Variational quantum Boltzmann machines

Abstract

This work presents a novel realization approach to quantum Boltzmann machines (QBMs). The preparation of the required Gibbs states, as well as the evaluation of the loss function’s analytic gradient, is based on variational quantum imaginary time evolution, a technique that is typically used for ground-state computation. In contrast to existing methods, this implementation facilitates near-term compatible QBM training with gradients of the actual loss function for arbitrary parameterized Hamiltonians which do not necessarily have to be fully visible but may also include hidden units. The variational Gibbs state approximation is demonstrated with numerical simulations and experiments run on real quantum hardware provided by IBM Quantum. Furthermore, we illustrate the application of this variational QBM approach to generative and discriminative learning tasks using numerical simulation.

1 Introduction

Boltzmann machines (BMs) (Ackley et al. 1985; Du and Swamy 2019) offer a powerful framework for modelling probability distributions. These types of neural networks use an undirected graph structure to encode relevant information. More precisely, the respective information is stored in bias coefficients and connection weights of network nodes, which are typically related to binary spin-systems and grouped into those that determine the output, the visible nodes, and those that act as latent variables, the hidden nodes. Furthermore, the network structure is linked to an energy function which facilitates the definition of a probability distribution over the possible node configurations by using a concept from statistical mechanics, i.e., Gibbs states (Boltzmann 1877; Gibbs 1902). The aim of BM training is to learn a set of weights such that the resulting model approximates a target probability distribution which is implicitly given by training data. This setting can be formulated as discriminative as well as generative learning task (Liu and Webb 2010). Applications have been studied in a large variety of domains such as the analysis of quantum many-body systems, statistics, biochemistry, social networks, signal processing, and finance; see, e.g., (Torlai et al. 2018; Carleo et al. 2018; Carleo and Troyer 2017; Nomura et al. 2017; Anshu et al. 2020; Melko et al. 2019; Hrasko et al. 2015; Tubiana et al. 2019; Liu et al. 2013; Mohamed and Hinton 2010; Assis et al. 2018). However, BMs are complicated to train in practice because the loss function’s derivative requires the evaluation of a normalization factor, the partition function, that is generally difficult to compute. Usually, it is approximated using Markov chain Monte Carlo methods which may require long runtimes until convergence (Carreira-Perpinan and Hinton 2005; Murphy 2012). Alternatively, the gradients could be estimated approximately using contrastive divergence (Hinton 2002) or pseudo-likelihood (Besag 1975) potentially leading to inaccurate results (Tieleman 2008; Sutskever and Tieleman 2010).

Quantum Boltzmann machines (QBMs) (Amin et al. 2018) are a natural adaption of BMs to the quantum computing framework. Instead of an energy function with nodes being represented by binary spin values, QBMs define the underlying network using a Hermitian operator, a parameterized Hamiltonian:

$$ H_{\theta}=\sum\limits_{i=0}^{p-1}\theta_{i}h_{i}, $$

with $\theta \in \mathbb {R}^{p}$ and $h_{i}=\bigotimes _{j=0}^{n-1}\sigma _{j, i}$ for σ_j,i ∈{I,X,Y,Z} acting on the j^th qubit. The network nodes are hereby characterized by the Pauli matrices σ_j,i. This Hamiltonian relates to a quantum Gibbs state, $\rho ^{\text {Gibbs}} = {e^{-H_{\theta }/\left (\text {k}_{\text {B}}\text {T}\right )}}/ {Z}$ with k_B and T denoting the Boltzmann constant and the system temperature, and $Z=\text {Tr}\left [e^{-H_{\theta }/\left (\text {k}_{\text {B}}\text {T}\right )}\right ]$. It should be noted that those qubits which determine the model output are referred to as visible and those which act as latent variables as hidden qubits. The aim of the model is to learn Hamiltonian parameters such that the resulting Gibbs state reflects a given target system. In contrast to BMs, this framework allows the use of quantum structures which are potentially inaccessible classically. Equivalently to the classical model, QBMs are suitable for discriminative as well as generative learning.

We present here a QBM implementation that circumvents certain issues which emerged in former approaches. The first paper on QBMs (Amin et al. 2018) and several subsequent works (Anschütz and Cao 2019; Kieferová and Wiebe 2017; Kappen 2020; Wiebe and Wossnig 2019) are incompatible with efficient evaluation of the loss function’s analytic gradients if the given model has hidden qubits and

$$ \exists j: \left[H_{\theta}, \frac{\partial H_{\theta}}{\partial\theta_{j}}\right] \neq 0. $$

Instead, the use of hidden qubits is either avoided, i.e., only fully visible settings are considered (Kieferová and Wiebe 2017; Kappen 2020; Wiebe and Wossnig 2019), or the gradients are computed with respect to an upper bound of the loss (Amin et al. 2018; Anschütz and Cao 2019; Kieferová and Wiebe 2017), which is based on the Golden-Thompson inequality (Thompson 1965; Golden 1965). It should be noted that training with an upper bound renders the use of transverse Hamiltonian components, i.e., off-diagonal Pauli terms, difficult and imposes restrictions on the compatible models.

Furthermore, we would like to point out that, in general, it is not trivial to evaluate a QBM Hamiltonian with a classical computer, i.e., using exact simulation with quantum Monte Carlo methods (Troyer and Wiese 2005), because the underlying Hamiltonian can suffer from the so-called sign-problem (Hangleiter et al. 2020; Okunishi and Harada 2014; Li et al. 2016; Alet et al. 2016; Li et al. 2015). As already discussed in Ortiz et al. (2001), evaluations on quantum computers can avoid this problem.

Our QBM implementation works for generic Hamiltonians H_𝜃 with real coefficients 𝜃 and arbitrary Pauli terms h_i, and furthermore, is compatible with near-term, gate-based quantum computers. The method exploits Variational Quantum Imaginary Time Evolution (McArdle et al. 2019; Yuan et al. 2019) (VarQITE), which is based on McLachlan’s variational principle (McLachlan 1964), to not only prepare approximate Gibbs states, $\rho _{\omega }^{\text {Gibbs}}$, but also to train the model with gradients of the actual loss function. During each step of the training, we use VarQITE to generate an approximation to the Gibbs state underlying H_𝜃 and to enable automatic differentiation for computing the gradient of the loss function which is needed to update 𝜃. This variational QBM algorithm (VarQBM) is inherently normalized which implies that the training does not require the explicit evaluation of the partition function.

We focus on training quantum Gibbs states whose sampling behavior reflects a classical probability distribution. However, the scheme could be easily adapted to an approximate quantum state preparation scheme by using a loss function which is based on the quantum relative entropy (Kieferová and Wiebe 2017; Kappen 2020; Wiebe and Wossnig 2019). Hereby, the approximation to ρ^Gibbs is fitted to a given target state ρ^data. Notably, this approach is not necessarily suitable for learning classical distributions. More precisely, we do not need to train a quantum state that captures all features of the density matrix ρ^data but only those which determine the sampling probability. It follows that fitting the full density matrix may impede the training.

The remainder of this paper is structured as follows. Firstly, we review classical BMs and VarQITE in Section 2. Then, we outline VarQBM in Section 3. Next, we illustrate the feasibility of the Gibbs state preparation and present QBM applications in Section 4. Finally, a conclusion and an outlook are given in Section 5.

2 Preliminaries

This section introduces the concepts which form the basis of our VarQBM algorithm. First, classical BMs are presented in Section 2.1. Then, we discuss VarQITE, the algorithm that VarQBM uses for approximate Gibbs state preparation, in Section 2.2.

