Abstract
Quantum Boltzmann machines (QBMs) are machine-learning models for both classical and quantum data. We give an operational definition of QBM learning in terms of the difference in expectation values between the model and target, taking into account the polynomial size of the data set. By using the relative entropy as a loss function, this problem can be solved without encountering barren plateaus. We prove that a solution can be obtained with stochastic gradient descent using at most a polynomial number of Gibbs states. We also prove that pre-training on a subset of the QBM parameters can only lower the sample complexity bounds. In particular, we give pre-training strategies based on mean-field, Gaussian Fermionic, and geometrically local Hamiltonians. We verify these models and our theoretical findings numerically on a quantum and a classical data set. Our results establish that QBMs are promising machine learning models.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Introduction
Machine learning (ML) research has developed into a mature discipline with applications that impact many different aspects of society. Neural network and deep learning architectures have been deployed for tasks such as facial recognition, recommendation systems, time series modeling, and for analyzing highly complex data in science. In addition, unsupervised learning and generative modeling techniques are widely used for text, image, and speech generation tasks, which many people encounter regularly via interaction with chatbots and virtual assistants. Thus, the development of new machine learning models and algorithms can have significant consequences for a wide range of industries and, more generally, society as a whole1.
Recently, researchers in quantum information science have asked the question of whether quantum algorithms can offer advantages over conventional machine learning algorithms. This has led to the development of quantum algorithms for gradient descent, classification, generative modeling, reinforcement learning, as well as many other tasks2,3,4,5,6. However, one cannot straightforwardly generalize results from the conventional ML realm to the quantum ML realm. One must carefully reconsider the data encoding, training complexity, and sampling in the quantum machine learning (QML) setting. For example, it is yet unclear how to efficiently embed large data sets into quantum states so that a genuine quantum speedup is achieved7,8. Furthermore, as quantum states prepared on quantum devices can only be accessed via sampling, one cannot estimate properties with arbitrary precision. This gives rise to new problems, such as barren plateaus9,10,11,12,13,14, that make the training of certain QML models challenging or even practically impossible.
In this work, we show that a particular quantum generative model, the quantum Boltzmann machine15,16,17,18 (QBM) without hidden units, does not suffer from these issues, and can be trained with polynomial sample complexity on future fault-tolerant quantum computers. QBMs are physics-inspired ML models that generalize the classical Boltzmann machines to quantum Hamiltonian ansätze. The Hamiltonian ansatz is defined on a graph where each vertex represents a qubit and each edge represents an interaction. The task is to learn the strengths of the interactions, such that samples from the quantum model mimic samples from the target data set. Quantum generative models of this kind could find use in ML for science problems, by learning approximate descriptions of the experimental data. QBMs could also play an important role as components of larger QML models19,20,21,22,23. This is similar to how classical Boltzmann machines can provide good weight initializations for the training of deep neural networks24. One advantage of using a QBM over a classical Boltzmann machine is that it is more expressive since the Hamiltonian ansatz can contain more general non-commuting terms. This means that in some settings, the QBM outperforms its classical counterpart, even for classical target data18. Here we focus on QBMs without hidden units since their inclusion leads to additional challenges12,25,26, and there exists no evidence that they are beneficial.
In order to obtain practically relevant results, we begin by providing an operational definition of QBM learning. Instead of focusing on an information-theoretic measure, we assess the QBM learning performance by the difference in expectation values between the target and the model. We require that the data set and model parameters can be efficiently stored in classical memory. This means that the number of training data points is polynomial in the number of features if the target is classical, or in the number of particles if the target is quantum. Thus, statistics computed from such data set have polynomial precision. We then employ stochastic gradient descent methods27,28 in combination with shadow tomography29,30,31 to prove that QBM learning can be solved with polynomially many evaluations of the QBM model. Each evaluation of the model requires the preparation of one Gibbs state, and therefore, in this paper, we refer to the sample complexity as the required number of Gibbs state preparations. We also prove that the required number of Gibbs samples is reduced by pre-training on a subset of the parameters. This means that classically pre-training a simpler QBM model can potentially reduce the (quantum) training complexity. For instance, we show that one can analytically pre-train a mean-field QBM and a Gaussian Fermionic QBM in closed form. In addition, we show that one can pre-train geometrically local QBMs with gradient descent, which has some improved performance guarantees. To the best of our knowledge, this is the first time these exactly solvable models are used for either training or pre-training QBMs.
It is important to note that the time complexity of preparing and estimating properties of Gibbs states at finite temperature is largely unknown, but can be exponential in the worst case32. For cases where the Gibbs state preparation is intractable, the polynomial sample complexity of QBM learning becomes irrelevant as obtaining one sample already takes exponential time. On the other hand, the polynomial sample complexity is a prerequisite for efficient learning which many other QML models do not share. In this work, we are only concerned with the sample complexity of the QBM learning problem, and we do not discuss any specific Gibbs sampling implementation. We shall mention in passing that Gibbs states satisfying certain locality constraints can be efficiently simulated by classical means, e.g., using tensor networks33,34. Furthermore, generic Gibbs states can be prepared and sampled on a quantum computer by a variety of methods (e.g., see refs. 35,36,37,38,39,40,41,42,43), potentially with a quadratic speedup over classical algorithms. There also exist hybrid quantum-classical variational algorithms for Gibbs state preparation44,45,46,47,48,49, but there are currently many open issues regarding their trainability9,11,14,50. In practice, QBM learning allows for great flexibility in model design and can be combined with almost any Gibbs sampler.
Throughout the paper, we compare QBM learning and a related, but crucially different, problem called Hamiltonian learning31,51,52,53,54,55. Learning the Hamiltonian of a physical system in thermal equilibrium can be thought of as performing quantum state tomography. This has useful applications, for example, in the characterization of quantum devices, but requires prior knowledge about the target system and suffers an unfavorable scaling with respect to the temperature53. In contrast, the aim of QBM learning is to produce a generative model of the data at hand, with the Hamiltonian playing the role of an ansatz.
Our setup and theoretical results are summarized in Fig. 1. We provide classical numerical simulation results in Figs. 2 and 3 which confirm our analytical findings. We then conclude the paper with a discussion of the generalization capability of QBMs and avenues for future work.
