Introduction

Sampling from a Gibbs-Boltzmann distribution has numerous practical applications across various fields, such as combinatorial optimization1, biology2, and machine learning3. A widely used approach is the Markov-Chain Monte Carlo (MCMC) method4 with the Metropolis-Hastings (M-H) acceptance probability5. In this scheme, we generate a proposal state from any probability distribution and decide whether the proposal state is accepted or not based on the M-H acceptance probability. This process is repeated until the samples can be considered to come from a stationary distribution. Many iterations are required to reach the stationary distribution for complex energy landscapes, such as spin glasses6. Therefore, faster convergence of MCMC is a crucial problem.

To achieve faster convergence of MCMC, a well-known approach is the cluster update7,8, which involves updating multiple spins simultaneously and is more efficient than local updates. Cluster update is a technique primarily developed in pure systems. However, they are ineffective in spin glass due to frustration, which yields many local minima in the energy landscape9. They cannot overcome the slow relaxation in low temperatures. For this reason, extended ensemble methods such as exchange Monte Carlo10 and parallel tempering11, as well as the efficient sequential Monte Carlo algorithm like population annealing12, have been proposed to overcome the slow relaxation in spin glasses.

Accelerating the dynamics of MCMC for spin glasses is a challenging task. In a previous study, machine learning has been applied to accelerate the dynamics of MCMC. One of the pioneering research is the self-learning Monte Carlo (SLMC) method13,14. In SLMC, an effective model trained on the training data has been introduced to accelerate the dynamics of MCMC. The proposal from the effective model has yielded the global update and brought about faster convergence of MCMC. This pioneering research has led to the development of Neural MCMC (NMCMC) algorithm15,16,17,18,19. In NMCMC, an autoregressive neural network has often been applied as an effective model. Although NMCMC could provide a computational speed-up over the MCMC with a local update for the ferromagnetic Ising model and two-dimensional (2d)-Edward Anderson model, NMCMC has not worked in the antiferromagnetic Potts model on a random graph20.

In recent studies, quantum and hybrid quantum-classical algorithms have been proposed to overcome the slow relaxation typically observed in MCMC. For an extensive review of quantum algorithms applied to sampling problems, see Ref21.. Mazzola et al.22 have introduced two distinct quantum approaches for sampling from the target distribution. The first approach has leveraged the formalism of reverse stochastic quantization, which interprets Langevin dynamics as imaginary-time Schrödinger evolution to prepare thermal equilibrium states. The second approach has utilized wavefunction collapse as a global update mechanism. In this method, a Gaussian-centered initial state, expressed as a linear combination of the low-lying eigenstates of the target Hamiltonian, is evolved using quantum phase estimation (QPE). Upon measurement, the system has collapsed onto one of these eigenstates, thereby enabling nonlocal transitions and accelerating exploration across metastable states. However, the QPE-based state proposal is computationally expensive in terms of quantum resources. To address this issue, an alternative approach known as quantum-enhanced MCMC (QEMCMC) has been proposed23. In QEMCMC, gate-based quantum computers generate new proposal states via quantum time evolution. The classical computer subsequently determines whether the proposal state is accepted or not. This hybrid algorithm has been shown numerically to converge faster than classical MCMC methods with uniform and local updates in small spin systems23. The proposal states had energy levels comparable to the current state with a larger Hamming distance than those generated by the local update. This property led to a broader exploration of the configuration space, thereby accelerating convergence speed. QEMCMC has subsequently been extended to a coarse-grained approach for larger size problems than the size of the quantum computer24, variational methods for adjusting hyperparameters25, and continuous systems26. In the recent theoretical analysis, QEMCMC has not yielded a quantum speed-up over classical sampling on the masked item sampling problem27. A bottleneck analysis has indicated the region where quantum improvement may exist28.