2.1 Boltzmann machines

Here, we will briefly review the original concept of classical BMs (Ackley et al. 1985). A BM represents a network model that stores the learned knowledge in connection weights between network nodes. More explicitly, the connection weights are trained to generate outcomes according to a probability distribution of interest, e.g., to generate samples which are similar to given training samples or to output correct labels depending on input data samples.

Typically, this type of neural network is related to an Ising-type model (Ising 1925; Peierls 1936) such that each node i corresponds to a binary variable z_i ∈{− 1,+ 1}. Now, the set of nodes may be split into visible and hidden nodes representing observed and latent variables, respectively. Furthermore, a certain configuration $z=\left \{v, h\right \}$ of all nodes—visible and hidden—determines an energy, which is given as

$$ E_{z = \left\{v, h\right\}}= -\sum\limits_{i}\tilde{\theta}_{i}z_{i} - \sum\limits_{i, j}\theta_{ij}z_{i}z_{j}, $$

with $\tilde {\theta }_{i}, \theta _{ij}\in \mathbb {R}$ denoting the weights and z_i representing the value taken by node i. It should be noted that the parameters 𝜃_ij correspond to the weights of connections between different nodes. More explicitly, if two nodes are connected in the network, then a respective term appears in the energy function. The probability to observe a configuration v of the visible nodes is defined as

$$ p^{BM}_{v} = \frac{\sum\limits_{n}e^{-E_{z = \left\{v, h\right\}/\text{k}_{\text{B}}\text{T}}}}{Z}, $$

(1)

where k_B is the Boltzmann constant, T the system temperature and Z the canonical partition function

$$ Z=\sum\limits_{z= \left\{v, h\right\}}e^{-E_{z}/\left( \text{k}_{\text{B}}\text{T}\right)}. $$

We would like to point out that BMs adopt a concept from statistical mechanics. Suppose a closed system that is in thermal equilibrium with a coupled heat bath at constant temperature. The possible configuration space is determined by the canonical ensemble, i.e., the probability for observing a configuration is given by the Gibbs distribution (Boltzmann 1877; Gibbs 1902) which corresponds to Eq. 1.

Now, the goal of a BM is to fit the target probability distribution p^data with p^BM. Typically, this training objective is achieved by optimizing the cross-entropy

$$ L = -\sum\limits_{v}p_{v}^{\text{data}}\log{p_{v}^{BM}}. $$

(2)

In theory, fully connected BMs have interesting representation capabilities (Ackley et al. 1985; Younes 1996; Fischer and Igel 2012a), i.e., they are universal approximators (Roux and Bengio 2010). However, in practice they are difficult to train as the optimization easily gets expensive. Thus, it has become common practice to restrict the connectivity between nodes which relates to Restricted Boltzmann machines (RBMs) (Montúfar 2018). Furthermore, several approximation techniques, such as contrastive divergence (Hinton 2002), have been developed to facilitate BM training. However, these approximation techniques typically still face issues such as long computation time due to a large amount of required Markov chain steps or poor compatibility with multimodal probability distributions (Murphy 2012). For further details, we refer the interested reader to Hinton (2012), Fischer and Igel (2012b), and Fischer (2015).

2.2 Variational quantum imaginary time evolution

Imaginary time evolution (ITE) (Magnus 1954) is an approach that is well known for (classical) ground-state computation (McArdle et al. 2019; Gupta et al. 2002; Auer et al. 2001) but—as suggested by some prior literature on ITE (Matsui 1998; Khalkhali and Marcolli 2008; Motta et al. 2020; McArdle et al. 2019; Yuan et al. 2019)—may also be used for Gibbs state preparation.

Suppose a starting state |ψ₀〉 and a time-independent Hamiltonian $H={\sum }_{i=0}^{p-1}\theta _{i}h_{i}$ with real coefficients 𝜃_i and Pauli terms h_i. Then, the normalized ITE propagates |ψ₀〉 with respect to H for time τ according to

$$ |\psi_{\tau}\rangle= C\left( \tau\right)e^{-H\tau}|\psi_{0}\rangle, $$

where $C\left (\tau \right ) = {1}/{\sqrt {\text {Tr}\left [e^{-2H\tau }|\psi _{0} \rangle \langle \psi _{0}|\right ]}}$ is a normalization. The differential equation that describes this evolution is the Wick-rotated Schrödinger equation:

$$ \frac{d|\psi_{\tau}\rangle}{d \tau} =-\left( H - E_{\tau} \right) |\psi_{\tau}\rangle, $$

(3)

where E_τ = 〈ψ_τ|H|ψ_τ〉 originates from the normalization of |ψ_τ〉. The terms in e^−Hτ, corresponding to small eigenvalues of H, decay slower than the ones corresponding to large eigenvalues. Due to the continuous normalization, the smallest eigenvalue dominates for $\tau \rightarrow \infty$. Thus, |ψ_τ〉 converges to the ground state of H given that there is some overlap between the ground and starting states. Furthermore, if ITE is only evolved to a finite time, $\tau = {1}/{2\left (\text {k}_{\text {B}}\text {T}\right )}$, then it enables the preparation of Gibbs states; see Section 3.1.

As introduced in McArdle et al. (2019) and Yuan et al. (2019), an approximate ITE can be implemented on a gate-based quantum computer by using McLachlan’s variational principle (McLachlan 1964). The basic idea of the method is to introduce a parameterized trial state |ψ_ω〉 and to project the temporal evolution of |ψ_τ〉 to the parameters, i.e., ω := ω(τ). We refer to this algorithm as VarQITE and, now, discuss it in more detail.

First, we define an input state |ψ_in〉 and a quantum circuit $V\left (\omega \right ) = U_{q}\left (\omega _{q}\right ){\cdots } U_{1}\left (\omega _{1}\right )$ with parameters $\omega \in \mathbb {R}^{q}$ to generate the parameterized trial state

$$ |\psi_{\omega}\rangle := V\left( \omega\right)|\psi_{\text{in}}\rangle. $$

Now, McLachlan’s variational principle

$$ \delta \left\lVert \left( d/d\tau + H - E_{\tau}\right) |\psi_{\omega}\rangle \right\rVert = 0 $$

(4)

determines the time propagation of the parameters ω(τ). This principle aims to minimize the distance between the right-hand side of Eq. 3 and the change d|ψ_ω〉/dτ. Equation 4 leads to a system of linear equations for $\dot {\omega }= d \omega / d\tau$, i.e.,

$$ \begin{array}{@{}rcl@{}} A\dot{\omega} = C \end{array} $$

(5)

with

$$ \begin{array}{ll} \mathit{A}_{pq}\left( \tau\right) &\!= \text{Re}\left( \text{Tr}\left[\frac{\partial V^{\dagger}\left( \omega\left( \tau\right)\right)}{\partial\omega\left( \tau\right)_{p}}\frac{\partial V\left( \omega\left( \tau\right)\right)}{\partial\omega\left( \tau\right)_{q}}\rho_{\text{in}}\right]\right) \\ C_{p}\left( \tau\right) & = -\sum\limits_{i}\theta_{i}\text{Re}\left( \text{Tr}\left[\frac{\partial V^{\dagger}\left( \omega\left( \tau\right)\right)}{\partial\omega\left( \tau\right)_{p}}h_{i}\mathit{V}\left( \omega\left( \tau\right)\right)\rho_{\text{in}}\right]\right), \end{array} $$

(6)

where $\text {Re}\left (\cdot \right )$ denotes the real part and ρ_in = |ψ_in〉〈ψ_in|. The vector C describes the derivative of the system energy 〈ψ_ω|H|ψ_ω〉 and A is proportional to the classical Fisher information matrix, a metric tensor that reflects the system’s information geometry (Koczor and Benjamin 2019). To evaluate A and C, we compute expectation values with respect to quantum circuits of a particular form which is illustrated and discussed in Appendix A.

