. The method of Claim 1, further comprising: prepare the quantum processor in an initial state defined by samples of the visible nodes, wherein the quantum processor is configured to execute a quantum circuit defined byits updated parameters on the initial statemeasuring from the quantum processor.

17. The method of Claim 13, wherein the initial values of the configuration parameters of the thermodynamic processor are chosen randomly.

18. The method of Claim 13, wherein the initial values of the configuration parameters of the quantum processor are chosen randomly.

19. The method of Claim 13, wherein the plurality of samples of the data state are measured from a quantum processor.

20. The method of Claim 13, wherein the plurality of samples of the data state are measured from a quantum sensor.

21. The method of Claim 13, wherein each of the plurality of visible nodes and each of the plurality of hidden nodes comprises a superconducting flux qubit.

22. The method of Claim 13, wherein updating the configuration parameters of the thermodynamic processor comprises gradient descent.

23. The method of Claim 13, wherein updating the configuration parameters of the quantum processor comprises gradient descent.EXI-00125 24. The method of any one of Claims 13 to 24, wherein each of the plurality of visible nodes and each of the plurality of hidden nodes comprises a flux qubit.

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