In addition to gate-based quantum computers, quantum time evolution can also be realized in a quantum annealer29,30,31,32,33. The quantum annealer is the physical implementation of quantum annealing (QA), a metaheuristic designed for solving combinatorial optimization problems34,35,36,37,38,39. The primary applications of the quantum annealer are broadly categorized into optimization and sampling. For optimization tasks, quantum annealers have been successfully applied to various combinatorial optimization problems40,41. For sampling tasks, quantum annealers have been used to generate samples from an approximate Gibbs–Boltzmann distribution governed by a hardware-specific effective temperature42,43,44. Ghamari et al45,46. has proposed a hybrid quantum-classical scheme for sampling dominant reaction pathways in molecular dynamics simulations from the corresponding Gibbs-Boltzmann distribution. The target Hamiltonian was constructed using a coarse-grained graph representation of conformational dynamics, where its nodes correspond to finite regions of configuration space and edges encode transition probabilities derived from Langevin dynamics. The quantum annealer was employed as a proposal generator in the MCMC subroutine, producing uncorrelated transition paths through quantum superposition and adiabatic evolution. This method has been extended in a follow-up study47, where the quantum annealer was combined with machine-learning-based exploration to simulate a rare millisecond-scale conformational transition in Bovine Pancreatic Trypsin Inhibitor protein. Recently, the quantum annealer has been applied to various quantum simulations at finite temperatures, such as three-dimensional spin glass48, a fully frustrated square-octagonal lattice49, and spin ice50. Hardware development of the quantum annealer enables the operation in a coherent regime51,52.

In this paper, we investigate the use of QA as a proposal generator within the MCMC subroutine. We focus on the intrinsic advantages of QA in producing low-energy configurations and efficiently escaping local minima. Inspired by the QEMCMC framework23, we propose quantum annealing-enhanced MCMC(QAEMCMC), in which QA is employed to generate proposal configurations. These are subsequently accepted or rejected using the M-H acceptance criterion. Since the resulting Markov chain satisfies the detailed balance condition, QAEMCMC guarantees convergence to the target Gibbs–Boltzmann distribution. While Ghamari et al45,47. utilized QA to sample entire transition paths across a coarse-grained kinetic graph derived from molecular dynamics simulations, our method directly operates in the original configuration space. Their domain-specific approach requires prior construction of a graph representation of the conformational landscape, whereas our QAEMCMC framework is model-agnostic and broadly applicable to general energy-based systems. Furthermore, unlike the approach by Giuseppe et al53., which used QA to generate training data for a neural network-based sampler, our method directly leverages the output distribution from QA as the proposal distribution. This bypasses the need for an additional training phase, enabling more transparent utilization of quantum fluctuations, particularly in low-temperature regimes. Following the benchmark setup in Ref23., we evaluate the performance of QAEMCMC on the Sherrington–Kirkpatrick (SK) model54, a paradigmatic spin glass system with a highly rugged energy landscape. We analyze the absolute spectral gap and thermodynamic observables, demonstrating that QAEMCMC achieves significantly enhanced spectral gaps and faster convergence compared to classical MCMC methods.

This paper is organized as follows. In the next section, we explain the classical MCMC method and QAEMCMC. In the following section, we present the numerical simulations of QAEMCMC for the SK model. The final section is dedicated to discussing and summarizing the results.

Markov-Chain Monte Carlo

MCMC starts from a trivial spin configuration \(\varvec{\sigma }\) and constructs a sequence of states, which is called a Markov chain, by updating the current state with the transition probability \(P(\varvec{\sigma }'|\varvec{\sigma })\). The transition probability is designed so that the stationary distribution corresponds to the target distribution \(\mu (\varvec{\sigma })\). Then, the resultant Markov chain can be regarded as samples from the target distribution if MCMC iterates sufficiently. The sufficient condition to converge the Markov chain to the target distribution is to satisfy the detailed balance condition :

$$\begin{aligned} P(\varvec{\sigma }'|\varvec{\sigma })\mu (\varvec{\sigma })=P(\varvec{\sigma }|\varvec{\sigma }')\mu (\varvec{\sigma }'), \end{aligned}$$
(1)

for all \(\varvec{\sigma }\ne \varvec{\sigma }'\).

The popular approach to construct \(P(\varvec{\sigma }'|\varvec{\sigma })\) is the M-H method5. The M-H method decomposes \(P(\varvec{\sigma }'|\varvec{\sigma })\) into the acceptance probability \(A(\varvec{\sigma }'|\varvec{\sigma })\) and the proposed probability \(Q(\varvec{\sigma }'|\varvec{\sigma })\) as \(P(\varvec{\sigma }'|\varvec{\sigma })=A(\varvec{\sigma }'|\varvec{\sigma })Q(\varvec{\sigma }'|\varvec{\sigma })\). At first, we sample the proposal state \(\varvec{\sigma }'\) from \(Q(\varvec{\sigma }'|\varvec{\sigma })\). Next, we decide whether the current proposal is accepted or rejected with the M-H probability as

$$\begin{aligned} A(\varvec{\sigma }'|\varvec{\sigma })= \textrm{min}\left( 1,\frac{\mu (\varvec{\sigma }')}{\mu (\varvec{\sigma })}\frac{Q(\varvec{\sigma }|\varvec{\sigma }')}{Q(\varvec{\sigma }'|\varvec{\sigma })}\right) . \end{aligned}$$
(2)

