Systems and methods for performing machine learning using quantum computers

Abstract

A system and method for performing machine learning using a quantum computer are provided. The method includes providing a model comprising a quantum Boltzmann machine having a Hamiltonian Ansatz including a set of operators and a set of parameters. The method further includes the step of performing a first training stage on the model with data from a target using a selected subset of the set of operators to obtain optimized values ​​for a subset of the set of parameters. The first training stage is performed on classical computing hardware to provide a partially trained model. The method further includes the step of performing a second training stage on the model with data from a target using a larger subset of the set of operators to obtain optimized values ​​for a larger subset of the set of parameters for the model. The second training stage is performed at least partially using quantum computer hardware. The optimized parameter values ​​from the first training stage are used to initialize the corresponding parameters for the second training stage.

Description

[Technical Field]

[0001] This application relates to a system and method for performing machine learning using a quantum computer. [Background technology]

[0002] Machine learning (ML) research has developed into a mature field with applications that impact diverse aspects of society. Neural networks and deep learning architectures are being deployed for tasks such as facial recognition, recommendation systems, and time series modeling, as well as for the analysis of highly complex data in science. Furthermore, unsupervised learning and generative modeling techniques are widely used for text, image, and speech generation tasks, which many people encounter daily through interactions with chatbots and virtual assistants. Thus, the development of new machine learning models and algorithms can have significant impacts on a wide range of industries, and more generally, on society as a whole (see reference [1]).

[0003] In recent years, researchers in quantum information science have begun investigating whether quantum algorithms implemented on quantum computing hardware can offer advantages over conventional machine learning algorithms implemented on classical computing devices. This has led to the development of quantum algorithms for computational tasks related to various aspects of ML, such as gradient descent, classification, generative modeling, and reinforcement learning, as well as many other tasks (see references [2]–[6]). Further examples of the development of quantum systems for use in ML can be found in U.S. Patent No. 11157828B2 and U.S. Patent Publication No. 20200279185A1. [Overview of the project] [Problems that the invention aims to solve]

[0004] However, in most cases, generalizing results from the conventional (classical) ML domain to the quantum ML domain is not straightforward. Rather, various factors in the quantum machine learning (QML) setting, such as data encoding, training complexity, and sampling, must be re-examined. For example, there are several problems related to how large datasets (which can occur in many ML contexts) can be efficiently embedded into quantum states so that true quantum acceleration can be achieved [7, 8]. Furthermore, since quantum states prepared in quantum devices can only be accessed through sampling, it is not possible to estimate their properties with arbitrary precision. One particular problem is the vanishing gradient in training variational quantum algorithms, also known as the Valen Plateau [9-13]. Thus, there remains interest in the further development of systems, including quantum computing platforms (also referred to herein as quantum hardware, quantum devices, quantum computers, quantum computing hardware, etc.), to provide enhanced support for ML. [Means for solving the problem]

[0005] The present invention is defined in the appended claims.

[0006] A system and method for performing machine learning using a quantum computer are provided. The method includes providing a model comprising a quantum Boltzmann machine having a Hamiltonian Ansatz including a set of operators and a set of parameters. The method further includes the step of performing a first training stage on the model with data from a target using a selected subset of the set of operators to obtain optimized values ​​for a subset of the set of parameters. The first training stage is performed on classical computing hardware to provide a partially trained model. The method further includes the step of performing a second training stage on the model with data from a target using a larger subset of the set of operators to obtain optimized values ​​for a larger subset of the set of parameters for the model. The second training stage is performed at least partially using quantum computer hardware. The optimized parameter values ​​from the first training stage are used to initialize the corresponding parameters for the second training stage.

[0007] The second training stage can be performed iteratively with a larger subset of operators and / or a larger subset of parameters in each iteration, providing a trained quantum Boltzmann machine in which the difference between the expected values ​​of the parameter values ​​of the target probability distribution and the parameter values ​​output by the model decreases. The training of the first and second stages, and optionally further iterations, can provide a trained QBM that more accurately represents the Hamiltonian. We demonstrate that the performance of incremental QBM learning can take advantage of recent and anticipated future advances in quantum computing hardware, as described below with reference to exemplary quantum computing hardware.

[0008] By performing a first training phase of a quantum Boltzmann machine (e.g., “pre-training” on a classical computing device), a second training phase (and any subsequent iterations) begins with parameters initialized to allow optimization in subsequent training phases. A computer system for implementing the present invention may comprise a classical binary computer coupled to a quantum computer, utilizing the resources of the classical computer for the first training phase and then leveraging the quantum computer's probabilistic representation of the quantum state of a target real-world quantum system for a second training phase in which the model is refined.

[0009] Hereinafter, various embodiments and implementations of this disclosure will be described in detail, simply as examples, with reference to the following figures. [Brief explanation of the drawing]

[0010] [Figure 1] Figure 1 is a rough schematic diagram of an example of a method disclosed herein for performing machine learning using a quantum computer. [Figure 2A] Figure 2A is a schematic diagram illustrating various results obtained by using an example of the method disclosed herein for performing machine learning (collectively referred to herein as Figure 2). [Figure 2B] Figure 2B is a schematic diagram illustrating various results obtained by using an example of the method disclosed herein for performing machine learning (collectively referred to herein as Figure 2). [Figure 2C] Figure 2C is a schematic diagram illustrating various results obtained by using an example of the method disclosed herein for performing machine learning (collectively referred to herein as Figure 2). [Figure 3] Figure 3 is a rough flowchart of an example of a method disclosed herein for performing machine learning using a quantum computer. [Figure 4] Figure 4 is a schematic diagram showing various hardware and software components of an example system described herein. [Figure 5] Figure 5 shows two plots representing the minimum eigenvalue of the Hessian matrix as a function of the number of qubits for (b) a 1D nearest neighbor Hamiltonian and (b) a fully connected Hamiltonian. [Modes for carrying out the invention]

[0011] overview A quantum Boltzmann machine (QBM) is a machine learning model that can be used with both classical and quantum data. The operational definition of QBM learning is presented in terms of the difference in expected values ​​between the model and the target, taking into account the polynomial size of the dataset.