The quantum Boltzmann machine (QBM) learning algorithm takes as input a data set of size polynomial in the number of features/qubits, and an ansatz with parameters θ and a set of m Hermitian operators {Hi}. In Definition 1, we provide an operational definition of the QBM learning problem where the model and target expectations must be close to within polynomial precision ϵ. A solution θopt is guaranteed to exist by Jaynes' principle. With Theorems 1 and 2, we establish that QBM learning can be solved by minimizing the quantum relative entropy S(η∥ρθ) with respect to θ using stochastic gradient descent (SGD). This requires a polynomial number of iterations T, each using a polynomial number of Gibbs state preparations, i.e., the sample complexity is polynomial. With Theorem 3, we prove that pre-training strategies that optimize a subset \({\theta }^{{{\rm{pre}}}}\) of the QBM parameters are guaranteed to lower the initial quantum relative entropy. After training, the QBM can be used to generate new synthetic data. Icons by SVG Repo, used under CC0 1.0 and adapted for our purpose.
a Quantum relative entropy \(S(\eta | {\rho }_{{\theta }^{{{\rm{pre}}}}})\) obtained after various pre-training strategies. We compare a mean-field (MF) model, a one-dimensional and two-dimensional geometrically local (GL) model, and a Gaussian Fermionic (GF) model to no pre-training/maximally mixed state. For the GL models, we stop the pre-training after the pre-training gradient is smaller than 0.01. We consider an 8-qubit target η as the Gibbs state \({e}^{{{{\mathcal{H}}}}_{{{\rm{XXZ}}}}}/Z\) of a one-dimensional XXZ model (Quantum Data), and a target η which coherently encodes the binary salamander retina data set (Classical Data). b Quantum relative entropy versus number of iterations for Quantum Data. The t < 0 iterations (gray area) show the reduction in relative entropy for GL 2D pre-training (red line). The t = 0 iteration corresponds to the pre-training results in panel (a). The t > 0 iterations show the training results in the absence of gradient noise, i.e., κ = ξ = 0.
Methods
We start by formally setting up the quantum Boltzmann machine (QBM) learning problem. We give the definitions of the target and model and describe how to assess performance by the precision of the expectation values. These definitions and assumptions are key to our results, and we introduce and motivate them here carefully. To further motivate our problem definition we compare to other related problems in the literature, such as quantum Hamiltonian learning.
We consider an n-qubit density matrix η as the target of the machine learning problem. If the target is classical, n could represent the number of features, e.g., the pixels in black-and-white pictures, or more complex features that have been extracted and embedded in the space of n qubits. If the target is quantum, n could represent the number of spin-\(\frac{1}{2}\) particles, but again, more complex many-body systems 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. We do not take that path here. Instead, we consider a setup where one only has access to classical information about the target. We have a data set \({{\mathcal{D}}}=\{{s}^{\mu }\}\) of data samples sμ and, crucially, we assume the data set can be efficiently stored in a classical memory: the amount of memory required to store each data sample is polynomial in n, and there are polynomially many samples. For example, sμ can be bitstrings of length n; this includes data sets like binary images and time series, categorical and count data, or binarized continuous data. As another example, the data may originate from measurements on a quantum system. In this case, sμ identifies an element of the positive operator-valued measure describing the measurement.
Next, we define the machine learning model which we use for data fitting. The fully visible QBM is an n-qubit mixed quantum state of the form
where \(Z={{\rm{Tr}}}[{e}^{{{{\mathcal{H}}}}_{\theta }}]\) is the partition function. The parameterized Hamiltonian is defined as
where \(\theta \in {{\mathbb{R}}}^{m}\) is the parameter vector, and {Hi} is a set of m Hermitian and orthogonal operators acting on the 2n-dimensional Hilbert space. For example, these could be n-qubit Pauli operators. As the true form of the target density matrix is unknown, the choice of operators {Hi} in the Hamiltonian is an educated guess. It is possible that, once the Hamiltonian ansatz is chosen, the space of QBM models does not contain the target, i.e., ρθ ≠ η, ∀ θ. This is called a model mismatch, and it is often unavoidable in machine learning. In particular, since we require m to be polynomial in n, ρθ cannot encode an arbitrary density matrix. Note that for fixed parameters, ρθ is also known as Gibbs state in the literature. Thus, we refer to an evaluation of the model as the preparation of a Gibbs state.
A natural measure to quantify how well the QBM ρθ approximates the target η is the quantum relative entropy
This measure generalizes the classical Kullback-Leibler divergence to density matrices. The relative entropy is exactly zero when the two densities are equal, η = ρθ, and S > 0 otherwise. In addition, when S(η∥ρθ) ≤ ϵ, by Pinsker’s inequality, all Pauli expectation values are within \({{\mathcal{O}}}(\sqrt{\epsilon })\). However, achieving this is generally not possible due to the model mismatch. In theory, one can minimize the relative entropy in order to find the optimal model parameters \({\theta}^{{{\rm{opt}}}}={{{\rm{argmin}}}}_{\theta}S(\eta \!\!\parallel \!\!{\rho}_{\theta })\). The form of the partial derivatives can be obtained analytically and reads
This is the difference between the target and model expectation values of the operators that we choose in the ansatz. A stationary point of the relative entropy is obtained when \({\langle {H}_{i}\rangle }_{{\rho }_{\theta }}={\langle {H}_{i}\rangle }_{\eta }\) for i ∈ {1, …, m}. Since S is strictly convex (see Supplementary Note 2), this stationary point is the unique global minimum.
It is difficult in practice to quantify how well the QBM is trained by means of relative entropy. In order to accurately estimate S(η∥ρθ), one requires access to the entropy of the target and the partition function of the model. Moreover, due to the model mismatch, the optimal QBM may have \(S(\eta \!\!\parallel \!\!{\rho }_{{\theta }^{{{\rm{opt}}}}}) \, > \, 0\), which is unknown beforehand. In this work, we provide an operational definition of QBM learning based on the size of the gradient ∇S(η∥ρθ).