This evaluation is compatible with arbitrary parameterized unitaries in $V\left (\omega \right )$ because all unitaries can be written as $U\left (\omega \right ) = e^{iM\left (\omega \right )}$, where $M\left (\omega \right )$ denotes a parameterized Hermitian matrix. Furthermore, Hermitian matrices can be decomposed into weighted sums of Pauli terms, i.e., $M\left (\omega \right ) = {\sum }_{p}m_{p}\left (\omega \right )h_{p}$ with $m_{p}\left (\omega \right )\in \mathbb {R}$ and $h_{p}=\overset {n-1}{\underset {j=0}{\bigotimes }}\sigma _{j, p}$ for σ_j,p ∈{I,X,Y,Z} (Nielsen and Chuang 2010) acting on the j^th qubit. Thus, the gradients of $U_{k}\left (\omega _{k}\right )$ are given by

$$ \frac{\partial U_{k}\left( \omega_{k}\right)}{\partial\omega_{k}} = \sum\limits_{p}i \frac{\partial m_{k,p}\left( \omega_{k}\right)}{\partial\omega_{k}}U_{k}\left( \omega_{k}\right)h_{k_{p}}. $$

This decomposition allows us to compute A and C with the techniques described in Somma et al. (2002), McArdle et al. (2019), and Yuan et al. (2019). Furthermore, it should be noted that Eq. 5 is often ill-conditioned and may, thus, require the use of regularized regression methods; see Section 4.1.

Now, we can use, e.g., an explicit Euler method to evolve the parameters as

$$ \omega\left( \tau\right) = \omega\left( 0\right) + \sum\limits_{j = 1}^{\tau/\delta\tau} \dot{\omega}\left( {~}_{j} {\delta\tau}\right){\delta\tau}. $$

3 Quantum Boltzmann Machine algorithm

A QBM is defined by a parameterized Hamiltonian $H_{\theta }={\sum }_{i=0}^{p-1}\theta _{i}h_{i}$ where $\theta \in \mathbb {R}^{p}$ and $h_{i}=\bigotimes _{j=0}^{n-1}\sigma _{j, i}$ for σ_j,i ∈{I,X,Y,Z} acting on the j^th qubit. Equivalently to classical BMs, QBMs are typically represented by an Ising model (Ising 1925), i.e., a 2-local system (Bravyi et al. 2006) with nearest-neighbor coupling that is defined with regard to a particular grid. In principle, however, any Hamiltonian compatible with Boltzmann distributions could be used.

In contrast to BMs, the network nodes, given by the Pauli terms σ_j,i, do not represent the visible and hidden units. These are defined with respect to certain sub-sets of qubits. More explicitly, those qubits which determine the output of the QBM are the visible qubits, whereas the others correspond to the hidden qubits. Now, the probability to measure a configuration v of the visible qubits is defined with respect to a projective measurement Λ_v = |v〉〈v|⊗ I on the quantum Gibbs state

$$ \rho^{\text{Gibbs}} = \frac{e^{-H_{\theta}/\left( \text{k}_{\text{B}}\text{T}\right)}}{Z} $$

with $Z=\text {Tr}\left [e^{-H_{\theta }/\left (\text {k}_{\text {B}}\text {T}\right )}\right ]$, i.e., the probability to measure |v〉 is given by

$$ p_{v}^{\text{QBM}} = \text{Tr}\left[{\Lambda}_{v}\rho^{\text{Gibbs}} \right]. $$

For the remainder of this work, we assume that Λ_v refers to projective measurements with respect to the computational basis of the visible qubits. Thus, the configuration v is determined by v_i ∈{0,1}. It should be noted that this formulation does not require the evaluation of the configuration of the hidden qubits.

Our goal is to train the Hamiltonian parameters 𝜃 such that the sampling probabilities of the corresponding ρ^Gibbs reflect the probability distribution underlying given classical training data. For this purpose, the same loss function as described in the classical case (see Eq. 2) can be used

$$ L = -\sum\limits_{v}p_{v}^{\text{data}}\log{p_{v}^{\text{QBM}}}, $$

(7)

where $p_{v}^{\text {data}}$ denotes the occurrence probability of item v in the training dataset.

To enable efficient training, we want to evaluate the derivative of L with respect to the Hamiltonian parameters. Unlike existing QBM implementations, VarQBM facilitates the use of analytic gradients of the loss function given in Eq. 7 for generic QBMs. The presented algorithm involves the following steps. First, we use VarQITE to approximate the Gibbs state; see Section 3.1 for further details. Then, we compute the gradient of L to update the parameters 𝜃 with automatic differentiation, as is discussed in Section 3.2. The parameters are trained with a classical optimization routine where one training step consists of the Gibbs state preparation with respect to the current parameter values and a consecutive parameter update, as illustrated in Fig. 1.

Fig. 1

The VarQBM training includes the following steps. First, we need to fix the Pauli terms for H_𝜃 and choose initial parameters 𝜃. Then, VarQITE is used to generate $\rho _{\omega }^{\text {Gibbs}}$ and compute ∂ω/∂𝜃. The quantum state and the derivative are needed to evaluate $p_{v}^{\text {QBM}}$ and $\partial p_{v}^{\text {QBM}}/\partial \theta$. Now, we can find ∂L/∂𝜃 to update the Hamiltonian parameters with a classical optimizer

In the remainder of this section, we discuss Gibbs state preparation with VarQITE in Section 3.1 and VarQBM in more detail in Section 3.2.

3.1 Gibbs state preparation with VarQITE

The Gibbs state ρ^Gibbs describes the probability density operator of the configuration space of a system in thermal equilibrium with a heat bath at constant temperature T (Gibbs 2010). Originally, Gibbs states were studied in the context of statistical mechanics but, as shown in (Pauli 1927), the density operator also facilitates the description of quantum statistics.

Gibbs state preparation can be approached from different angles. Hereby, different techniques not only have different strengths but also different drawbacks. Some schemes (Temme et al. 2011; Yung and Aspuru-Guzik 2012; Poulin and Wocjan 2009) use quantum phase estimation (Abrams and Lloyd 1999) as a subroutine, which is likely to require error-corrected quantum computers. Other methods enable the evaluation of quantum thermal averages (Motta et al. 2020; Brandão and Kastoryano 2019; Kastoryano and Brandão 2016) for states with finite correlations. However, since QBM-related states may exhibit long-range correlations, these methods are not the first choice for the respective preparation. A thermalization-based approach is presented in Anschütz and Cao (2019), where the aim is to prepare a quantum Gibbs state by coupling the state register to a heat bath given in the form of an ancillary quantum register. Correct preparation requires a thorough study of suitable ancillary registers for a generic Hamiltonian as the most useful ancilla system is not a priori known. Furthermore, variational Gibbs state preparation methods which are based on the fact that Gibbs states minimize the free energy of a system at constant temperature have been presented in Wu and Hsieh (2019) and Chowdhury et al. (2020). The goal is to fit a parameterized quantum state that minimizes the free energy. Here, the difficulty is to estimate the von Neumann entropy in every step of the state preparation. An algorithm which aims to enable efficient von Neuman entropy estimation is given in Chowdhury et al. (2020). However, it requires the application of quantum amplitude estimation (Brassard et al. 2002), as well as matrix exponentiation of the input state, and thus is not well suited for near-term quantum computing applications.