[0012] In other words, a QBM functions as a model trained to emulate a target. The target, in fact, defines a system and its associated behavior. Generally, the target is unknown in itself, and samples of its system behavior can be taken from it. QBM learning (training) involves taking samples of the target and corresponding samples from the QBM (model), and updating the model so that the samples from the QBM (model) more closely match the samples of the target.

[0013] This specification demonstrates that machine learning solutions can be obtained using up to a polynomial number of Gibbs states (which can be considered as providing samples for the model) via stochastic gradient descent. One implication of this finding is that there is no Valen plateau in QBM learning (such as those without hidden units). It is also shown that pre-training a subset of QBM parameters can lower the sample complexity limit. Various pre-training strategies are proposed based on mean-field, Gaussian-fermion, and geometrically local Hamiltonians (additional models supporting training on classical computers are also available). The models and theoretical insights proposed herein are numerically validated on quantum and classical datasets. The results presented herein demonstrate that QBMs can provide promising machine learning models for training on current and future quantum devices.

[0014] In some implementations, a Hamiltonian Ansatz is prepared that is very well suited to a particular quantum computing device. After exhausting all available classical computing resources during the first training phase (also referred to herein as pre-training), the model can be scaled up to continue training on the quantum computing device and further improve overall performance. As quantum hardware continues to mature, it will support the execution of deeper circuits and further increases in model size. Incremental QBM training strategies can be designed to follow a quantum hardware roadmap toward training larger and more expressive quantum machine learning models.

[0015] Introduction As described herein, systems and methods have been developed for training quantum Boltzmann machines (QBMs) to obtain optimal parameter values ​​[10-12]. QBM training helps generate models that emulate target datasets and address some of the problems identified on implementing ML in quantum environments. QBMs can be viewed as a generalization of classical Boltzmann machines, which are a form of stochastic neural network having nodes linked by weighted connections.

[0016] In particular, the QBM is a physics-inspired ML model that generalizes the classical Boltzmann machine to a quantum Hamiltonian ansatz (ansatz can be viewed as a trial solution to a given problem). Thus, the QBM can be seen as providing a specific general-purpose type of ML model, and the Hamiltonian ansatz specializes the system to a given problem by defining, for example, input parameters for the QBM.

[0017] A (quantum) Hamiltonian Ansatz can be defined on a graph where each vertex represents a qubit (or quantum dit) and each edge represents an interaction (roughly speaking, a qubit is the quantum computing counterpart to a hardware bit in conventional / classical machines, and a quantum dit can represent a multilevel system). The task is to learn the strength (weights) of the interaction so that samples from the output quantum state of the QBM mimic samples obtained from a target dataset. In this approach, the QBM may be trained using polynomial sample complexity on a quantum computer. The effectiveness and advantages of such an approach will increase with the rapid development of quantum computing platforms (such as hardware systems that support a larger number of qubits and implement error detection or correction for fault tolerance).

[0018] The development of this type of quantum generative model [14–17] is expected to be useful in machine learning to address scientific problems (for example) by learning approximate descriptions of experimental data. QBMs can also play an important role as components of larger QML models [18–22] (similar to how classical BMs can provide good weight initialization for training deep neural networks

[23] ). One advantage of using QBMs over classical BMs is that QBMs are more expressive because their Hamiltonian Ansatz can include more general non-commutative terms. This means that in some settings, QBMs perform better than classical BMs, even for classical target data

[17] .

[0019] To help obtain results with good practical applicability, an operational definition of QBM learning is adopted. Rather than focusing on information-theoretic measures, QBM learning performance is evaluated by the difference in expected values ​​between the target and the model. This takes into account that the (classical) target dataset contains a polynomial number of data samples and therefore has a polynomial precision. It is shown that by employing stochastic gradient descent [26, 27] in combination with shadow tomography [28-30], this problem can be solved using a polynomial number of evaluations of the QBM model. Each evaluation of the model requires the preparation of one Gibbs state, and therefore the sample complexity is called the number of Gibbs state preparations required.

[0020] The Gibbs states used for QBM learning are prepared and sampled on a quantum computer in various ways (see, e.g., [31-36]). For this purpose, we will focus on sampling complexity rather than a specific Gibbs sampling implementation.

[0021] In practice, QBM learning allows for greater flexibility in model design, and therefore greater time complexity. Furthermore, the number of Gibbs samples required for QBM learning can be improved by pre-training a subset of the QBM parameters, as shown below. In other words, (quantum) training complexity can potentially be reduced by classically pre-training simpler models. For example, mean-field QBMs and Gaussian-fermion QBMs can be analytically pre-trained. Additionally, geometrically local QBMs can be pre-trained using gradient descent, which, as shown below, provides some improved performance assurance. As described herein, these precisely solvable models can be used for training and / or pre-training QBMs. Furthermore, classical numerical simulation results supporting the analytical findings are also presented.

[0022] Definition of the problem We begin by formally defining the quantum Boltzmann machine (QBM) learning problem and providing definitions of the target and model, as well as a description of how to evaluate performance based on the accuracy of the expected value. These definitions and assumptions, along with their motivations, are introduced below to help derive the results described herein. Furthermore, the definition of the problem described herein is compared with other related problems in the literature, such as quantum Hamiltonian learning.

[0023] As a target for a machine learning problem, we consider an n-qubit density matrix η. If the target is classical, n can represent a feature, such as the number of pixels in a black and white photograph, or a more complex feature that is extracted and embedded in the space of n qubits. If the target is quantum, n can represent a spin-1 / 2 particle, but again, a more complex many-body system can be embedded in the space of n qubits. In the literature, it is often assumed that algorithms have direct and simultaneous access to copies of η, but this assumption is not adopted herein. Instead, a setting is considered in which access to classical information about the target is restricted. N independent data samples s μ Dataset D = {sμ Assuming that the dataset can be efficiently stored in classical memory, the amount of memory required to store each data sample is a polynomial of n, and there are a polynomial number of samples. For example, s μ This can be a bit string. This includes datasets such as binary images and time series data, categorical and count data, and binary sequential data. As another example, the data may be from measurements in a quantum system. In this case, s μ This identifies the elements of the positive prime measure that describe the measured value.