Definition 1 (QBM learning problem)
Given a polynomial-space data set {sμ} obtained from an n-qubit target density matrixη, a target precisionϵ > 0, and a fully visible QBM with Hamiltonian\({{{\mathcal{H}}}}_{\theta }={\sum }_{i = 1}^{m}{\theta }_{i}{H}_{i}\), find a parameter vectorθsuch that with high probability
The expectation values of the target can be obtained from the data set in various ways. For example, for the generative modeling of a classical binary data set one can define a pure quantum state and obtain its expectation values (see Kappen18 and Supplementary Note 5). For the modeling of a target quantum state (density matrix) one can estimate expectation values from the outcomes of measurements performed in different bases. Due to the polynomial size of the data set, we can only compute the properties of the target (and model) to finite precision. For example, suppose that sμ are data samples from some unknown probability distribution P(s) and that we are interested in the sample mean. An unbiased estimator for the mean is \(\hat{s}=\frac{1}{M}\mathop{\sum }_{\mu = 1}^{M}{s}^{\mu }\). The variance of this estimator is σ2/M, where σ2 is the variance of P(s). By Chebyshev’s inequality, with high probability, the estimation error is of order \(\sigma /\sqrt{M}\). The polynomial size of the data set implies that the error decreases polynomially in general. In light of this, we say the QBM learning problem is solved for any precision ϵ > 0 in Eq. (5). The idea is that the expectation values of the QBM and target should be close enough that one cannot distinguish them without enlarging the data set.
An exact solution to the QBM learning problem always exists by Jaynes’ principle56: given a set of target expectations \(\{{\langle {H}_{i}\rangle }_{\eta }\}\) there exists a Gibbs state \({\rho }_{{\theta }^{{{\rm{opt}}}}}={e}^{{\sum }_{i}{\theta }_{i}^{{{\rm{opt}}}}{H}_{i}}/Z\) such that \(| {\langle {H}_{i}\rangle }_{{\rho }_{{\theta }^{{{\rm{opt}}}}}}-{\langle {H}_{i}\rangle }_{\eta }| =0,\forall i\). We refer to the model corresponding to Jaynes’ solution θopt as the optimal model. In QBM learning, we try to get as close as possible to this optimal model by minimizing the difference in expectation values. As shown in Supplementary Note 3, any solution to the QBM learning problem implies a bound on the optimal relative entropy, namely
A similar result can be derived from Proposition A2 in Rouzé and França31. This means that if one can solve the QBM learning problem to precision \(\epsilon \le \frac{{\epsilon }^{{\prime} }}{2\parallel \theta -{\theta }^{{{\rm{opt}}}}{\parallel }_{1}}\), one can also solve a stronger learning problem based on the relative entropy to precision \({\epsilon }^{{\prime} }\). However, this requires bounding ∥θ − θopt∥1.
We further motivate our problem definition by highlighting important differences with a related problem called quantum Hamiltonian learning31,51,52,53,54,55. This problem can be understood as trying to find a parameter vector ∥θ − θ*∥1 ≤ ϵ, where θ* is the true parameter vector defining a target Gibbs state η. This is, in fact, a stronger version of the QBM learning problem because finding the true parameters, θ*, is more demanding than getting ϵ-close to the expectation values of the optimal model from Jaynes’ solution. In realistic machine learning settings, we do not know the exact form of η a priori, and Hamiltonian learning is, in effect, equivalent to quantum state tomography. This task has an exponential sample-complexity lower bound29. Moreover, even if we do know the exact form, minimizing the distance to the true parameters, ∥θ − θ*∥1, can make the problem significantly more complicated than QBM learning, and potentially impractical to solve. Consider a single-qubit target \(\eta ={e}^{{{{\mathcal{H}}}}^{* }}/{{\rm{Tr}}}[{e}^{{{{\mathcal{H}}}}^{* }}]\), with Hamiltonian \({{{\mathcal{H}}}}^{* }={\theta }^{* }{\sigma }^{z}\). We can easily see that near \({\langle {\sigma }^{z}\rangle }_{\eta }=1-\epsilon\) the parameter θ* diverges. For example, if θ* = 2.64 then \({\langle {\sigma }^{z}\rangle }_{\eta }=0.99\) and if θ* = 3.8 then \({\langle {\sigma }^{z}\rangle }_{\eta }=0.999\), from which we see that the optimal parameter is very sensitive, \({{\mathcal{O}}}(1)\), to small changes, \({{\mathcal{O}}}(1{0}^{-2})\), in expectation values. In these cases, Hamiltonian learning is much harder than simply finding a QBM whose expectation value is within ϵ. A similar argument can be made for high-temperature targets near the maximally mixed state. Here the QBM problem is trivial: since all target expectations are \({\langle {H}_{i}\rangle }_{\eta }\approx {2}^{-n}{{\rm{Tr}}}[{H}_{i}]\), any model sufficiently close to the maximally mixed state solves the QBM learning problem. In contrast, Hamiltonian learning insists on estimating the unique true parameters.
Let us now discuss a method for solving QBM learning. We approach the problem by iteratively minimizing the quantum relative entropy, Eq. (3), by stochastic gradient descent (SGD)28. This method requires access to a stochastic gradient \({\hat{g}}_{{\theta }^{t}}\) computed from a set of samples at time t. Recall that our gradient has the form given in Eq. (4). The expectations with respect to the target are estimated from a random subset of the data, often called a mini-batch. The mini-batch size is a hyper-parameter and determines the precision ξ of each target expectation. Similarly, the QBM model expectations are estimated from measurements, for example, using classical shadows29,30 of the Gibbs state \({\rho }_{{\theta }^{t}}\) approximately prepared on a quantum device43. The number of measurements is also a hyper-parameter and determines the precision κ of each QBM expectation. We assume that the stochastic gradient is unbiased, i.e., \({\mathbb{E}}[\, {\hat{g}}_{{\theta }^{t}}]=\nabla S(\eta \!\!\parallel \!\!{\rho }_{\theta }){| }_{\theta = {\theta }^{t}}\), and that each entry of the vector has bounded variance. At iteration t, SGD updates the parameters as
where γt is called the learning rate.
Results
Sample complexity
Stochastic gradient descent solves the QBM learning problem with polynomial sample complexity. We state this in the following theorem, which is the main result of our work.
Theorem 1 (QBM training)
Given a QBM defined by a set ofn-qubit Pauli operators \({\{{H}_{i}\}}_{i = 1}^{m}\), a precisionκ for the QBM expectations, a precision ξ for the data expectations, and a target precision ϵ such that \({\kappa }^{2}+{\xi }^{2}\ge \frac{\epsilon }{2m}\). After
iterations of stochastic gradient descent on the relative entropy S(η∥ρθ) with constant learning rate \({\gamma }^{t}=\frac{\epsilon }{4{m}^{2}({\kappa }^{2}+{\xi }^{2})}\), we have
where \({\mathbb{E}}[\cdot ]\) denotes the expectation with respect to the random variable θt. Each iteration t ∈ {0, …, T} requires
preparations of the Gibbs state \({\rho }_{{\theta }^{t}}\), and the success probability of the full algorithm is λ. Here,\({\delta }_{0}=S(\eta \!\!\parallel \!\!{\rho }_{{\theta }^{0}})-S(\eta \! \! \parallel \! \! {\rho }_{{\theta }^{{{\rm{opt}}}}})\) is the relative entropy difference with the optimal model \({\rho }_{{\theta }^{{{\rm{opt}}}}}\).