In contrast to these Gibbs state preparation schemes, VarQITE is compatible with near-term quantum computers, and neither is limited to states with finite correlations nor requires ambiguous ancillary systems. In the following, we discuss how VarQITE can be utilized to generate an approximation of the Gibbs state ρ^Gibbs for a generic n −qubit Hamiltonian $H_{\theta }={\sum }_{i=0}^{p-1}\theta _{i}h_{i}$ with $\theta \in \mathbb {R}^{p}$ and $h_{i}=\bigotimes _{j=0}^{n-1}\sigma _{j, i}$ for σ_j,i ∈{I,X,Y,Z} acting on the $j_{i}^{\text {th}}$ qubit.

First, we need to choose a suitable variational quantum circuit $V\left (\omega \right )$, $\omega \in \mathbb {R}^{q}$, and set of initial parameters ω(0) such that the initial state is

$$ |\psi_{0}\rangle = V\left( \omega\left( 0\right)\right)|0\rangle^{\otimes 2n} = |\phi^{+}\rangle^{\otimes n} $$

(8)

where $|\phi ^{+}\rangle = \frac {1}{\sqrt {2}}\left (|00\rangle +|11\rangle \right )$ represents a Bell state. We define two n-qubit sub-systems a and b such that the first and second qubits of each |ϕ⁺〉 are in a and b, respectively. Accordingly, an effective 2n-qubit Hamiltonian $H_{\text {eff}} = H_{\theta }^{a}$ + I^b, where H_𝜃 and I act on sub-system a and b, is considered. It should be noted that tracing out sub-system b from |ψ₀〉 results in an n-dimensional maximally mixed state:

$$ \text{Tr}_{b}\left[|\phi^{+}\rangle^{\otimes n} \right] = \frac{1}{2^{n}}I. $$

Now, the Gibbs state approximation $\rho _{\omega }^{\text {Gibbs}}$ can be generated by propagating the trial state with VarQITE with respect to H_eff for $\tau = 1/2\left (\text {k}_{\text {B}}\text {T}\right )$. The resulting state

$$ |\psi_{\omega}\rangle = V\left( \omega\left( \tau\right)\right)|0\rangle^{\otimes 2n} $$

gives an approximation for the Gibbs state of interest

$$ \rho_{\omega}^{\text{Gibbs}} = \text{Tr}_{b}\left[|\psi\left( {\omega\left( \tau\right)}\right)\rangle\langle \psi\left( {\omega\left( \tau\right)}\right)| \right] \approx \frac{e^{-H_{\theta}/\left( \text{k}_{\text{B}}\text{T}\right)}}{Z} $$

by tracing out the ancillary system b. We would like to point out that the VarQITE propagation relates ω to 𝜃 via the energy derivative C given in Eq. 6.

Notably, this is an approximate state preparation scheme that relies on the representation capabilities of |ψ_ω〉. However, since the algorithm is employed in the context of machine learning, we do not necessarily require perfect state preparation. The noise may even improve the training, as discussed, e.g., in (Noh et al. 2017).

McLachlan’s variational principle is not only the key component for Gibbs state preparation. It also enables the QBM training with gradients of the actual loss function for generic Pauli terms in H_𝜃, even if some of the qubits are hidden. Further details are given in Section 3.2.

3.2 Variational QBM

In the following, VarQBM and the respective utilization of McLachlan’s variational principle and VarQITE are discussed. We consider training data that takes at most 2ⁿ different values and is distributed according to a discrete probability distribution p^data. The aim of a QBM is to train the parameters of H_𝜃 such that the sampling probability distribution of the corresponding $\rho _{\omega }^{\text {Gibbs}}= e^{-H_{\theta }/\left (\text {k}_{\text {B}}\text {T}\right )} / Z$ for |v〉,v ∈ 0,…,2ⁿ − 1 with

$$ p_{v}^{\text{QBM}} = \text{Tr}\left[{\Lambda}_{v}\rho_{\omega}^{\text{Gibbs}}\right], $$

approximates p^data. The QBM model is trained to represent p^data by minimizing the loss, given in Eq. 7, with respect to the Hamiltonian parameters 𝜃, i.e.,

$$ \underset{\theta}{\min} L = \underset{\theta}{\min} \left( -\sum\limits_{v}p_{v}^{\text{data}}\log{p_{v}^{\text{QBM}}}\right). $$

Now, VarQBM facilitates gradient-based optimization with the derivative of the actual loss function

$$ \begin{array}{ll} \frac{\partial L}{\partial\theta_{i}} = \frac{\partial\left( - \sum\limits_{v}p_{v}^{\text{data}}\log{p_{v}^{\text{QBM}}}\right)}{\partial\theta_{i}} = - \sum\limits_{v}p_{v}^{\text{data}}\frac{\partial p_{v}^{\text{QBM}}/\partial\theta_{i}}{p_{v}^{\text{QBM}}} \end{array} $$

(9)

by using the chain rule, i.e., automatic differentiation. More precisely, the gradient of L can be computed by using the chain rule for

$$ \begin{array}{@{}rcl@{}} \begin{array}{ll} \frac{\partial p_{v}^{\text{QBM}}}{\partial\theta_{i}} = \frac{\partial p_{v}^{\text{QBM}}}{\partial\omega\left( \tau\right)}\frac{\partial\omega\left( \tau\right)}{\partial\theta_{i}} =\sum\limits_{k=0}^{q-1} \frac{\partial p_{v}^{\text{QBM}} }{\partial\omega_{k}\left( \tau\right)}\frac{\partial\omega_{k}\left( \tau\right)}{\partial\theta_{i}}. \end{array} \end{array} $$

(10)

Firstly, $\partial p_{v}^{\text {QBM}}/\partial \omega _{k}\left (\tau \right )=\partial \text {Tr}\left [{\Lambda }_{v}\rho _{\omega }^{\text {Gibbs}}\right ] / \partial \omega _{k}\left (\tau \right )$ can be evaluated with quantum gradient methods discussed in Farhi and Neven (2018), Mitarai et al. (2018), Dallaire-Demers and Killoran (2018), Schuld et al. (2019), and Zoufal et al. (2019) because the term has the following form:

$$ \partial \text{Tr}\left[\hat{O}|\phi\left( \alpha\right)\rangle\langle\phi\left( \alpha\right)|\right] / \partial\alpha. $$

Secondly, ${\partial \omega _{k}\left (\tau \right )}/{\partial \theta _{i}}$ is evaluated by computing the derivative of Eq. 5 with respect to the Hamiltonian parameters:

$$ \frac{\partial A\dot{\omega}\left( \tau\right)}{\partial{\theta_{i}}} = \frac{\partial C}{\partial{\theta_{i}}}. $$

This gives the following system of linear equations:

$$ \begin{array}{@{}rcl@{}} \label {eq:sle_derivative} A\left( \frac{\partial\dot{\omega}\left( \tau\right)}{\partial{\theta_{i}}}\right) = \frac{\partial C}{\partial{\theta_{i}}} - \left( \frac{\partial A}{\partial{\theta_{i}}}\right)\dot{\omega}\left( \tau\right). \end{array} $$

(11)

Equation 11 is just as Eq. 5 prone to being ill- conditioned. Thus, the use of regularization schemes may be required.