[0024] Next, we define the machine learning model used for data fitting in this specification. A fully visible QBM[10~12] is an n-qubit mixed quantum state of the following form:

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[0025] QBMρ θ A natural measure for quantifying how well QBMρ approximates the target η is the quantum relative entropy

[19] . S(η||ρ θ ) = Tr[η log η] - Tr[η log ρ θ (3) This measure generalizes the classical Kullback-Leibler divergence to density matrices. When the two densities are equal (η = ρ θ ), the relative entropy becomes exactly zero, and S > 0 otherwise. Furthermore, when S(η||ρ θ ) ≤ ε, by Pinsker's inequality, all possible Pauli expectation values are

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[0026] Theoretically, the optimal model parameter θ opt = argmin θ S(η||ρ θ ) can be found by minimizing the relative entropy S(η||ρ θ ). The form of the partial derivative of the relative entropy can be calculated analytically and is given as follows.

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[0027] Quantifying how well a QBM is trained by relative entropy actually presents several problems. S(η||ρ θ Accurate estimation of ) generally requires access to the target entropy and the model's partition function. Due to the expected model mismatch because we are selecting m operators from an exponentially large number of candidate operators, the optimal QBM is

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[0028] Definition 1 (QBM learning problem): n-qubit target density matrix η, target precision ε>0, and Hamiltonian

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[0029] A solution to the QBM learning problem always exists by Jaynes' principle

[41] , namely, the set of target expectations { <H i > η Given},

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[0030] The expected value of a target can be obtained from a dataset in various ways. For example, in classical generative modeling of binary datasets, a pure quantum state can be defined and its expected value obtained (see Appendix E). In modeling the target quantum state (density matrix), the expected value can be estimated from the results of measurements performed under different basis states.

[0031] As shown in Appendix C3, the solution to the QBM learning problem implies the boundedness of the optimal relative entropy. That is,

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[0032] result In this example, we use stochastic gradient descent (SGD)

[27] to approach the QBM learning problem by iteratively minimizing the quantum relative entropy (see equation (3)). This is calculated from the sample set at time t.

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[36] .

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[0033] The stochastic gradient is unbiased, i.e.

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[0034] This method allows us to solve the QBM learning problem using polynomial sample complexity. This is stated by the following theorem, which is a key aspect of the approach described herein.

[0035] Theorem 1 (QBM training). QBM is κ2 +ξ 2 A set of n-qubit Pauli operators such that ≤ ε / 2m

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[0036] The success probability is the probability that the QBM expectation value is correctly determined. This is a free parameter that can be set to determine the number of measurements to be performed in the experiment.

[0037] A detailed proof of this theorem is given in Appendix C2 and involves carefully combining three key observations and results. First, any QBMρ θ The quantum relative entropy for is,

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[29] , which yields the result in a Pauli observable H i ≡P i Restricted, and therefore, ||H i ||2=1. This can be extended to general binary observables

[28] with multiple logarithmic overhead by comparing it with equation (10) (see Appendix C2). Furthermore, for k-local Pauli observables, the results can be obtained by using a classical shadow

[42] composed of random measurements, or by using a pure thermal shadow

[36] .

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[0038] By combining equations (8) and (10), the final number N of cast preparations can be obtained. tot It can be seen that ≥T×N scales polynomially to m, which is the number of terms in the QBM Hamiltonian. Under classical memory assumptions, m ∈ O(poly(n)) is the only possible number. This means that the number of measurements required to solve the QBM learning scales polynomially to the number of qubits (features). As a result, there is no Valen plateau in the optimization landscape for this problem, and the Valen plateau of the loss function f(θ) is the vanishing of its gradient E[∇ θ The variance var[∇] is defined by f(θ) = 0 and decreases exponentially in the gradient.θ f(θ)] <O(2 -n It is also defined by [another definition].

[0039] The following theorem is proven in Appendix C2.

[0040] Theorem 2 (α-strongly convex QBM training). QBM, S(η||ρ θ ) is an α-strongly convex Hamiltonian Ansatz H θ κ 2 +ξ 2 The learning rate is defined by the accuracy κ relative to the QBM expectation, the accuracy ξ relative to the data expectation, and the target accuracy ε, all of which are ≥ε / 2m.

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[0041] The sample limit in Theorem 1 depends on δ0, which is the relative entropy difference between the initial QBM and the optimal QBM. This means that if the initial relative entropy can be reduced, the limit on the QBM learning sample complexity can also be tightened. In this regard, it is shown that δ0 can be reduced by pre-training a subset of the Hamiltonian-Ansatz parameters. Thus, pre-training reduces the number of steps required to reach the global minimum.

[0042] Theorem 3 (QBM pre-training). Relative entropy S(η||ρ θ To minimize ), the target η and QBM model

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[0043] Appendix D.1 provides a detailed proof of Theorem 3, which applies to any method in which it is possible to minimize relative entropy with respect to a subset of parameters. All other parameters are fixed to specific intrinsic values, usually (but not limited to) zero, and pre-trained to the maximum mixed state.

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[0044] Figure 1 shows a rough schematic diagram of an example of a method disclosed herein for performing machine learning using a quantum computer. Figure 1 includes three boxes: the box on the left is the input, the box on the right is the output, and the box in the middle is the process used to derive the output from the input.

[0045] In particular, the input data contains two components. The first component is the QBM associated with the Hamiltonian Ansatz. This first component actually represents the ML model to be trained. The second component contains a set of data values ​​(samples) that represent the Hamiltonian expectation for the target. For example, the data may represent measurements performed on the target quantum system. This set of samples has a polynomial size with respect to the size of the QBM (corresponding to the number of qubits used in the quantum-based implementation of the QBM). This polynomial scaling converges using an accessible level of computing resources.

[0046] The output data (right image) corresponds to the QBM and Ansatz-Hamiltonian models shown as input data (left image), and similarly shows the results after training the QBM mode using the target dataset shown in the input data. The right image shows the new sample output, which is the sample output provided by the trained QBM.

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[0047] The central box in Figure 1 represents training a QBM model based on a dataset from the target. This training is performed using stochastic gradient descent (SGD) based on the relative entropy level between the target and the model, in this example. Thus, Figure 1 shows a column θ of the model, which substantially represents the continuous generation of the trained QBM model. 0 , θ 1 , ..., θ T This shows that the minimum value occurs when the difference between the expected value of the model Hamiltonian and the expected value of the target Hamiltonian is smaller than the set threshold ε for each sample i (see Definition 1). θ opt For this, ε=0 is the exact solution given by Jaynes' principle. Theoretically, this is the best solution that can be realized in SGD, but in reality, SGD cannot be made arbitrarily close, and θ T It is sufficient to achieve a fixed (specified) precision ε>0 corresponding to this.