A detailed proof of this theorem is given in Supplementary Note 3. It consists of carefully combining three important observations and results. First, we show that the quantum relative entropy for any QBM ρθ is L-smooth with \(L=2m{\max }_{j}\parallel \!\!{H}_{j}{\parallel }_{2}^{2}\), which follows from upper bounding the relative entropy and quantum belief propagation57. A similar result can be found in Rouzé and França31, Proposition A4. This is then combined with SGD convergence results from the machine learning literature27,28 to obtain the number of iterations T. Finally, we use sampling bounds from quantum shadow tomography to obtain the number of preparations N. In this last step, we focus on the shadow tomography protocol in Huang et al.30, which restricts our result to Pauli observables Hi ≡ Pi, thus ∥Hi∥2 = 1. It is possible to extend this to generic two-outcome observables29,58 with a polylogarithmic overhead compared to Eq. (10). Furthermore, for k-local Pauli observables, we can improve the result to
with classical shadows constructed from randomized measurement59 or by using pure thermal shadows43.
By combining Eqs. (8) and (10), we see that the final number of Gibbs state preparations Ntot ≥ T × N scales polynomially with m, the number of terms in the QBM Hamiltonian. By our classical memory assumption, we can only have \(m\in {{\mathcal{O}}}({{\rm{poly}}}(n))\). This means that the number of required measurements to solve QBM learning scales polynomially with the number of qubits. Consequently, there are no barren plateaus in the optimization landscape for this problem, resolving an open question from the literature26,50 for QBMs without hidden units. Furthermore, also the stronger problem in Eq. (6) can be solved without encountering an exponentially vanishing gradient as we show in Supplementary Note 3.
We remark, however, that our result does not imply a solution to the stricter Hamiltonian learning problem with polynomially many samples. In order to achieve this one needs some stronger conditions. For example, using α-strong convexity for the specific case of geometrically local models, as shown in Anshu et al.53. In Supplementary Note 2, we investigate generalizing this by looking at generic Hamiltonians made of low-weight Pauli operators. We perform numerical simulations using differentiable programming60 and find evidence indicating that, even in this case, the quantum relative entropy is α-strong convex. This means we could potentially achieve an improved sample complexity for QBM learning. We prove the following theorem in Supplementary Note 3.
Theorem 2 (α-strongly convex QBM training)
Given a QBM defined by a Hamiltonian ansatz\({{{\mathcal{H}}}}_{\theta }\)such thatS(η∥ρθ) isα-strongly convex, a precision κ for the QBM expectations, a precision ξ for the data expectations, and a target precisionϵ such that\({\kappa }^{2}+{\xi }^{2}\ge \frac{\epsilon }{2m}\). After
iterations of stochastic gradient descent on the relative entropyS(η∥ρθ) with learning rate\({\gamma }^{t}\le \frac{1}{4{m}^{2}}\), we have
Each iteration requires the number of samples given in Eq. (10).
Finally, we observe that the sample bound in Theorem 1 depends on δ0, the relative entropy difference of the initial and optimal QBMs. This means that if we can lower the initial relative entropy with an educated guess, we also tighten the bound on the QBM learning sample complexity. In this respect, we prove that we can reduce δ0 by pre-training a subset of the parameters in the Hamiltonian ansatz. Thus, pre-training reduces the number of iterations required to reach the global minimum.
Theorem 3 (QBM pre-training)
Assume a targetηand a QBM model\({\rho }_{\theta }={e}^{{\sum }_{i = 1}^{m}{\theta }_{i}{H}_{i}}/Z\) for which we like to minimize the relative entropyS(η∥ρθ). Initializingθ0 = 0 and pre-trainingS(η∥ρθ) with respect to any subset of\(\tilde{m}\le m\) parameters guarantees that
where \({\theta }^{{{\rm{pre}}}}=[{\chi }^{{{\rm{pre}}}},{0}_{m-\tilde{m}}]\) and the vector \({\chi }^{{{\rm{pre}}}}\) of length \(\tilde{m}\) contains the parameters for the terms \({\{{H}_{i}\}}_{i = 1}^{\tilde{m}}\) at the end of pre-training. More precisely, starting from\({\rho }_{\chi }={e}^{\mathop{\sum }_{i = 1}^{\tilde{m}}{\chi }_{i}{H}_{i}}/Z\) and minimizingS(η∥ρχ) with respect toχ ensures Eq. (14) for any \(S(\eta \!\!\parallel \!\!{\rho }_{{\chi }^{{{\rm{pre}}}}})\le S(\eta \!\!\parallel \!\!{\rho }_{{\chi }^{0}})\).
We refer to Supplementary Note 4 for the proof of this theorem. It applies to any method that is able to minimize the relative entropy with respect to a subset of the parameters. Crucially, all the other parameters are fixed to zero, and the pre-training needs to start from the maximally mixed state \({\mathbb{I}}/{2}^{n}\). Note that the maximally mixed state is not the most ‘distant’ state from the target state, and there exist states for which \(S(\eta \!\!\parallel \!\!{\rho }_{\theta }) > S(\eta \!\!\parallel \!\!{\mathbb{I}}/{2}^{n})\). As an example of pre-training, one could use SGD as described above, and apply updates only to the chosen subset of parameters. With a suitable learning rate, this ensures that pre-training lowers the relative entropy compared to the maximally mixed state \(S(\eta \!\!\parallel \!\!{\mathbb{I}}/{2}^{n})\). As a consequence, one can always add additional linearly independent terms to the QBM ansatz without having to retrain the model from scratch. The performance is guaranteed to improve, specifically toward the global optimum, due to the strict convexity of the relative entropy. By Eq. (14) and Pinsker’s inequality, the trace-distance to all the other observables reduces as well. This is in contrast to other QML models, which do not have a convex loss function. It is particularly useful if one can pre-train a certain subset classically before training the full model on a quantum device. For instance, in Supplementary Note 4, we present mean-field and Gaussian Fermionic pre-training strategies with closed-form expressions for the optimal subset of parameters.