Now, solving for $\partial \dot {\omega }\left (\tau \right )/\partial {\theta _{i}}$ in every time step of the Gibbs state preparation enables the use of, e.g., an explicit Euler method to get

$$ \begin{array}{ll} \frac{\partial \omega_{k}\left( \tau \right)}{\partial_{\theta_{i}}} &= \frac{\partial\omega_{k}\left( \tau-\delta\tau\right)}{\partial_{\theta_{i}}}+\frac{\partial\dot{\omega}_{k}\left( \tau-\delta\tau\right)}{\partial_{\theta_{i}}}\delta\tau \\ &=\frac{\partial \omega_{k}\left( 0\right)}{\partial_{\theta_{i}}} + \sum\limits_{j=1}^{\tau/\delta\tau}\frac{\partial \dot{\omega}_{k}\left( j\delta\tau\right)}{\partial_{\theta_{i}}}\delta\tau. \end{array} $$

(12)

We discuss the structure of the quantum circuits used to evaluate $\partial _{\theta _{i}}A$ and $\partial _{\theta _{i}}C$, in Appendix A.

In principle, the gradient of the loss function could also be approximated with a finite difference method. If the number of Hamiltonian parameters is smaller than the number of trial state parameters, this requires less evaluation circuits. However, given a trial state that has less parameters than the respective Hamiltonian, the automatic differentiation scheme presented in this section is favorable in terms of the number of evaluation circuits. A more detailed discussion on this topic can be found in Appendix B.

An outline of the Gibbs state preparation and evaluation of ${\partial \omega _{k}\left (\tau \right )}/{\partial _{\theta _{i}}}$ with VarQITE is presented in Algorithm 1.

Now, using a classical optimizer, such as truncated Newton (Dembo and Steihaug 1983) or Adam (Kingma and Ba 2015), allows the parameters 𝜃 to be updated according to ∂L/∂𝜃 from Eq. 9. The VarQBM training is illustrated in Fig. 1.

4 Results

In this section, the Gibbs state preparation with VarQITE is demonstrated using numerical simulation as well as the quantum hardware provided by IBM Quantum (IBM Q Experience 2020). Furthermore, we present numerically simulated QBM training results for a generative and a discriminative learning task. First, aspects which are relevant for the practical implementation are discussed in Section 4.1. Next, experiments of quantum Gibbs state preparation with VarQITE are shown in Section 4.2. Then, we illustrate the training of a QBM with the goal to generate a state which exhibits the sampling behavior of a Bell state (see Section 4.3), and to classify fraudulent credit card transactions (Section 4.4).

4.1 Methods

To begin with, we discuss the choice of a suitable parameterized trial state consisting of $V\left (\omega \right )$ and |ψ_in〉. Most importantly, the initial state |ψ_in〉 must not be an eigenstate of $V\left (\omega \right )$ as this would imply that the circuit could only act trivially onto the state and |ψ_in〉 as well as $V\left (\omega \left (0\right )\right )$ must be chosen such that the initial trial state suffices the form given in Eq. 8. As there is currently little known about what types of Hamiltonians map particularly well to what types of trial states, this work employs generic hardware-efficient Ansätze. These are chosen such that the number of hidden nodes and the entanglement between the qubits are not too large as these might lead to barren plateaus (Marrero et al. 2020). Furthermore, the state needs to be able to represent a sufficiently accurate approximation of the target state. If we have to represent, e.g., a non-symmetric Hamiltonian, the chosen trial state needs to be able to generate non-symmetric states. Numerical experiments indicate that states which exhibit complex correlations require Ansätze with higher depth, i.e., more parameters. Moreover, $V\left (\omega \right )$ should not exhibit too much symmetry as this may lead to a singular A which in turn causes ill-conditioning of Eq. 5. Assume, e.g., that all entries of C are zero and, thus, that Eq. 5 is homogeneous. If A is singular, infinitely many solutions exist and it is difficult for the algorithm to estimate which path to choose. If A is non-singular, the solution is $\dot {\omega } = 0$ and the evolution stops although we might have only reached a local extreme point. Another possibility to cope with ill-conditioned systems of linear equations are least-squares methods in combination with regularization schemes. We test Tikhonov regularization (Tikhonov et al. 1995) and Lasso regularization (Tibshirani 2011) with an automatic parameter evaluation based on L-curve fitting (Hansen 2000), as well as an 𝜖-perturbation of the diagonal, i.e., $A\rightarrow A+\epsilon I$. It turns out that all regularization methods perform similarly well.

The results discussed in this section employ Tikhonov regularization. Furthermore, we use trial states which are parameterized by Pauli-rotation gates. Therefore, the gradients of the QBM probabilities with respect to the trial state parameters

$$ \frac{ \partial p_{v}^{\text{QBM}}}{\partial\omega_{k}\left( \tau\right)}=\frac{\partial \text{Tr}\left[{\Lambda}_{v}\rho_{\omega}^{\text{Gibbs}}\right]} {\partial\omega_{k}\left( \tau\right)} $$

can be computed using a π/2 −shift method which is, e.g., described in Zoufal et al. (2019). All experiments employ an additional qubit |0〉_add and parameter ω_add to circumvent a potential phase mismatch between the target |ψ_τ〉 and the trained state $| \psi \left (\omega \left (\tau \right )\right )\rangle$ (McArdle et al. 2019; Yuan et al. 2019; Koczor and Benjamin 2019) by applying

$$ R_{Z}\left( \omega_{\text{add}}\right)|0\rangle_{\text{add}}. $$

Notably, the additional parameter increases the dimension of A and C by 1. The effective temperature, which in principle acts as a scaling factor on the Hamiltonian parameters, is set to $\left (\text {k}_{\text {B}}\text {T}\right ) = 1$ in all experiments.

4.2 Gibbs state preparation

We verify VarQITE is able to generate suitable approximations to Gibbs states by illustrating the convergence of the fidelity between the prepared and the target state for several example Hamiltonians. This experiment is run for a one- and a two-qubit Hamiltonian using simulation as well as an actual quantum computer and for Hamiltonians with more qubits using only a simulated backend. The results for the latter are given in Appendix C.

The Gibbs state preparation for the following Hamiltonians

$$ \begin{array}{@{}rcl@{}} H_{1} &=& 1.0 Z, \\ H_{2} &=& 1.0 ZZ - 0.2 ZI - 0.2 IZ + 0.3 XI + 0.3 IX. \end{array} $$

corresponding to

$$ \begin{array}{@{}rcl@{}} \rho^{\text{Gibbs}}_{1} &=& \left( \begin{array}{cc} 0.12 & 0. \\ 0. & 0.88\\ \end{array}\right) , \\ \rho^{\text{Gibbs}}_{2} &=& \left( \begin{array}{cccc} 0.10 & -0.06 & -0.06 & 0.01 \\ -0.06 & 0.43 & 0.02 & -0.05 \\ -0.06 & 0.02 & 0.43 & -0.05 \\ 0.01 & -0.05 & -0.05 & 0.05\\ \end{array}\right) \end{array} $$

is executed for 10 time steps on an ideal simulator and the ibmq_johannesburg 20-qubit backend. The results are computed using the parameterized quantum circuit shown in Fig. 2. Notably, readout error-mitigation (Aleksandrowicz et al. 2019; Dewes et al. 2012; Stamatopoulos et al. 2020) is used to obtain the final results run on real quantum hardware. Figure 3 depicts the results considering the fidelity between the trained and the target Gibbs states for each time step. It should be noted that the fidelity for the quantum backend evaluations employs state tomography. The plots illustrate that the method approximates the states, we are interested in reasonably well and that also the real quantum hardware achieves fidelity values over 0.99 and 0.96, respectively.