[0048] As discussed herein, the procedure shown in Figure 1 utilizes pre-training on a classical computer to reduce relative entropy, thereby facilitating subsequent full (quantum-based) training, which can be initialized according to the lower relative entropy configuration generated by the pre-training.

[0049] The central box in Figure 1 represents the operational definition of the QBM learning problem for expected value, i.e.

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[0050] Therefore, Figure 1 shows a configuration in which the input to the problem is a dataset of polynomial size with respect to the number of features / qubits, and an Ansatz for a QBM model with parameter θ. Definition 1 provides an operational definition of a QBM learning problem in which the model and target expected values ​​must be close within polynomial precision ε. By Jayne's principle, the solution θ opt It is guaranteed that such a thing exists. By Theorems 1 and 2, the quantum relative entropy S(η||ρ) with respect to θ can be obtained using SGD. θ It is established that QBM learning can be solved by minimizing ). This involves preparing Gibbs states with a polynomial number. In Theorem 3, the parameter θ of the QBM pre It has been shown that a pre-training strategy that optimizes a subset of guarantees a decrease in the initial quantum relative entropy. The algorithm provides a solution θ for the problem in steps T of polynomials. T It outputs the following. A trained QBM can be used, for example, to generate new synthetic data.

[0051] Numerical experiment To further investigate the theoretical findings above, numerical experiments were conducted on QBM learning for datasets consisting of quantum and classical sources. First, according to Theorem 3, the initial relative entropy was obtained by QBM pre-training.

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[0052] In contrast, the GL model is Hamiltonian Ansatz.

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[0053] Figures 2A, 2B, and 2C (collectively referred to herein as Figure 2) are schematic diagrams illustrating various results obtained by using an example of the method disclosed herein for performing machine learning. Figure 2A shows various forms of pre-trained initial relative entropy using the model for two 8-qubit problems.

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[0054] In the left portion of Figure 2A, pre-training is performed using quantum data (e.g., data generated by quantum hardware), while in the right portion of Figure 2A, pre-training is performed using classical data. For quantum data, an 8-qubit target is used as the Gibbs state of a one-dimensional XXZ model.

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[0055] As mentioned above, Figure 2A shows the maximum mixed state S(η||ρ) without prior training. θ The results starting from =0 are also shown. In all cases, as can be seen from Figure 2A, pre-training results in a decrease in the initial relative entropy for subsequent training of the model in quantum hardware. This decrease is particularly strong for classical data. For quantum data, the situation is a little more complex; the decrease in relative entropy is relatively small for mean-field based pre-training, but it is much more pronounced for other forms of pre-training shown in Figure 2A.

[0056] Therefore, for both targets (quantum data and classical data), all pre-training strategies are

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[0057] The effect of using a pre-trained model as a starting point for QBM learning using accurate gradient descent is as follows:

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[0058] In Figure 2B, the decay of quantum relative entropy (y-axis) is plotted against the number of learning iterations (t, x-axis) for training starting from various pre-training strategies as in Figure 2A. θ 0 Define this as the parameter vector at the end of pre-training,

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[0059] The performance of the MF pre-trained model (middle line) is better than the top line corresponding to the untrained case in all iterations, but the improvement is relatively small. Using a 2D GL model (bottom line) for pre-training yields a much greater improvement.

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[0060] Figure 2C plots the maximum expected error (comparison between the model and the target) on the y-axis. This phase of training is performed on noise-free simulated quantum hardware and plotted against the number of SGD iterations (x-axis). Classical input data is used (according to the components on the right side of Figure 2A), and two different noise intensities are compared. The lower line corresponds to smaller noise (0.01), and the upper line corresponds to larger noise (0.05). Learning rate γ = ε / (2m 2 (κ 2 +ξ 2The )) is used. The dashed line indicates the target accuracy ε = 0.1. The expected values ​​of the Gibbs state for the 1D quantum XXZ model in the external field [45, 46] and the expected value for the classical salamander retina dataset

[47] are used as targets. Details of these models and how to calculate the expected values ​​for the classical data are given in Appendix E.

[0061] The limit on the number of SGD updates according to equation (8) of Theorem 1 was numerically verified. This involved considering data from a classical salamander retinal target with eight variables and a fully connected QBM model with eight qubits. As mentioned above, Figure 2C shows two different noise intensities κ. 2 +ξ 2 We compare training using these settings. These settings were implemented by adding Gaussian noise, but in reality (not in simulation), the noise intensity is determined by the number of data points and the number of Gibbs state measurements in the quantum device. Using standard Monte Carlo estimation, each update is of size 1 / ξ 2 Mini-batch of data samples and number of measurements 1 / κ 2 (Assuming these measurements can be performed without additional hardware noise). A mini-batch of size 1 and a single measurement may be usable, provided there is no bias in the expected value of the Gibbs state. For either noise intensity, the desired target accuracy ε=0.1 is 10 4 This was obtained within steps. This is the limit of the number of steps in Theorem 1, which is the worst-case scenario: O(10) 9 It is well within the range of ).

[0062] Discussion and Conclusion An operational definition of quantum Boltzmann machine (QBM) learning has been developed, and it has been shown that this problem can be solved by polynomially many preparations of quantum Gibbs states. To prove the relevant limits, the properties of quantum relative entropy are used in combination with the performance guarantee of stochastic gradient descent (SGD). There are no assumptions about the form of the QBM Hamiltonian other than that it consists of polynomially many terms. This is in contrast to some prior studies [38, 39] that have considered the somewhat related Hamiltonian learning problem only for geometrically local models. In this context, strong convexity is required to relate the optimal Hamiltonian parameters to the expectation values ​​of the Gibbs states. In the machine learning setting described herein, the form of the target Hamiltonian is not known in advance. Therefore, learning the exact parameters is not so important, and instead the focus is directly on the expectation values. For this reason, the limits for the approach described herein relate only to the L-smooth relative entropy and can be applied to all types of QBMs that do not have hidden units.