Numerical experiments
To verify our theoretical findings, we perform numerical experiments of QBM learning on data sets constructed from a quantum and a classical source. In particular, we use expectation values of the Gibbs state of a 1D quantum XXZ model in an external field61,62 and expectation values of a classical salamander retina data set63. How to obtain the target state expectation values for both cases is explained in Supplementary Note 5.
First we focus on reducing the initial relative entropy \(S(\eta \!\!\parallel \!\!{\rho }_{{\theta }^{0}})\) by QBM pre-training, following Theorem 3. We consider mean-field (MF), Gaussian Fermionic (GF), and geometrically local (GL) models as the pre-training strategies. The Hamiltonian ansatz of the MF model consists of all possible one-qubit Pauli terms \({\{{H}_{i}\}}_{i = 1}^{3n}={\{{\sigma }_{i}^{x},{\sigma }_{i}^{y},{\sigma }_{i}^{z}\}}_{i = 1}^{n}\) in Eq. (2), hence it has 3n parameters. The Hamiltonian of the GF model is a quadratic form \({{{\mathcal{H}}}}_{\theta }^{{{\rm{GF}}}}={\sum }_{i,j}{\tilde{\theta }}_{ij}{\vec{C}}_{i}^{{\dagger} }{\vec{C}}_{j}\) of Fermionic creation and annihilation operators, where \({\tilde{\theta }}_{ij}\) is the 2n × 2n Hermitian parameter matrix, which has n2 free parameters. Here \({\overrightarrow{C}}^{{\dagger} }=[{c}_{1}^{{\dagger} },\ldots ,{c}_{n}^{{\dagger} },{c}_{1},\ldots ,{c}_{n}]\) with the operators satisfying {ci, cj} = 0 and \(\{{c}_{i}^{{\dagger} },{c}_{j}\}={\delta }_{ij}\), where {A, B} = AB + BA is the anti-commutator. The advantage of the MF and GF pre-training is that there exists a closed-form solution given target expectation values \({\langle {H}_{i}\rangle }_{\eta }\) (see Supplementary Note 4).
In contrast, the GL models are defined with a Hamiltonian ansatz
for which, in general, the parameter vector \(\vec{\theta }\equiv \{\lambda ,\sigma \}\) cannot be found analytically. Here the sum 〈i, j〉 imposes some constraints on the (geometric) locality of the model, i.e., summing over all possible nearest neighbors in some d-dimensional lattice. In particular, we choose one- and two-dimensional locality constraints suitable to the assumptions given in previous work53,54. In these specific cases the relative entropy is strongly convex, thus pre-training has the improved performance guarantees from Theorem 2.
In Fig. 2a, we show the relative entropy \(S(\eta \!\!\parallel \!\!{\rho }_{{\theta }^{{{\rm{pre}}}}})\) obtained after pre-training with these models for 8-qubit problems. For MF and GF, we use the closed-form solutions, and for GL, we use exact gradient descent with learning rate of \(\gamma =1/\tilde{m}\). As a comparison, we also show the result obtained for the QBM without pre-training, i.e., for the maximally mixed state S(η∥ρθ=0). We observe that for both targets, all pre-training strategies are successful in obtaining a lower \(S(\eta \!\! \parallel \!\!{\rho }_{{\theta }^{{{\rm{pre}}}}})\) compared to the maximally mixed state. For the GL 1D ansatz, the target quantum state is contained within the QBM model space, which means that the relative entropy is zero after pre-training. This shows that having knowledge about the target (e.g., the fact that it is one-dimensional) could help inform QBM ansatz design and significantly reduce the complexity of QBM learning. The GF model, which has completely different terms in the ansatz, manages to reduce \(S(\eta \!\!\parallel \!\!{\rho }_{{\theta }^{{{\rm{pre}}}}})\) by a factor of ≈5 for the quantum target and ≈4 for the classical target. By the Jordan-Wigner transformation, we can express the target XXZ model in a Fermionic basis. In this representation, the target only has a small perturbation compared to the model space of the GF model. This could explain the good performance of pre-training with the GF model.
Next, we investigate the effect of using pre-trained models as starting point for QBM learning with exact gradient descent. We extend all pre-trained models with additional linearly independent terms in the ansatz
where now, compared to Eq. (15), any qubit can be connected to any other qubit, and we do not have a constraint on the geometric locality. This is the fully connected QBM Hamiltonian used in previous work18,43. We now consider data from the quantum target \(\eta ={e}^{{{{\mathcal{H}}}}_{{{\rm{XXZ}}}}}/Z\) for 8 qubits.
In Fig. 2b, we show the quantum relative entropy during training when starting from different pre-trained models \({\rho }_{{\theta }^{0}}:={\rho }_{{\theta }^{{{\rm{pre}}}}}\). The initial parameter vector θ0 is the one obtained at the end of pre-training on quantum data in panel (a). We do not include gradient noise in this simulation, i.e., κ = ξ = 0, and we use a learning rate of γ = 1/(2m). The performance of the MF pre-trained model (orange line) is better than that of the vanilla model (blue line) for all t. However, the improvement is modest. Starting from a GL 2D model (red line) has a much more significant effect, with \(S(\eta \!\!\parallel \!\! {\rho }_{{\theta }^{t}})\) being an order of magnitude smaller than that of the vanilla model at all iterations t. This pre-training strategy requires very few gradient descent iterations (dark gray area). This could be stemming from the strong convexity of this particular pre-training model. In general, the benefits of pre-training need to be assessed on a case-by-case basis, as the size of the improvement depends on the particular target and the particular pre-training model used. Importantly, we only proved Theorem 1 for a learning rate of \(\gamma =\min \left\{\frac{1}{L},\frac{\epsilon }{4{m}^{2}({\kappa }^{2}+{\xi }^{2})}\right\}\). Therefore, choosing a larger learning rate might reduce the benefits of pre-training.
Lastly, we numerically confirm our bound on the number of SGD iterations (Eq. (8) in Theorem 1). We consider data from the classical salamander retina target with 8 features and a fully connected QBM model on 8 qubits. In Fig. 3, we compare training with two different noise strengths κ2 + ξ2. We implemented these settings by adding Gaussian noise, effectively simulating the noise strength determined by the number of data points in the data set and the number of measurements of the Gibbs state on a quantum device. Using the standard Monte Carlo estimate, at each update iteration, we require a mini-batch of data samples of size \(\,\frac{1}{{\xi }^{2}}\), and a number of measurements \(\frac{1}{{\kappa }^{2}}\) (assuming these measurements can be performed without additional hardware noise). We could even use mini batches of size 1 and a single measurement, as long as the expectation values are unbiased. For both noise strengths, we obtain the desired target precision of ϵ = 0.1 within 40000 iterations. This is well within the bound of order 109 on the number of iterations obtained from Theorem 1, which is the worst-case scenario. For the interested reader, in Supplementary Note 3, we provide additional numerical evidence confirming the scaling behavior of our theorems.