Fig. 2

The depicted circuits illustrate the initial trial state for the Gibbs state preparation of a $\rho ^{\text {Gibbs}}_{1}$ and b $\rho ^{\text {Gibbs}}_{2}$ using VarQITE

Fig. 3

Fidelity between trained and target Gibbs states with VarQITE for a $\rho ^{\text {Gibbs}}_{1}$ and b $\rho ^{\text {Gibbs}}_{2}$ trained with an ideal simulator and real quantum hardware, i.e., the ibmq_johannesburg 20-qubit backend. Each simulation uses 10 time steps

4.3 Generative learning

Now, the results from an illustrative example of a generative QBM model are presented. More explicitly, the QBM is trained to mimic the sampling statistics of a Bell state $(|00\rangle + |11\rangle )/\sqrt {2}$, which is a state that exhibits non-local correlations. Numerical simulations show that the distribution can be trained with a fully visible QBM which is based on the following Hamiltonian:

$$ H_{\theta}= \theta_{0} ZZ + \theta_{1} IZ + \theta_{2} ZI. $$

We draw the initial values of the Hamiltonian parameters 𝜃 from a uniform distribution on $\left [-1, 1\right ]$. The optimization runs on an ideal simulation of a quantum computer using AMSGrad (Reddi et al. 2018) with initial learning rate 0.1, maximum number of iterations 200, first momentum 0.7, and second momentum 0.99 as optimization routine. The Gibbs state preparation uses the initial trial state shown in Fig. 4 and 10 steps per state preparation.

Fig. 4

We train a QBM to mimic the sampling behavior of a Bell state. The underlying Gibbs state preparation with VarQITE uses the illustrated parameterized quantum circuit to prepare the initial trial state. The first two qubits represent the target system and the last two qubits are ancillas needed to generate the maximally mixed state as starting state for the evolution

The training is run 10 times using different randomly drawn initial parameters. The averaged values of the loss function as well as the distance between the target distribution $p^{\text {data}}=\left [0.5, 0., 0., 0.5\right ]$ and the trained distribution p^QBM with respect to the ℓ₁ norm are illustrated over 50 optimization iterations in Fig. 5. The plot shows that loss and distance converge toward the same values for all sets of initial parameters. Likewise, the trained parameters 𝜃 converge to similar values. Furthermore, Fig. 6 illustrates the target probability distribution and for the best and worst of the trained distributions. The plot reveals that the model is able to train the respective distribution very well.

Fig. 5

The figure illustrates the training progress of a fully visible QBM model which aims to represent the measurement distribution of a Bell state. The green function corresponds to the loss and the pink function represents the distance between the trained and target distributions with respect to the ℓ₁ norm at each step of the iteration. Both measures are computed for 10 different random seeds. The points represent the mean and the error bars the standard deviation of the results

Fig. 6

The figure illustrates the sampling probability of the Bell state (blue), as well as the best (pink) and worst (purple) probability distributions achieved from 10 different random seeds

We verify the scalability of this method by training equivalent QBM models for 3 and 4 qubit Greenberg-Horne-Zeilinger (GHZ) states (Greenberger et al. 1989). The respective results are presented in Appendix D.

4.4 Discriminative learning

QBMs are not only applicable for generative but also for discriminative learning. We discuss the application to a classification task: the identification of fraudulent credit card transactions.

To enable discriminative learning with QBMs, we use the input data points x as bias for the Hamiltonian weights. More explicitly, the parameters of the Hamiltonian

$$ H_{\theta}\left( x\right)=\sum\limits_{i}f_{i}\left( \theta, x\right)h_{i} $$

(13)

are given by a function $f_{i}\left (\theta , x\right )$ which maps 𝜃 and x to a scalar in $\mathbb {R}$. Now, the respective loss function reads:

$$ \begin{array}{ll} L = -\sum\limits_{x}p_{x}^{\text{data}}\sum\limits_{v}p_{v|x}^{\text{data}}\log{p_{v|x}^{\text{QBM}}} \end{array} $$

with

$$ p_{v|x}^{\text{QBM}} = \text{Tr}\left[{\Lambda}_{v}\rho\left( x\right)_{\omega}^{\text{Gibbs}}\right], $$

where $\rho \left (x\right )_{\omega }^{\text {Gibbs}}$ denotes the approximate Gibbs state corresponding to $H_{\theta }\left (x\right )$. The model encodes the class labels in the measured output configuration of the visible qubits v of $\rho \left (x\right )_{\omega }^{\text {Gibbs}}$. Now, the aim of the training is to find Hamiltonian parameters 𝜃 such that, given a data sample x, the probability of sampling the correct output label from $\rho \left (x\right )_{\omega }^{\text {Gibbs}}$ is maximized.

The training is based on 500 artificially created credit card transactions (Altman 2019) with about 15% fraudulent instances. To avoid redundant state preparation, the training is run for all unique item instances in the dataset and the results are averaged according to the item’s occurrence counts. The dataset includes the following features: location (ZIP code), time, amount, and Merchant Category Code (MCC) of the transactions. To facilitate the training, the features of the given dataset are discretized and normalized as follows. Using k-means clustering, each of the first three features are independently discretized to 3 reasonable bins. Furthermore, we consider MCCs < 10,000 and group them into 10 different categories. The discretization is discussed in more detail in Table 1. Furthermore, for each feature, we map the values x to $x^{\prime } = \frac {x-\mu }{\sigma }$ with μ denoting the mean and σ denoting the standard deviation.

Table 1 The table discusses the binning of a transaction fraud dataset which is used to train a discriminative QBM model

The complexity of this model demands a Hamiltonian that has sufficient representation capabilities. We investigate a diagonal Hamiltonian:

$$ \begin{array}{ll} H^{(0)}_{\theta}\left( x\right) =& f^{(2)}_{0}\left( \theta, x\right) ZZ + f^{(2)}_{1}\left( \theta, x\right)ZI \\ &+f^{(2)}_{2}\left( \theta, x\right) IZ, \end{array} $$

(14)

a Hamiltonian with off-diagonal terms whose parameters are fixed to 0.1

$$ \begin{array}{ll} H^{(1)}_{\theta}\left( x\right) =& f^{(1)}_{0}\left( \theta, x\right) ZZ + f_{1}^{(1)}\left( \theta, x\right)ZI \\ &+f^{(1)}_{2}\left( \theta, x\right) IZ + 0.1 XI + 0.1 IX, \end{array} $$

(15)

and a Hamiltonian with parameterized off-diagonals

$$ \begin{array}{ll} H^{(2)}_{\theta}\left( x\right) =& f^{(2)}_{0}\left( \theta, x\right) ZZ + f^{(2)}_{1}\left( \theta, x\right)ZI \\ &+f^{(2)}_{2}\left( \theta, x\right) IZ + f^{(2)}_{3}\left( \theta, x\right)XI \\&+ f^{(2)}_{4}\left( \theta, x\right) IX, \end{array} $$

(16)

where $f^{(j)}_{i}\left (\theta , x\right ) =\overset {\rightarrow }{\theta }^{(j)}_{i} \cdot \overset {\rightarrow }{x}$ corresponds to the dot product of the vector corresponding to the data item $\overset {\rightarrow }{x}$ and a parameter vector $\overset {\rightarrow }{\theta }^{(j)}_{i}$ of equal length. Additionally, the first and second qubits of all systems correspond to hidden and visible qubits, respectively.