For a binary system \(\varvec{\sigma }\in \{1,-1\}^N\) where N is the system size, the local update is often applied. This involves choosing a random spin index \(i\in [1, N]\) of the current spin configuration and flipping it as \(\sigma _i' \leftarrow -\sigma _i\). The mathematical representation of \(Q(\varvec{\sigma }'|\varvec{\sigma })\) for the local update can be described as

$$\begin{aligned} Q_{\textrm{local}}(\varvec{\sigma }'|\varvec{\sigma })={\left\{ \begin{array}{ll}\frac{1}{N} \quad & (d(\varvec{\sigma }',\varvec{\sigma })=1)\\ 0\quad \quad & (otherwise) \end{array}\right. } , \end{aligned}$$
(3)

where \(d(\varvec{\sigma }',\varvec{\sigma })=\sum _{i=1}^N|\sigma _i'-\sigma _i|\) is the Hamming distance between the current and proposed states. In the local update, Eq. (2) is easily calculated since the ratio of the Boltzmann factor \(\mu (\varvec{\sigma })/\mu (\varvec{\sigma }')\) can be efficiently computed. It is noted that the local update often fails to escape local minima, and the resultant acceptance probability is very low in low-temperature regions4. Another common approach is the uniform update, which employs uniform sampling from

$$\begin{aligned} Q_{\textrm{uniform}}(\varvec{\sigma })= \frac{1}{2^N}. \end{aligned}$$
(4)

In the case that the proposed state is independent of the current state, the M-H acceptance probability can be extended as

$$\begin{aligned} A(\varvec{\sigma }'|\varvec{\sigma })= \textrm{min}\left( 1,\frac{\mu (\varvec{\sigma }')}{\mu (\varvec{\sigma })}\frac{Q(\varvec{\sigma })}{Q(\varvec{\sigma }')}\right) , \end{aligned}$$
(5)

which is called as Metroplized independent sampling55. It is noted that a simple calculation confirms the detailed balance condition with Eq. (5). While the uniform update may give rise to the proposal states with a large Hamming distance and lead to escape from the local minima, the acceptance probability becomes low. As a result, the small acceptance probability deteriorates the convergence rate of the Markov chain.

Mixing time \(\tau _\epsilon\) is utilized to quantify the convergence rate of the Markov chain. It is defined as the minimum number of iterations needed for the probability distribution obtained by the Markov chain to get within \(\epsilon>0\) of \(\mu (\varvec{\sigma })\) in total variation distance for any initial distribution as

$$\begin{aligned} (1-\delta ^{-1})\ln 2\epsilon \le \tau _\epsilon \le -\delta ^{-1}\ln \left( \epsilon \textrm{min}_{\varvec{\sigma }}\mu (\varvec{\sigma })\right) \end{aligned}$$
(6)

where \(\delta\) is the absolute spectral gap as \(\delta = 1- \textrm{max}_{\lambda \ne 1}\lambda\) and \(\lambda\) is the eigenvalues of \(P(\varvec{\sigma }'|\varvec{\sigma })\)4. Equation. (6) shows that the mixing time is bounded by the absolute spectral gap, and a larger spectral gap results in a shorter mixing time. The absolute spectral gap depends on the choice of \(Q(\varvec{\sigma }'|\varvec{\sigma })\) in the M-H method. Choosing a proposal probability that increases the absolute spectral gap leads to faster convergence of MCMC.