The target η is made of 8 features from the classical salamander retina data set, the model \({\rho }_{{\theta }^{t}}\) is a quantum Boltzmann machine (QBM). When computing expectation values for the gradient, we allow precision ξ for the target and precision κ for the QBM. We compare the combined noise strength of 0.01 (blue line) to 0.05 (orange line). We aim for a maximum error of ϵ = 0.1 (red dashed line) and use a learning rate of \(\gamma =\frac{\epsilon }{4{m}^{2}({\kappa }^{2}+{\xi }^{2})}\). SGD converges within a number of iterations consistent with Theorem 1.
Generalization
Before concluding our work, in this section, we briefly comment on the generalization capability of the fully connected QBM to which our convergence results apply. In particular, we connect with a result proved by Aaronson64, which concerns the learnability of quantum states.
Theorem 4 (Theorem 1.1 in ref. 64)
Let η be a 2n-dimensional mixed state, and let \({{\mathcal{D}}}\) be any probability measure over two-outcome measurements. Then given samples E1, …, Em drawn independently from \({{\mathcal{D}}}\), with probability at least 1 − ν, the samples have the following generalization property: any hypothesis stateρ such that \(| {\langle {E}_{i}\rangle }_{\rho }-{\langle {E}_{i}\rangle }_{\eta }| \le \zeta \varepsilon\) for all i ∈ {1, …, m} will also satisfy
provided we took
for some large enough constant C.
In other words, when a hypothesis state ρ matches the expectation values of m randomly sampled measurement operators from distribution \({{\mathcal{D}}}\), then, with high probability, most expectation values of the other measurement operators from \({{\mathcal{D}}}\) match up to error ε. Comparing to Definition 1, we can identify the learned QBM ρθ as a specific hypothesis state that matches the expectation values of operators \({\{{H}_{i}\}}_{i = 1}^{m}\) of the target η. Our convergence results show how many Gibbs state samples are required to obtain such a hypothesis state. Thus, our QBM learning result in Theorem 1, in a sense, complements Theorem 4 and provides a concrete algorithm for finding the hypothesis ρ with \({{\mathcal{O}}}({{\rm{poly}}}(n))\) samples. One distinction is that in QBM learning, we already assume access to a fixed set of measurement operators which defines the Hamiltonian ansatz, instead of randomly sampling operators from a distribution.
In order to say something about generalization, we now consider a scenario in which the user is interested in solving the QBM learning problem for some fixed (potentially exponentially large) number of two-outcome measurement operators \({\{{H}_{i}\}}_{i = 1}^{K}\). Then by defining \({{\mathcal{D}}}\) as the uniform distribution over the set \({\{{H}_{i}\}}_{i = 1}^{K}\), and constructing the QBM ansatz by sampling m < K operators from \({{\mathcal{D}}}\), we can directly apply Theorem 4 for generalization. In particular, defining the ansatz in this way ensures that our trained QBM will, with high probability, generalize to any other observable sampled from \({{\mathcal{D}}}\), including ones that are not in the ansatz.
Conclusions
In this paper, we give an operational definition of quantum Boltzmann machine (QBM) learning. We show that this problem can be solved with polynomially many preparations of quantum Gibbs states. To prove our bounds, we use the properties of the quantum relative entropy in combination with the performance guarantees of stochastic gradient descent (SGD). We do not make any assumption on the form of the Hamiltonian ansatz, other than that it consists of polynomially many terms. This is in contrast with earlier works that looked at the related Hamiltonian learning problem for geometrically local models53,54. There, strong convexity is required in order to relate the optimal Hamiltonian parameters to the Gibbs state expectation values. In our machine learning setting, we do not know the form of the target Hamiltonian a priori. Therefore, we argue that learning the exact parameters is irrelevant, and one should focus directly on the expectation values. Our bounds only require L-smoothness of the relative entropy and apply to all types of QBMs without hidden units.
We also show that our theoretical sampling bounds can be tightened by lowering the initial relative entropy of the learning process. We prove that pre-training on any subset of the parameters is guaranteed to perform better than (or equal to) the maximally mixed state. This is beneficial if one can efficiently perform the pre-training, which we show is possible for mean-field, Gaussian Fermionic, and geometrically local QBMs. We verify the performance of these models and our theoretical bounds with classical numerical simulations. From this, we learn that knowledge about the target (e.g., its dimension, degrees of freedom, etc.) can significantly improve the training process. Furthermore, we find that our generic bounds are quite loose, and in practice, one could get away with a much smaller number of samples.
This brings us to interesting avenues for future work. One possibility consists of tightening the sample bound by going beyond the plain SGD method that we have used here. This could be done by adding momentum, by using other advanced update schemes28,65, or by exploiting the convexity of the relative entropy. While we believe this can improve the \({{\mathcal{O}}}({{\rm{poly}}}(m,\frac{1}{\epsilon }))\) scaling in our bounds, we note that it does not change the main conclusion of our paper: the QBM learning problem can be solved with polynomially many preparations of Gibbs states. Another important direction is the investigation of different ansätze and their performance. Generative models are often assessed in terms of training quality66, but generalization capabilities have been recently investigated by both classical47,67 and quantum68,69 machine learning researchers. For the case of QBMs, we employ an alternative definition of generalization64 and show how it can be used to construct ansätze. The numerical investigation of this method is an interesting venue for future work.
Our pre-training result could be useful for implementing QBM learning on near-term and early fault-tolerant quantum devices. To this end, one would use a quantum computer as a Gibbs sampler. There exists a plethora of quantum algorithms (e.g., see refs. 36,38,40,41,42) that prepare Gibbs states with a quadratic improvement in time complexity over the best existing classical algorithms. Moreover, the use of a quantum device gives an exponential reduction in space complexity in general. One can also sidestep the Gibbs state preparation and use algorithms that directly estimate Gibbs state expectation values, e.g., by constructing classical shadows of pure thermal quantum states43. This reduces the number of qubits and, potentially, the circuit depth.