The training uses a conjugate gradient optimization routine with a maximum iteration number of 100.

The initial values for the Hamiltonian parameters are drawn from a uniform distribution on $\left [-1, 1\right ]$.

Given a test dataset consisting of 250 instances, with about 10% fraudulent transactions, the Gibbs states, corresponding to the unique items of the test data, are approximated using VarQITE with the trained parameters 𝜃 and the trial state shown in Fig. 7. To predict the labels of the data instances, we sample from the states $\rho _{\omega }^{\text {Gibbs}}$ and choose the label with the highest sampling probability. These results are, then, used to evaluate the accuracy (Acc), recall (Rec), precision (Pre), and F₁ score. It should be noted that we choose a relatively simple quantum circuit to keep the simulation cost small. However, it can be expected that a more complex parameterized quantum circuit might lead to further improvement in the training results.

Fig. 7

Given a transaction instance, the measurement output of the QBM labels it as being either fraudulent or valid. The underlying Gibbs state preparation with VarQITE uses the illustrated parameterized quantum circuit as initial trial state. The first qubit is the visible unit that determines the QBM output, the second qubit represents the hidden unit, and the last two qubits are ancillas needed to generate the maximally mixed state as starting state for the evolution

The resulting values are compared to a set of standard classifiers defined in a scikit-learn (Pedregosa et al. 2011) classifier comparison tutorial (Classifier Comparison 2020) as well as a classifier consisting of a restricted Boltzmann machine and a logistic regression (RBM) (Restricted Boltzmann Machine for classification 2020); see Table 2. The respective classifiers are used with the hyperparameters defined in the tutorials. Neither the linear SVM nor the RBM classifies any test data item as fraudulent, and thus, the classifier sets precision and recall score to 0. The comparison reveals that all variational QBMs perform similarly well to the classical classifiers considering accuracy and are competitive regarding precision. In terms of recall and F₁ score, the VarQBM models without off-diagonal elements and with fixed off-diagonal elements perform well. Notably, the addition of fixed-off diagonal terms in the Hamiltonian seems not to improve the model performance compared to a Hamiltonian without these terms in this case. Finally, the VarQBM with trained off-diagonal elements achieves the best results in terms of recall and F₁ score.

Table 2 This table presents performance measures for scikit-learn standard classifiers, as well as the trained variational QBM models

5 Conclusion and outlook

This work presents the application of McLachlan’s variational principle to facilitate VarQBM, a variational QBM algorithm, that is compatible with generic Hamiltonians and can be trained using analytic gradients of the actual loss function even if some of the qubits are hidden. Suppose a sufficiently powerful variational trial state, the presented scheme, is not only compatible with local but also long-range correlations and for arbitrary system temperatures.

We outline the practical steps for utilizing VarQITE for Gibbs state preparation and verify that it can train states which are reasonably close to the target using simulation as well as real quantum hardware. Moreover, applications to generative learning and classification are discussed and illustrated with further numerical results. The presented model offers a versatile framework which facilitates the representation of complex structures with quantum circuits.

An interesting question for future research to conduct is a thorough analysis of trial state Ansätze which would help to improve our understanding of the model’s representation capabilities and enable more informed choices of Ansatz states. Furthermore, QBMs are not limited to the presented applications. They could also be utilized to train models for data from experiments with quantum systems. This is a problem that has recently gained interest; see e.g., Gentile et al. (2020). Additionally, they might be employed for combinatorial optimization. Classical BMs have been investigated in this context (Spieksma 1995) and developing and analyzing quantum algorithms for combinatorial optimization is an active area of research (Farhi et al. 2014; Barkoutsos et al. 2020). All in all, there are many possible applications which still have to be explored.

Appendices

Appendix: 1: Evaluation of A, C, and their gradients

The elements of the matrix A and the vector C (see Eq. 6) are of the following form:

$$ \text{Re}\left( e^{i\alpha}\text{Tr}\left[U^{\dagger}V\rho_{\text{in}}\right]\right) $$

(17)

with $\text {Re}\left (\cdot \right )$ denoting the real part and ρ_in = |ψ_in〉〈ψ_in|. As discussed in McArdle et al. (2019), Yuan et al. (2019), and Somma et al. (2002), such terms can be computed by sampling the expectation value of an observable Z with respect to the quantum circuit shown in Fig. 8.

Fig. 8

Quantum circuit to evaluate $\text {Re}\left (e^{i\alpha }\text {Tr}\left [U^{\dagger }V\rho _{\text {in}}\right ]\right )$ with ρ_in = |ψ〉〈ψ_in|

Notably, the phase e^iα in the first qubit is needed to include phases which may occur from gate derivatives. In our case, α needs to be set to 0 respectively π/2 when computing the terms of A or C. More precisely, the first qubit is initialized by an H gate for A and H followed by an S gate for C. These phases come from the fact that the trial states, used in this work, are constructed via Pauli rotations, i.e., $U\left (\omega \right ) = R_{\sigma _{l}}\left (\omega \right )$ with σ_l ∈{X,Y,Z}, which leads to

$$ \frac{\partial U\left( \omega\right)}{\partial\omega} = -\frac{i}{2}{\sigma_{l}}R_{\sigma_{l}}\left( \omega\right). $$

(18)

Furthermore, this method can be applied for the evaluation of ∂A/∂𝜃 and ∂C/∂𝜃, i.e., the respective terms can be written in the form of Eq. 17. More precisely,

$$ \begin{array}{ll} &\partial_{\theta_{i}}A_{p,q}\left( \tau\right)=\sum\limits_{s} \frac{\partial\omega_{s}\left( \tau\right)}{\partial_{\theta_{i}}}\text{Re}\left( \text{Tr}\left[\left( \frac{\partial^{2} V^{\dagger}\left( {\omega\left( \tau\right)}\right)}{\partial\omega_{p}\left( \tau\right) \partial\omega_{s}\left( \tau\right)}\frac{\partial V\left( {\omega\left( \tau\right)}\right)}{\partial\omega\left( \tau\right)_{q}}\right. \right.\right.\\ &~~~~~~~~~~~~~~~~~~~~~\left.\left.\left. + \frac{\partial V^{\dagger}\left( {\omega\left( \tau\right)}\right)}{\partial\omega\left( \tau\right)_{p}}\frac{\partial^{2} V\left( {\omega\left( \tau\right)}\right)}{\partial\omega_{q}\left( \tau\right)\partial\omega_{s}\left( \tau\right)}\right)\rho_{\text{in}}\right]\right) \end{array} $$

and

$$ \begin{array}{ll} &\partial_{\theta_{j}}C_{p} = \\ &-\text{Re}\left( \text{Tr}\left[\frac{\partial V^{\dagger}\left( {\omega\left( \tau\right)}\right)}{\partial\omega\left( \tau\right)_{p}}h_{j}V\left( {\omega\left( \tau\right)}\right)\rho_{\text{in}}\right]\right)\\ &\left. - \sum\limits_{i,s}\theta_{i}\frac{\partial\omega_{s}\left( \tau\right)}{\partial_{\theta_{j}}}\text{Re}\left( \text{Tr}\left[\left( \frac{\partial V^{\dagger}\left( {\omega\left( \tau\right)}\right)}{\partial\omega_{p}\left( \tau\right)}h_{i}\frac{\partial V\left( {\omega\left( \tau\right)}\right)}{\partial\omega_{s}\left( \tau\right)} \right.\right. \right.\right.\\ &+ \left.\left. \left.\frac{\partial^{2} V^{\dagger}\left( {\omega\left( \tau\right)}\right)}{\partial\omega_{p}\left( \tau\right)\partial\omega_{s}\left( \tau\right)} h_{i}V\left( {\omega\left( \tau\right)}\right)\right)\rho_{\text{in}} \right] \right). \end{array} $$

Hereby, α must be set to π/2 for all terms in ∂A/∂𝜃 and the first term in ∂C/∂𝜃 and 0 for the remaining terms in ∂C/∂𝜃. This is achieved with the same gates as mentioned before.