QA-enhanced MCMC

In QA, the Hamiltonian can be constructed as

$$\begin{aligned} \hat{H}=\mathcal {A}(s(t))\hat{H}_0(\varvec{\hat{\sigma }}^z)-\mathcal {B}(s(t))\sum _{i=1}^N\hat{\sigma }_i^x \end{aligned}$$
(7)

where \(\hat{\sigma }_i^{z}\) and \(\hat{\sigma }_i^{x}\) are the Pauli operators acting on the site i, and \(\hat{H}_0(\varvec{\hat{\sigma }}^z)\) is the target Hamiltonian whose eigenvalues correspond to the energy of the Gibbs-Boltzmann distribution

$$\begin{aligned} \mu (\varvec{\sigma })=\frac{1}{Z}{\langle {\varvec{\sigma }}|} \exp (- \beta \hat{H}_0(\varvec{\hat{\sigma }}^z)){|{\varvec{\sigma }}\rangle }. \end{aligned}$$
(8)

The partition function is defined as \(Z=\sum _{\varvec{\sigma }}{\langle {\varvec{\sigma }}|} \exp (- \beta \hat{H}_0(\varvec{\hat{\sigma }}^z)){|{\varvec{\sigma }}\rangle }\) and \(\beta =1/T\) is the inverse temperature. The annealing schedule function \(\mathcal {A}(s(t))\) and \(\mathcal {B}(s(t))\) interpolate two Hamiltonians as a function of annealing parameter s(t). We adopt the linear schedule as \(\mathcal {A}(s(t))=s(t)\) and \(\mathcal {B}(s(t))=1-s(t)\). In the vanilla QA, the initial state is the superposition state, which is the ground state of the second term in Eq. (7), and evolves following the Schrödinger dynamics. The annealing parameter s(t) increases monotonically from \(s(t=0)=0\) to \(s(t=\tau )=1\). The annealing time \(\tau\) determines the period of total time evolution. If the dynamics evolves adiabatically, the final state concentrates on the ground state of \(\hat{H}_0(\varvec{\hat{\sigma }}^z)\) in the limit of \(\tau \rightarrow \infty\).

In fast annealing schedules, the time evolution of QA fails to follow the adiabatic path and instead populates excited states through diabatic transitions. As a result, the final state deviates from the ground state of the target Hamiltonian and contains contributions from various excited eigenstates. The classical spin configuration \(\varvec{\sigma }\) is sampled from the final wavefunction \({|{\psi }\rangle }\) with probability

$$\begin{aligned} Q_{\textrm{QA}}(\varvec{\sigma }) = |{\langle {\varvec{\sigma }|\psi }\rangle }|^2. \end{aligned}$$
(9)

For large but finite annealing times, the probability distribution \(Q_{\textrm{QA}}(\varvec{\sigma })\) tends to concentrate around the ground state of the target Hamiltonian, allowing us to obtain low-energy configurations. We exploit these diabatic transitions in QA39 to construct proposal states within the MCMC subroutine. According to the standard protocol of the M-H method, the proposed state is sampled from Eq. (9) and then decided whether it is accepted or not. This QA update and acceptance or rejection steps are iterated. We refer to the above procedure as QA-enhanced MCMC (QAEMCMC). The schematic diagram of the QAEMCMC protocol is plotted in Fig. 1.

Results

In this section, we show the results of the SK model as

$$\begin{aligned} \hat{H}_0(\hat{\varvec{\sigma }}^z)=\sum _{i< j}J_{ij}\hat{\sigma }_i^z\hat{\sigma }_j^z+\sum _{i=1}^Nh_i\hat{\sigma }_i^z \end{aligned}$$
(10)

where \(J_{ij}\) and \(h_i\) are sampled from the Gaussian distribution with zero mean and unit variance. The model is the same as used in the previous studies23,25.

Fig. 1
figure 1

The schematic diagram of the QAEMCMC protocol.

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Fig. 2
figure 2

(a): The absolute spectral gap as a function of temperature for the fixed \(N=10\). Each marker represents different update schemes. The histogram of the optimized annealing time for (b): intermediate temperature and (c): low temperature regions.

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Fig. 3
figure 3

The system size dependence of the absolute spectral gap for the fixed \(T=1\) with different update schemes.