Finally, our results open the door to novel methods for the incremental learning of QBMs driven by the availability of both training data and quantum hardware. For example, one could select a Hamiltonian ansatz that is very well suited for a particular quantum device. After exhausting all available classical resources during the pre-training, one enlarges the model and continues the training on the quantum device, which is guaranteed to improve the performance. As the quantum hardware matures, it allows us to execute deeper circuits and to further increase the model size. Incremental QBM training strategies could be designed to follow the quantum hardware roadmap, toward training ever larger and more expressive quantum machine learning models.
Data availability
The data that support the findings of this study are available at Zenodo70.
Code availability
The code used to create the figures in this paper is available from the authors upon reasonable request.
References
Bommasani, R. et al. On the opportunities and risks of foundation models. Preprint at https://doi.org/10.48550/arXiv.2108.07258 (2021).
Dunjko, V. & Briegel, H. J. Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Rep. Prog. Phys. 81, 074001 (2018).
Ciliberto, C. et al. Quantum machine learning: a classical perspective. Proc. R. Soc. A Math. Phys. Eng. Sci. 474, 20170551 (2018).
Benedetti, M., Lloyd, E., Sack, S. & Fiorentini, M. Parameterized quantum circuits as machine learning models. Quantum Sci. Technol. 4, 043001 (2019).
Lamata, L. Quantum machine learning and quantum biomimetics: a perspective. Mach. Learn. Sci. Technol. 1, 033002 (2020).
Cerezo, M., Verdon, G., Huang, H.-Y., Cincio, L. & Coles, P. J. Challenges and opportunities in quantum machine learning. Nat. Comput. Sci. 2, 567–576 (2022).
Aaronson, S. Read the fine print. Nat. Phys. 11, 291–293 (2015).
Hoefler, T., Häner, T. & Troyer, M. Disentangling hype from practicality: on realistically achieving quantum advantage. Commun. ACM 66, 82–87 (2023).
McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018).
Grant, E., Wossnig, L., Ostaszewski, M. & Benedetti, M. An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019).
Cerezo, M., Sone, A., Volkoff, T., Cincio, L. & Coles, P. J. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nat. Commun. 12, 1791 (2021).
Ortiz, C., Kieferová, M. & Wiebe, N. Entanglement-induced barren plateaus. PRX Quantum 2, 040316 (2021).
Martín, E., Plekhanov, K. & Lubasch, M. Barren plateaus in quantum tensor network optimization. Quantum 7, 974 (2023).
Rudolph, M. S. et al. Trainability barriers and opportunities in quantum generative modeling. Preprint at https://doi.org/10.48550/arXiv.2305.02881 (2023).
Amin, M. H., Andriyash, E., Rolfe, J., Kulchytskyy, B. & Melko, R. Quantum Boltzmann machine. Phys. Rev. X 8, 021050 (2018).
Benedetti, M., Realpe-Gómez, J., Biswas, R. & Perdomo-Ortiz, A. Quantum-assisted learning of hardware-embedded probabilistic graphical models. Phys. Rev. X 7, 041052 (2017).
Kieferová, M. & Wiebe, N. Tomography and generative training with quantum Boltzmann machines. Phys. Rev. A 96, 062327 (2017).
Kappen, H. J. Learning quantum models from quantum or classical data. J. Phys. A Math. Theor. 53, 214001 (2020).
Benedetti, M., Realpe-Gómez, J. & Perdomo-Ortiz, A. Quantum-assisted Helmholtz machines: a quantum-classical deep learning framework for industrial datasets in near-term devices. Quantum Sci. Technol. 3, 034007 (2018).
Khoshaman, A. et al. Quantum variational autoencoder. Quantum Sci. Technol. 4, 014001 (2018).
Crawford, D., Levit, A., Ghadermarzy, N., Oberoi, J. S. & Ronagh, P. Reinforcement learning using quantum Boltzmann machines. Quantum Info. Comput. 18, 51–74 (2018).
Wilson, M., Vandal, T., Hogg, T. & Rieffel, E. G. Quantum-assisted associative adversarial network: applying quantum annealing in deep learning. Quantum Mach. Intell. 3, 19 (2021).
Wang, L., Sun, Y. & Zhang, X. Quantum deep transfer learning. N. J. Phys. 23, 103010 (2021).
Hinton, G. E., Osindero, S. & Teh, Yee-Whye A fast learning algorithm for deep belief nets. Neural Comput. 18, 1527–1554 (2006).
Wiebe, N. & Wossnig, L. Generative training of quantum Boltzmann machines with hidden units. Preprint at https://doi.org/10.48550/arXiv.1905.09902 (2019).
Kieferova, M., Carlos, O. M. & Wiebe, N. Quantum generative training using Rényi divergences. Preprint at https://doi.org/10.48550/arXiv.2106.09567 (2021).
Khaled, A. & Richtárik, P. Better theory for SGD in the nonconvex world. Preprint at https://doi.org/10.48550/arXiv.2002.03329 (2020).
Garrigos, G. & Gower, R. M. Handbook of convergence theorems for (stochastic) gradient methods. Preprint at https://doi.org/10.48550/arXiv.2301.11235 (2023).
Aaronson, S. Shadow tomography of quantum states. SIAM J. Comput. 49, STOC18–368–STOC18–394 (2020).
Huang, H.-Y., Kueng, R. & Preskill, J. Information-theoretic bounds on quantum advantage in machine learning. Phys. Rev. Lett. 126, 190505 (2021).
Rouzé, C. & França, D. S. Learning quantum many-body systems from a few copies. Quantum 8, 1319 (2024).
Bravyi, S., Chowdhury, A., Gosset, D. & Wocjan, P. Quantum Hamiltonian complexity in thermal equilibrium. Nat. Phys. 18, 1367–1370 (2022).
Alhambra, Á. M. & Cirac, J. I. Locally accurate tensor networks for thermal states and time evolution. PRX Quantum 2, 040331 (2021).
Kuwahara, T., Alhambra, Á. M. & Anshu, A. Improved thermal area law and quasilinear time algorithm for quantum Gibbs states. Phys. Rev. X 11, 011047 (2021).
Poulin, D. & Wocjan, P. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Phys. Rev. Lett. 103, 220502 (2009).
Temme, K., Osborne, T. J., Vollbrecht, K. G., Poulin, D. & Verstraete, F. Quantum metropolis sampling. Nature 471, 87–90 (2011).
Kastoryano, M. J. & Brandao, F. G. S. L. Quantum Gibbs samplers: the commuting case. Commun. Math. Phys. 344, 915–957 (2016).