Appendix: 2: Complexity analysis

To compute the gradient ∂L/∂𝜃 of the loss function, given in Eq. 7, we could use either a numerical finite differences method (Kardestuncer 1975), or the analytic, automatic differentiation approach that is presented in this paper. In the following, we discuss the number of circuits that have to be evaluated for those gradient implementations for a trial state with q parameters, an n-qubit Hamiltonian with p parameters, and VarQITE for Gibbs state preparation using t steps. In near-term quantum devices, the run of every circuit requires compilation, queuing for a position to be run on quantum hardware and, preferably, error mitigation/correction. Thus, the number of circuits strongly impacts the execution time and resources.

The number of circuits that need to be evaluated for Gibbs state preparation with VarQITE are $\varTheta \left (tq^{2} \right )$ and $\varTheta \left (tqp \right )$ for A and C, respectively. Therefore, the overall number of circuits is $\varTheta \left (tq(q+p)\right )$. Now, computing the gradient with forward finite differences reads:

$$ \frac{\partial L}{\partial \theta} \approx \frac{L\left( \theta + \epsilon \right) - L\left( \theta \right)}{\epsilon}, $$

for 0 < 𝜖 ≪ 1. For this purpose, VarQITE must be run once with 𝜃 and p times with an 𝜖-shift which leads to a total number of $\varTheta \left (tpq(q+p)\right )$ circuits.

The automatic differentiation gradient, given in Eq. 9, corresponds to:

$$ \begin{array}{ll} \frac{\partial L}{\partial\theta} = - \sum\limits_{v}\sum\limits_{k=0}^{q-1}\frac{p_{v}^{\text{data}}}{\left\langle {\Lambda}_{v}\right\rangle} \frac{\partial \left\langle {\Lambda}_{v} \right\rangle}{\partial \omega_{k}} \frac{\partial \omega_{k}} {\partial \theta} \end{array} $$

with $\langle {\ldots } \rangle = \text {Tr}\left [\rho _{\omega }^{Gibbs}\ldots \right ]$. VarQITE needs to be run once to prepare $\rho _{\omega }^{\text {Gibbs}}$. Furthermore, the evaluation of ∂ω_k/∂𝜃 requires that ∂A/∂𝜃 and ∂C/∂𝜃 are computed for every step of the Gibbs state preparation. This leads to $\varTheta \left (tq^{2}(q+p)\right )$ circuits. The resulting overall complexity of the number of circuits is $\varTheta \left (tq^{2}(q+p)\right )$. The results are summarized in Table 3. Automatic differentiation is more efficient than finite differences if q < p. For q > p, on the other hand, focusing mainly on computational complexity, one should rather use finite differences. Considering, e.g., a k-local Ising model that corresponds to a Hamiltonian with $\mathcal {O}\left (n^{k}\right )$ parameters. Suppose that we can find a reasonable variational n-qubit trial state with $\mathcal {O}\left (n\right )$ layers of parameterized and entangling gates, which results in $q = \mathcal {O}\left (n^{2}\right )$ parameters, then automatic differentiation would outperform finite differences for k > 2.

Table 3 Comparing the number of circuits needed to train a QBM with VarQITE using either finite differences or automatic differentiation

Appendix: 3: Numerical Gibbs state preparation

In order to verify that VarQITE enables approximate Gibbs state preparation, we conduct exact simulation experiments for Hamiltonians with n > 2 qubits. The states are prepared for the following Hamiltonians:

$$ \begin{array}{@{}rcl@{}} H_{3} &=& 2. \cdot ZZI + 1.\cdot IZZ - 0.5 \cdot IZI \\ H_{4} &=& 2. \cdot ZZII + 1.\cdot IZZI - 0.5 \cdot IZIZ \\ H_{5} &=& 2. \cdot ZZIII + 1.\cdot IZZII - 0.5 \cdot IZIIZ, \end{array} $$

temperature k_BT = 1 and a discretization of 10 time steps. All Gibbs states are prepared using a shallow, depth 2 Ansatz that consists of parameterized R_Y and R_Z gates as well as CX gates. Figure 9 illustrates the respective quantum circuit for H₃, whereby the given parameters prepare the initial state. The states for H₄ and H₅ are prepared equivalently.

Fig. 9

This quantum circuit is used as Ansatz for the Gibbs state preparation of H₃. The given parameter set prepares the initial state, i.e., tracing out q₃ to q₅ results in a maximally mixed state

The effectiveness of VarQITE is confirmed with a plot (see Fig. 10), of the fidelities between the trained and the target Gibbs states throughout the simulated time evolution. It should be noted that all final states approximate the targets with fidelity > 0.936.

Fig. 10

This figure illustrates the fidelity between the state that is prepared using VarQITE and the target Gibbs state preparation for H₃, H₄, and H₅ during each step of the simulated imaginary time evolution

Appendix: 4: Generative QBM training for larger systems

To investigate the trainability of QBMs for larger systems, we extend the learning setting discussed in Section 4.3 to n = {3, 4} qubits. More explicitly, given a 3- and 4-qubit Greenberger-Horne-Zeilinger (GHZ) state, a QBM model is trained to represent the respective sampling probabilities of these states:

$$ \begin{array}{@{}rcl@{}} p^{\text{data}}_{3} \!&= &\!\left[0.5, 0., 0., 0., 0., 0., 0., 0.5 \right] \\ p^{\text{data}}_{4} \!&= &\!\left[0.5, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,\right.\\&&\left. 0., 0., 0.5 \right] \end{array} $$

using Hamiltonians of the form

$$ H_{\theta}=\sum\limits_{i=0}^{n-1}\theta_{i}Z_{i} + \sum\limits_{i=0}^{n-1}\sum\limits_{j=0}^{n-1}\theta_{ij}Z_{i}Z_{j}, $$

where Z_i acts on the i^th qubit. The temperature is again set to k_BT = 1 and the time is discretized into 10 steps. Optimizer and initial states are chosen equivalently to the ones from Section 4.3. Notably, the number of qubits in the Ansatz state shown in Fig. 4 needs to match to the number of qubits in the respective Hamiltonian. Figure 11 illustrates the progress of the loss function as well as the ℓ₁ norm between the trained and the target distributions for 30 training steps. Both measures show a good convergence behavior and thus indicate that the method scales well.

Fig. 11

The figure illustrates the training progress of a fully visible QBM model which aims to represent the measurement distribution of a 3- and a 4-qubit GHZ state. The green and pink dots (crosses) correspond to the loss and the distance, respectively, between the trained and target distributions with respect to the ℓ₁ norm at each step of the iteration for the 3 (4)qubit GHZ state