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We exhibit the absolute spectral gap \(\langle \delta \rangle\) for \(N=10\) as a function of temperature T in Fig. 2 (a). The bracket \(\langle \cdot \rangle\) represents the average for the 100 random instances. The error bars and shaded regions represent the standard deviation. For each instance and T, we compute the eigenvalues of the transition probability. In the QA update, we adopt continuous-time QA dynamics without employing Trotter decomposition. The time-dependent Schrödinger equation is numerically integrated using the default ordinary differential equation solver for unitary dynamics provided in QuTiP56,57. For each instance and temperature, \(\tau\) is optimized to maximize \(\delta\) with the default TPE-based sampler using Optuna58. Our heuristic tuning strategy proceeds in two stages: first, we perform 40 preliminary trials over a broad search range, \(\tau \in [10^{-4}, 10^{3}]\). Then, for intermediate temperatures \(0.1< T < 10\), we refine the search by initializing around the best values found in the preliminary stage. In the high-temperature regime \(T \ge 10\), we fix \(\tau = 10^{-4}\) based on prior knowledge of the target distribution and empirical trends observed during the preliminary exploration.

In this region, the Gibbs-Boltzmann distribution approaches the uniform distribution. The proposal distribution of the QA update with small \(\tau\) approximately yields the uniform distribution since few transitions of the states from the initial superposition states occur. As a result, the uniform and QA updates take the largest \(\langle \delta \rangle\) because the proposal distribution is similar to the target distribution. For the local update, \(\langle \delta \rangle\) shows the parabolic behavior. This behavior is reflected in the structure of eigenvalues of the transition probability and was also discussed in the previous study23. In the intermediate region \(1\le T <10\), the QA update takes the larger \(\langle \delta \rangle\) than those from the uniform and local updates. Figure 2 (b) exhibits the histogram of the optimized \(\tau\) with \(N=10\) in the intermediate temperature. The broader histogram shows the strong dependence of the optimized \(\tau\) on each instance. In this region, the target distribution is a multi-modal distribution dominated by several low-energy states. That leads to the difficulty in optimization of \(\tau\). As we increase the temperature, the histogram is concentrated on the small \(\tau\). This observation is consistent with the results that the target distribution can be approximated by short annealing times and yields a large spectral gap in the high-temperature region. At low \(T<1\), the uniform and local updates do not work due to the existence of the stable local minima and lead to a small \(\langle \delta \rangle\). This difficulty is alleviated by the QA update, enabling us to escape from the local minima. Since \(Q_{\textrm{QA}}(\varvec{\sigma })\) approximates the target distribution well, \(\langle \delta \rangle \simeq 1.0\) can be obtained. Figure 2 (c) shows the histogram of the optimized \(\tau\) with \(N=10\) in the low-temperature region. The lower the temperature is, the greater frequency is concentrated around the large \(\tau =10^3\). The Gibbs-Boltzmann distribution is dominated by the ground state in the low-temperature region. The original usage of QA for obtaining the ground state matches the situation in this low temperature.

Table 1 The fitting exponents of the results in Fig. 3.
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Figure 3 shows the dependence of \(\langle \delta \rangle\) on N for the fixed \(T=1\). Compared to \(\langle \delta \rangle\) obtained from the uniform and local updates, \(\delta\) from the QA update decreases slowly. We fit \(\langle \delta \rangle\) as \(\langle \delta \rangle \sim 2^{-\alpha N}\) and exhibit the fitting exponents in Table 1. More than doubled quantum enhancement exists for the exponents. From the results of \(\langle \delta \rangle\), mixing time obtained by the QA update becomes smaller than those obtained by the other updates. The QA update reduces the effect of slow relaxation at a low temperature and accelerates the MCMC convergence.