Chowdhury, A. N. & Somma, R. D. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quantum Info. Comput. 17, 41–64 (2017).
Anschuetz, E. R. & Cao, Y. Realizing quantum Boltzmann machines through eigenstate thermalization. Preprint at https://doi.org/10.48550/arXiv.1903.01359 (2019).
Holmes, Z., Muraleedharan, G., Somma, R. D., Subasi, Y. & Şahinoğlu, B. Quantum algorithms from fluctuation theorems: thermal-state preparation. Quantum 6, 825 (2022).
Chen, C.-F., Kastoryano, M. J., Brandão, F. G. S. L. & Gilyén, A. Quantum thermal state preparation. Preprint at https://doi.org/10.48550/arXiv.2303.18224 (2023).
Zhang, D., Bosse, J. L. & Cubitt, T. Dissipative quantum Gibbs sampling. Preprint at https://doi.org/10.48550/arXiv.2304.04526 (2023).
Coopmans, L., Kikuchi, Y. & Benedetti, M. Predicting Gibbs-state expectation values with pure thermal shadows. PRX Quantum 4, 010305 (2023).
Wu, J. & Hsieh, T. H. Variational thermal quantum simulation via thermofield double states. Phys. Rev. Lett. 123, 220502 (2019).
Chowdhury, A. N., Low, G. H. & Wiebe, N. A variational quantum algorithm for preparing quantum Gibbs states. Preprint at https://doi.org/10.48550/arXiv.2002.00055 (2020).
Liu, J.-G., Mao, L., Zhang, P. & Wang, L. Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits. Mach. Learn. Sci. Technol. 2, 025011 (2021).
Zhao, S. et al. Bias and generalization in deep generative models: an empirical study. Preprint at https://doi.org/10.48550/arXiv.1811.03259 (2018).
Consiglio, M. et al. Variational Gibbs state preparation on noisy intermediate-scale quantum devices. Phys. Rev. A 110, 012445 (2024).
Huijgen, O., Coopmans, L., Najafi, P., Benedetti, M. & Kappen, H. J. Training quantum Boltzmann machines with the β-variational quantum eigensolver. Mach. Learn. Sci. Technol. 5, 025017 (2024).
Anschuetz, E. R. & Kiani, B. T. Quantum variational algorithms are swamped with traps. Nat. Commun. 13, 7760 (2022).
Wiebe, N., Granade, C., Ferrie, C. & Cory, D. G. Hamiltonian learning and certification using quantum resources. Phys. Rev. Lett. 112, 190501 (2014).
França, D. S., Brandão, F. G.S L. & Kueng, R. Fast and robust quantum state tomography from few basis measurements. In Proc. 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021), Vol. 197, 7:1–7:13 (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2021).
Anshu, A., Arunachalam, S., Kuwahara, T. & Soleimanifar, M. Sample-efficient learning of interacting quantum systems. Nat. Phys. 17, 931–935 (2021).
Haah, J., Kothari, R. & Tang, E. Learning quantum Hamiltonians from high-temperature Gibbs states and real-time evolutions. Nat. Phys. 20, 1027–1031 (2024).
Onorati, E., Rouzé, C., França, D. S. & Watson, J. D. Efficient learning of ground & thermal states within phases of matter. Preprint at https://doi.org/10.48550/arXiv.2301.12946 (2023).
Jaynes, E. T. Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957).
Hastings, M. B. Quantum belief propagation: an algorithm for thermal quantum systems. Phys. Rev. B 76, 201102 (2007).
Bădescu, C. & O’Donnell, R. Improved quantum data analysis. In Proc. 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2021), 1398–1411 (Association for Computing Machinery, 2021).
Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).
Baydin, A. G., Pearlmutter, B. A., Radul, A. A. & Siskind, J. M. Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res. 18, 1–43 (2018).
Heisenberg, W. Zur theorie des ferromagnetismus. Z. Phys. 49, 619–636 (1928).
Franchini, F. An Introduction to Integrable Techniques for One-Dimensional Quantum Systems (Springer, 2017).
Tkačik, G. et al. Searching for collective behavior in a large network of sensory neurons. PLoS Comput. Biol. 10, 1–23 (2014).
Aaronson, S. The learnability of quantum states. Proc. R. Soc. A Math. Phys. Eng. Sci. 463, 3089–3114 (2007).
Foster, D. J. et al. The complexity of making the gradient small in stochastic convex optimization. PMLR 99, 1319–1345 (2019).
Riofrio, C. A. et al. A characterization of quantum generative models. ACM Trans. Quantum Comput. 5, 12 (2024).
Thanh-Tung, H. & Tran, T. Toward a generalization metric for deep generative models. Preprint at https://doi.org/10.48550/arXiv.2011.00754 (2020).
Du, Y., Tu, Z., Wu, B., Yuan, X. & Tao, D. Power of quantum generative learning. Preprint at https://doi.org/10.48550/arXiv.2205.04730 (2022).
Gili, K., Mauri, M. & Perdomo-Ortiz, A. Generalization metrics for practical quantum advantage in generative models. Phys. Rev. Appl. 21, 044032 (2024).
Coopmans, L. & Benedetti, M. Data for figures of manuscript entitled: “On the sample complexity of quantum Boltzmann machine learning”. Zenodo https://doi.org/10.5281/zenodo.12761198 (2024).
Acknowledgements
We thank Guillaume Garrigos and Robert M. Gower for pointing us to the literature regarding stochastic gradient descent. We thank Samuel Duffield, Yuta Kikuchi, Mark Koch, Enrico Rinaldi, Matthias Rosenkranz, and Oscar Watts for helpful discussions and for their feedback on an earlier version of this manuscript.
Author information
Authors and Affiliations
Contributions
L.C. and M.B. contributed equally to this work.
Corresponding authors
Ethics declarations
Competing interests
L.C. and M.B. are affiliated with Quantinuum Ltd and the results presented in this paper are subject to a pending patent application filed by Quantinuum Ltd with Application Number PCT/GB2024/051593.
Peer review
Peer review information
This manuscript has been previously reviewed in another Nature Portfolio journal. The manuscript was considered suitable for publication without further review at Communications Physics.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Coopmans, L., Benedetti, M. On the sample complexity of quantum Boltzmann machine learning. Commun Phys 7, 274 (2024). https://doi.org/10.1038/s42005-024-01763-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005-024-01763-x
- Springer Nature Limited