[0063] Furthermore, it is shown herein that the theoretical sampling limit can be made tighter by reducing the initial relative entropy of the learning process. Typically, QBM learning is started from the maximum mixed state, i.e., a state with no prior information. It is shown herein that pre-training on any subset of parameters may behave better (or at least as well) than the maximum mixed state. This is beneficial when pre-training can be performed efficiently, as is the case for mean-field, Gaussian fermions, and geometrically local QBMs, as shown herein. The performance and theoretical limits of these models are verified by classical numerical simulations. These simulations also show that knowledge about the target (e.g., its dimensions, degrees of freedom, etc.) can significantly improve the training process. Moreover, the general limits adopted herein are quite loose, and it has been found that in practice, it may be feasible to use far fewer samples.

[0064] In some implementations, the sample limit can be made tighter by going beyond the simple SGD method described above. This can be done in various ways, such as by adding momentum, using other advanced updating schemes [27, 48], and / or by utilizing the convexity of relative entropy. This is within the limits of the inventors.

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[0065] Another interesting point concerns the training performance of different Ansatz. Generative models are often evaluated in terms of training quality

[49] , while generalization ability has recently been studied by researchers in both classical machine learning [50, 51] and quantum machine learning [52, 53]. In the case of QBMs, generalization may pave the way for further development.

[0066] The operations and results described herein can be generalized to QBM models with hidden units. This generalization may include demonstrating the L-smoothness of relative entropy for more general and challenging settings, and positive results provide the ability to train highly expressive models. In this regard, it should be noted that the results presented here already hold for the special case of QBMs with fixed hidden units, as this problem boils down to the problem described above.

[0067] The pre-training results described herein may be useful for implementing QBM learning on short-term and early fault-tolerant quantum devices. For this purpose, quantum computers can be used as Gibbs samplers. There are many quantum algorithms that produce Gibbs states with a square improvement in time complexity compared to the best existing classical algorithms (see, e.g., [31-35]). Furthermore, the use of quantum devices generally provides an exponential reduction in spatial complexity. For example, the authors of reference

[54] implemented a 2-qubit Gibbs state for an antiferromagnetic Ising model Hamiltonian on the Aspen-1 quantum computer. It is expected that improved quantum processing devices with higher gate fidelity and a higher number of qubits, e.g. (but not limited to) Quantinuum's system model H2 or Aspen-M-3, may be able to prepare similar Gibbs states, possibly for more complex Hamiltonians (i.e., those with more operators in the Ansatz). A further possibility is to avoid preparing the Gibbs state by constructing a classical shadow of a pure thermal quantum state, for example, and using an algorithm that directly estimates the expectation value of the Gibbs state. This reduces the number of qubits and, in some cases, the depth of the circuit.

[0068] The results presented here support various methods for incremental learning QBMs that are implemented using the availability of both training data and quantum hardware. For example, one can select a Hamiltonian Ansatz that is very well suited to a particular quantum device. After exhausting all available classical resources during pre-training, the model can be scaled up, and then training can continue on the quantum device, thus improving overall performance. As quantum hardware matures, it will enable the execution of deeper circuits and support further increases in model size. Incremental QBM training strategies can be designed to follow a quantum hardware roadmap toward training larger and more expressive quantum machine learning models.

[0069] Exemplary Implementation Forms The results presented here support the development of methods for incremental learning by QBM, which is performed by the availability of both training data and quantum hardware. For example, a Hamiltonian Ansatz that is very well suited to a particular quantum device can be selected. During the pre-training phase, after exhausting all available classical resources on selected components of the model (e.g., by selecting a subset of operators and parameters), the model is scaled up and training continues on the quantum device, thereby ensuring improved performance (compared to the output at the end of the pre-training phase). As quantum hardware continues to develop further, it will enable the execution of deeper circuits and further increases in model size. Incremental QBM training strategies can be designed to follow a quantum hardware roadmap toward training larger and more expressive quantum machine learning models.

[0070] Figure 3 shows a rough flowchart of an example of a method disclosed herein for performing machine learning using a quantum computer. Operation 310 includes providing a model comprising a quantum Boltzmann machine having an Ansatz-Hamiltonian which includes a set of operators and a set of parameters.

[0071] A quantum Boltzmann machine with an Ansatz-Hamiltonian may be further provided with a target expectation for performing the first training stage. For example, QBMρ with an Ansatz-Hamiltonian θ is an operator

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[0072] Operation 320 runs a first training phase on the model using a selected subset of operators on data from the target, obtaining optimized values ​​for a subset of parameters. The first training phase is performed on classical computing hardware to provide a partially trained model.

[0073] In the first training stage, operators that can be trained classically are

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[0074] Operation 330 runs a second training phase on the model using the complete set of operators on data from the target, obtaining optimized values ​​for a larger subset of the set of parameters for the model. The second training phase is performed on quantum computer hardware to provide a further trained model. The optimized parameter values ​​saved from the first phase can be used to initialize the corresponding parameters for the second training phase.

[0075] A larger subset of the set of parameters for a model may, in some implementations, include the complete set of parameters for the model. Therefore, the second training phase may include the entire set of parameters for the model (intrinsically, the first training phase does not include training for all parameters in the set, as this would prevent the second training phase from including a larger subset).

[0076] The second training phase is t=1~T q1 This can be repeated up to T q1 represents the maximum number of iterations (if convergence does not occur beforehand). In this second training phase, relative entropy can be optimized for all parameters in the model and target by calculating the expectation value of the Gibbs state in the quantum device. Before performing this optimization, the Ansatz-Hamiltonian is extended with an additional set of operators and parameters (extending the model). These additional operators and parameters are those that were not included in their respective subsets during the pre-training phase (and therefore not yet incorporated into the model).

[0077] Therefore, the second phase involves, for example, using thermal shadows

[29] to determine the expected value of Gibbs states in quantum devices.

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[0078] Depending on the available quantum computing resources, in the third training phase, the above approach can be further developed so that the ansatz Hamiltonian is further extended using a set of (orthogonal) operators [Mathematics] and parameters [Mathematics] . The parameters of the extended QBM are initialized as [Mathematics] , where θ opt are the optimal parameters obtained at the end of the previous quantum optimization loop. Additional target expectation values [Mathematics] are calculated and used for training.

[0079] In this further developed form, the third training phase may be repeated over t = 1 to T q2 , where T q2represents the maximum number of iterations. Then, each iteration obtains the expected value of the required Gibbs state in the quantum device, thereby determining the relative entropy S(η||ρ) with respect to the parameter λ of all extended QBMs. λ This includes optimization of ).

[0080] In some implementations, the second (and / or third, if applicable) training phase can be performed in a hybrid system that includes both quantum computing hardware and classical computing hardware. For example, Gibbs states for a quantum Boltzmann machine can be used to provide samples for machine learning. The Gibbs states may be prepared and sampled in the quantum computing hardware, while the parameters for the model may be maintained in the classical computing hardware. Various other configurations of the hybrid system may also be used for the second and / or third training phases.

[0081] Figure 4 is a schematic diagram showing various hardware and software components of an example of a machine learning system 400 described herein. In particular, system 400 comprises a classical computing platform 410 and a quantum computing platform 450. The classical computing platform 410 may comprise a known form of digital computer, including one or more processors for executing program instructions and memory for storing program instructions and data. The quantum computing platform may comprise one or more known forms of quantum computers. It should be understood that the components and configurations shown in Figure 4 are shown as examples only and not as limitations.

[0082] Figure 4 shows three specific components implemented using a classical computing platform 410: the Hamiltonian Ansatz 415, the optimization program 420, and the set of target data 480. The Hamiltonian Ansatz 415 is structured according to a quantum Boltzmann machine (QBM) and represents a model related to, for example, a complex physical system. The Hamiltonian 415 incorporates a set of operators and a set of parameters. Machine learning involves determining values ​​for the set of parameters such that the output of the model, represented by the expectation of the operators, mimics (approximates) the modeled system represented by the target data 480.

[0083] The classical computing platform 410 further includes an optimization (minimization) program 420, for example, a program that performs stochastic gradient descent (SGD). Roughly speaking, the optimization program 420 can obtain samples represented by the expectation values ​​of operators in the Hamiltonian Ansatz 415 for comparison with the training data, i.e., the target data 480. The optimization program uses the results of these comparisons to update the parameters of the Hamiltonian Ansatz 415 to decrease the quantum relative entropy. The optimization program 420 performs multiple iterations of this machine learning process to arrive at a configuration of model parameters with low (minimum) quantum relative entropy.

[0084] The first stage of the process (pre-training) is performed only on a classical computing device 410. Such a device may not have sufficient processing power to perform the entire optimization procedure. Therefore, as described herein, pre-training may be performed, for example, with respect to a subset of model parameters. The remaining parameters (those not included in the subset) may be held at a fixed value, such as zero. Using a subset of parameters for optimization, such as with SGD, typically reduces the computational resources used for this second training stage.

[0085] The second stage of the process (after pre-training) involves the use of the quantum computing platform 450. The quantum computing platform 450 includes a quantum circuit 452 associated with one or more qubits 455 to support the computations performed on the quantum computing platform 450. The quantum computing platform 450 also includes a QBM425 associated with a Hamiltonian Ansatz. This Hamiltonian in the quantum computer 450 generally corresponds to the Hamiltonian Ansatz 415 in the classical computing device 410, particularly with respect to the relevant model, but is adapted to function on different hardware platforms as shown in Figure 4. For example, the QBM425 can be implemented using the quantum circuit 452 of the quantum computing platform 450.

[0086] In the example in Figure 4, the optimization program 420 is used to control the optimization procedure in the second stage as well as in the first stage. Thus, the second stage can be considered hybrid in that it involves computing operations on both the classical computing platform 410 and the quantum computing platform 450. The optimization program 420 (such as SGD) can provide parameters to the QBM 425, compare the QBM output with the training data (target data 480), and then use this to determine machine learning updates.

[0087] By measuring the physical properties of a QBM (425) prepared on a quantum device (450), an optimizer (SGD) can search the entire parameter space in parallel to find the parameter values ​​with the minimum relative entropy. This ability may offer the possibility of performing machine learning on the quantum computer 450 that is computationally unfeasible (or computationally expensive) on a classical computer 410. For example, a second phase of the search can be performed using a larger subset (or the complete set) of parameters for the model. Thus, the approach described herein leverages the various properties and features of classical and quantum computing devices to support an efficient approach for machine learning on complex systems.

[0088] This specification has disclosed various implementations and embodiments. It should be understood that these implementations and embodiments are not intended to be exhaustive. Those skilled in the art will recognize many possible variations and modifications of these implementations and embodiments included within the scope of this disclosure. It will also be understood that features of certain implementations and embodiments can typically be incorporated into other implementations and embodiments (unless the context clearly indicates otherwise). In summary, the various implementations and embodiments disclosed herein are illustrative and not limiting, and the scope of the invention is defined by the appended claims.

[0089] appendix Appendix A: Preparation: Some useful mathematical facts and relationships Here, we identify some useful mathematical facts and relationships, and derive some useful results that will be used in the proofs later in the appendix.

[0090] 1.Convex Definition 2 (convex). Multivariable function

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[0091] The following lemma can be deduced from the standard definition of convexity (see reference

[27] ). Lemma 1. Let f be twice continuously differentiable. Then, f is

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[0092] Lemma 2. If f is α-strongly convex, then f is α-Polyak-Lojasiewicz. The strong convexity of a function can be tested as follows.

[0093] Let Lemma 3.f be twice continuously differentiable. Then,

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[27] ).

[0094] Lemma 4 (Descent Lemma)

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[0095] 2. Derivative of Matrix Exponential The derivative of the matrix exponential e with respect to a certain parameter H is given by Duhamel's formula.

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[0097] 1. Strong convexity To show that S is (strongly) convex, we can use Lemma 1 above. First, the Hessian ∇ of the quantum relative entropy with respect to the parameters of the QBM. 2 We show that S is positive semi-definite. Then, we show that S is ∇S(η||ρ θ* For )=0, there is one unique global optimal solution θ * We apply this lemma by showing that it has [a certain property]. From the text, QBM Hamiltonian H θ =Σ i θ i H i This is the ermitian (generally a non-commutative operator H). i Remember that this is a sum over ). Using the derivative of the matrix exponent in equation (A12),

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[0098] Next, it can be immediately shown that it is positive semi-definite and satisfies equation (A2). Any vector

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[38] , we prove strong convexity by contradiction: ∇S(η||ρ θ* )=0 for the parameter θ * Assume we have found one set of H. Then, from equation (B1), all H i In contrast,

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[41] ). * Note that we can always find at least one of all H i In contrast,

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[0099] 2. Strong convexity Lemma 3 can be used to show that S is α-strongly convex. To the best of my knowledge, there is no literature that proves that the quantum relative entropy of Gibbs states is generally strongly convex. On the other hand, this property has been proven for certain classes of Hamiltonians. Anshu et al. (see reference

[38] ) have proven strong convexity for k local Hamiltonians defined on a finite-dimensional lattice. They,

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[39] ) have shown strong convexity for a more general class of low-crossing Hamiltonians. Low-crossing Hamiltonians have terms that act nontrivially on only a certain number of qubits, each of which nontrivially intersects with a certain number of other terms. In this section, we use differentiable programming

[44] to find the smallest eigenvalue λ of the Hessian. min (∇ 2 We numerically analyze S) to find evidence for strong convexity. 1D nearest neighbor Hamiltonian

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[0100] The coefficients are sampled uniformly and randomly at [-μ,μ], where μ is the scale parameter, which determines the maximum size of the random parameter in the coefficient vector. Figure 5 shows the minimum eigenvalue (y-axis) of the Hessian as a function of the number of qubits n (x-axis), and shows the median of 25 random examples for (a) the one-dimensional nearest neighbor Hamiltonian and (b) the fully connected Hamiltonian. The scale parameter μ determines the maximum size of the random parameter. In all cases, the minimum eigenvalue decreases as the number of qubits increases, but appears to converge to a positive value (constant balance) for larger values ​​of n (especially in the case of the fully connected Hamiltonian (b)). The fully connected Hamiltonian is given by m∈O(n 2 It has ) parameters, which gives smaller eigenvalues ​​than a one-dimensional Hamiltonian with m∈O(n) parameters instead. These results provide evidence that it is strongly convex with respect to α, which decreases polynomially with respect to the system size.

[0101] 3. L-Smooth Quantum relative entropy S(η||ρ) θ We will show that ) is an L-smooth function of θ. To do this, we need the upper bound of the largest eigenvalue of the Hessian in equation B2. We will start with the following property.

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[0102] Appendix C: Convergence results of stochastic gradient descent for training quantum Boltzmann machines This appendix first reviews useful results from the machine learning literature, and then proves Theorems 1 and 2 in the main text that precede the appendix. Several upper bounds on relative entropy in the context of QBM learning are also discussed.

[0103] 1. Review of convergence results for stochastic gradient descent. We will begin by describing three convergence results from the SGD literature. It is L-smooth (definition 5).

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[26] proved the following convergence result for SGD.

[0104] Lemma 5 (re-presentation of Corollary 1 in

[26] ): Precision ε>0 and step size

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[0105] Lemma 6 (a re-publication of Lemma 3 in

[26] )

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[26] proved the following SGD convergence result.

[0106] Lemma 7 (repeating Corollary 2 in

[26] ) precision ε>0 and

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[0107] 2. Proofs of Theorems 1 and 2 in the text I will prove Theorem 1. For completeness, I will repeat it here.

[0108] Theorem 1 (QBM training)

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[0109] Proof. Quantum relative entropy is

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[29] , for example, m different Pauli operators {H i The expected value of} is ρ θ of

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[0110] Now, I will give the proof of Theorem 2, but here I will state Theorem 2 again. Theorem 2 (α-strongly convex QBM training) S(η||ρ θ ) is an α-strongly convex Hamiltonian Ansatz H θ ,

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[0111] Proof.

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[0112] 3. Achieving the desired precision in quantum relative entropy as defined in Theorem 1. This section examines a scenario where the user is more interested in obtaining a certain degree of accuracy regarding quantum relative entropy than in accuracy regarding the difference in expected values. Again, due to possible model mismatches, the optimal

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[0113] Appendix D: Pre-training This appendix first proves Theorem 3 in the main text, and then discusses various pre-trained models.

[0114] 1. The proof of the theorem guaranteed improvement through prior training. For the sake of completeness, we will begin by reiterating the theorem from the main text.

[0115] Theorem 3 (QBM pre-training): Relative entropy S(η||ρ) θ To minimize ), the target η and the QBM model

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[0116] 2. Pre-training methods Here, we discuss possible pre-trained models and strategies for optimizing them. Of the models discussed in this text, we focus on 1) the mean-field model, 2) the Gaussian-fermion model, and 3) the nearest-neighbor quantum spin model. The advantage of the first two models is that they can be trained analytically. While analytical training is not possible for the nearest-neighbor model, it satisfies the assumption of locality as described in references

[38] and

[39] , and therefore possesses strongly convex relative entropy.

[0117] 2a. Mean-field quantum Boltzmann machine The mean-field QBM is defined using a parameterized Hamiltonian.

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[0118] 2b. Gaussian-fermion quantum Boltzmann machine The Gaussian fermion QBM is a parameterized quadratic fermion Hamiltonian.

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[56] ). In particular, the Gaussian fermion QBM gradient is reduced to the difference between the target and model correlation matrices.

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[0119] 2c. Geometrically local quantum Boltzmann machine The final type of restricted QBM model to discuss is the geometrically local QBM. We consider the same Hamiltonian (equation (16)) for the general fully connected 2-local QBM, but with further constraints on the locality of the Pauli operators. In particular, we focus on the nearest-neighbor model on a d-dimensional lattice, for example, on a one-dimensional chain where each Pauli operator acts only on two adjacent qubits. In complete generality, the parameterized QBM Hamiltonian is:

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[38] and

[57] show that quantum relative entropy is strongly convex. Thus, optimization is guaranteed to converge quickly to a global optimal value, recalling Theorem 2. However, this involves obtaining the expectation value of the Gibbs state of the geometrically local Hamiltonian, which can be done on a quantum computer or, in some cases, classically on a tensor network (see references

[58] and

[59] ).

[0120] Appendix E: Construction of the expected value of the target state This appendix reviews methods for embedding classical data into target density matrices η. It follows the approach described in reference

[17] for quantum spin models. It also shows how to extend this form to fermion quantum models required for pre-training Gaussian fermion QBMs. Finally, it describes two different targets used in the numerical simulations in the main text.

[0121] 1. Classical Data Coding Following the approach in reference

[17] , an N-bit string

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[17] . Now, this encoding is fermion QBM (that is, the Hamiltonian-Ansatz term is fermion generating operator

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[0122] 2. Data used in numerical simulations in the main text For the numerical simulations in this text, we use two different targets η: 1) a target composed of a quantum source, and 2) a classical dataset embedded in η using the encoding described above. For the quantum source, we use the Hamiltonian of the XYZ model.

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[47] . This dataset consists of bitstring data of various features of the response of salamander retina cells. We select the first eight features and truncate their data into the first 10 data records. Then, using the procedure outlined above, we obtain the expected value <H i > η It constitutes.

[0123] 1 Please note that this procedure applies only to Pauli operators; therefore, from now on, we will refer to the H of the QBM Hamiltonian. i We define this as a Pauli operator. In this text, we discuss how this result can be generalized to other types of operators by using other shadow tomography protocols.

[0124] 2 Note that, depending on the specific parameter δ0 and the free parameters κ and ξ, there may be other cases of Lemma 7. Then, following the same steps shown here, we get a slightly different number of steps for a polynomial in n.

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Claims

1. A method for performing machine learning using quantum computing hardware, The steps include providing a model that includes a quantum Boltzmann machine having a Hamiltonian Ansatz that includes a set of operators and a set of parameters, A step of performing a first training phase on the model with data from a target using a selected subset of the set of operators to obtain optimized values ​​for a subset of the set of parameters, wherein the first training phase is performed on classical binary computing hardware to provide a partially trained model. A step of performing a second training phase on the model with data from the target using a larger subset of the set of operators to obtain optimized values ​​for a larger subset of the set of parameters for the model, wherein the second training phase is performed using quantum computer hardware, and the optimized parameter values ​​from the first training phase are used to initialize the corresponding parameters for the second training phase. A method that includes this.

2. The method according to claim 1, comprising the step of iterating the second training stage in each iteration with a larger subset of operators and / or a larger subset of parameters to provide a trained quantum Boltzmann machine in which the difference in expected values ​​between the target and the model is iteratively reduced.

3. The method according to claim 1 or 2, wherein the first training step trains the model using quantum relative entropy between the model and the target.

4. The method according to claim 3, wherein the gradient of the quantum relative entropy is determined with respect to the expected values ​​for the model and the target.

5. The method according to any one of claims 1 to 4, wherein the first training stage is performed using a mean-field (MF) model, a one-dimensional or two-dimensional geometrically local (GL) model, and / or a Gaussian fermion (GF) model.

6. The method according to any one of claims 1 to 5, wherein parameters not within the selected subset of the operator are kept at zero during the first training stage.

7. The method according to any one of claims 1 to 6, wherein a subset of the operators and a subset of the parameters are selected to use substantially all of the computing resources from the classical computer hardware.

8. The method according to any one of claims 1 to 7, further comprising the step of extending the Hamiltonian Ansatz from the subset of operators and the subset of parameters in the first stage to the larger subset of operators and the larger subset of parameters in the second stage.

9. The method according to any one of claims 1 to 8, wherein the second training stage is performed on the quantum computing hardware with respect to all of the parameters.

10. The method according to any one of claims 1 to 9, wherein the second training stage comprises optimizing the quantum relative entropy with respect to all the parameters by calculating the Gibbs expectation on the quantum computing hardware.

11. The method according to claim 10, wherein the first training step is to train the model using quantum relative entropy between the model and the target to provide the partially trained model comprising a quantum Boltzmann machine, and the second training step is to sample the partially trained model by preparing Gibbs states and calculating Gibbs expectations, wherein each sampling of the model comprises preparing Gibbs states and calculating Gibbs expectations on the quantum computing hardware.

12. The method according to any one of claims 1 to 11, wherein the second training stage includes performing a stochastic gradient descent.

13. The method according to any one of claims 1 to 12, wherein the second training stage comprises T iterations, each containing N samples, and N × T polynomially scales to the number of terms in the QBM Hamiltonian.

14. The method according to any one of claims 1 to 13, further comprising the step of extending the Hamiltonian Ansatz with at least one other set of operators and a set of parameters for a third training stage, wherein the at least one other set of operators and a set of parameters are optionally orthogonal.

15. The method according to claim 14, further comprising the step of initializing the parameters of the QBM with an extended Hamiltonian Ansatz using the optimal parameters obtained at the end of the previous quantum optimization loop.

16. The method according to any one of claims 1 to 15, wherein a gibbs state is used to provide a sample for machine learning.

17. The method according to any one of claims 1 to 16, wherein the Gibbs state used in the quantum Boltzmann machine is prepared and sampled in the quantum computing hardware, and the parameters are maintained in classical computing hardware.

18. A system comprising classical binary computing hardware and quantum computing hardware for performing machine learning, A machine learning system having a model that includes a quantum Boltzmann machine having a Hamiltonian Ansatz that includes a set of operators and a set of parameters, A first part of the machine learning system, which runs on the aforementioned classical binary computing hardware and is configured to provide a system partially trained with respect to a target, the first part being configured to use a selected subset of the set of operators to obtain values ​​optimized for a subset of the set of parameters, A second part of the machine learning system, which runs at least partially on the quantum computing hardware and is configured to provide a system trained with respect to the target, is configured to use a larger subset of the set of operators to obtain optimized values ​​for a larger subset of the set of parameters, the optimized parameter values ​​from the first stage being used to initialize the corresponding parameters used by the second part of the machine learning system. A system that includes this.

19. The system according to claim 18, wherein the quantum computing hardware is implemented using a plurality of qubits, and each of the plurality of qubits can be connected to any other qubit among the plurality of qubits.

20. We provide a quantum Boltzmann machine having a Hamiltonian Ansatz that includes a set of operators and a set of parameters. From a classical computing system, receive parameter values ​​optimized for a selected subset of the set of parameters associated with a selected subset of the operators of the quantum Boltzmann machine. The corresponding parameters of the quantum Boltzmann machine are initialized using the optimized parameter values ​​received from the classical computing system. The quantum Boltzmann machine is trained using a larger subset of the operators of the Hamiltonian Ansatz to optimize the parameter values ​​of the quantum Boltzmann machine for machine learning. A machine learning system equipped with quantum computing hardware configured in such a way.

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