To investigate the acceleration of the MCMC convergence, we demonstrate the dynamical behaviors of the averaged energy as a function of the Monte Carlo step (MCS) \(\tau _{\textrm{MCS}}\) represented as \(\bar{E}=\sum _{i=1}^{\tau _{\textrm{MCS}}}{\langle {\varvec{\sigma }^{(i)}}|} \hat{H}_0(\varvec{\hat{\sigma }}^z){|{\varvec{\sigma }^{(i)}}\rangle }/{\tau _{\textrm{MCS}}}\) where \(\varvec{\sigma }^{(i)}\) is the spin configuration generated by each Monte Carlo step in Fig. 4 (a). The experimental settings are as follows: \(N=10\) and \(T=1\). To demonstrate the performance of the QA update, we select a challenging problem instance used in Fig. 2. Its target distribution is multi-modal and dominated by five low energy states with \(\mu (\varvec{\sigma })> 0.1\), which is difficult to sample efficiently. Each line represents the averaged values computed from the independent 100 MCMC runs. The shaded region represents the standard deviation. The result obtained by the QA update converges the exact equilibrium energy \(\bar{E}_{\textrm{ex}}=\sum _{\varvec{\sigma }}\mu (\varvec{\sigma }) {\langle {\varvec{\sigma }}|} \hat{H}_0(\varvec{\hat{\sigma }}^z){|{\varvec{\sigma }}\rangle }\) faster than those by other updates. Figure 4 (b) shows the absolute error between the averaged energy and the exact equilibrium energy as \(| \ \bar{E} - \bar{E}_{\textrm{ex}}|\). The horizontal line denotes \(| \ \bar{E} - \bar{E}_{\textrm{ex}}|=0.1\). The QA update achieves \(| \ \bar{E} - \bar{E}_{\textrm{ex}}|=0.1\) at \(\tau _{\textrm{MCS}}\simeq 618\). The uniform and local updates reach at \(\tau _{\textrm{MCS}}\simeq 21232\) and \(\tau _{\textrm{MCS}}\simeq 7920\) respectively. The QA update gives a better estimator, at least more than about twelve times faster than the classical updates. Therefore, the QA update can realize efficient sampling of the low-energy states dominated by the Gibbs-Boltzmann distribution.

Fig. 4
figure 4

(a): The dynamics of the averaged energy obtained at \(T=1\) from MCMC. The horizontal axes represent the number of MCS. The horizontal line denotes the exact equilibrium energy. (b): The absolute error between the exact equilibrium energy and the empirical averages. The horizontal line represents \(\Delta E =0.1\).

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Fig. 5
figure 5

(a): The Monte Carlo step dependence of the total variation distance between the Gibbs-Boltzmann distribution and the empirical probability distribution with fixed \(T=1\) and \(N=10\). (b): the system size dependence of the total variation distance with fixed \(T=1\) and \(\tau _{\textrm{MCS}}=10^5\).

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In addition to the efficacy of the sampling of the low-energy states, whether the empirical distribution \(Q_{\textrm{emp}}(\varvec{\sigma })\) obtained by MCMC converges to the target distribution \(P_{\textrm{ex}}\) is also important in MCMC. We compute the total variation distance between \(P_{\textrm{ex}}(\varvec{\sigma })\) and \(Q_{\textrm{emp}}(\varvec{\sigma })\) as

$$\begin{aligned} TV(P_{\textrm{ex}},Q_{\textrm{emp}})=\frac{1}{2}\sum _{\varvec{\sigma }}|P_{\textrm{ex}}(\varvec{\sigma })-Q_{\textrm{emp}}(\varvec{\sigma })|_1. \end{aligned}$$
(11)

The experimental settings are the same as those in Fig. 4. Figure 5 (a) demonstrates the dependence of \(TV(P_{\textrm{ex}},Q_{\textrm{emp}})\) on \(\tau _{\textrm{MCS}}\). The scaling of \(TV(P_{\textrm{ex}},Q_{\textrm{emp}})\) follows the power law decay as a function of \(\tau _{\textrm{MCS}}\) shown in Table 2. The QA update gives larger exponents than the local updates. Even though the difference in the exponents between the QA update and the uniform update for \(\tau _{\textrm{MCS}}\) is small, the constant gap exists. Therefore, the QA update converges to the target distribution faster than other updates. Figure 5 (b) exhibits the dependence of \(TV(P_{\textrm{ex}},Q_{\textrm{emp}})\) on N with \(\tau _{\textrm{MCS}}=10^5\). The scaling exponents fitted by \(2^{\gamma N}\) are shown in Table 2. The QA update yields smaller exponents than the classical updates. The difference between the QA and other updates increases as we increase N. For the large N, the QA update is effective. The acceleration of the MCMC convergence by the QA update can be seen from the dynamical simulations.

Table 2 The scaling exponents of the total variation distance for the data in Fig. 5.
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Fig. 6
figure 6

The dependence of the cumulative probability distribution on (a): the Hamming distance and (b): the difference of the absolute energy between the current configurations and the proposed ones for the fixed \(N=10\) and \(T=1\). Each symbols represent the different update schemes. (c): the acceptance probability for each update. The error bar represents the standard deviation.

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Finally, we investigate the features of samples obtained from different updates with the cumulative probability of the observables \(\mathcal {O}(\varvec{\sigma },\varvec{\sigma }')\). For each Markov chain, we compute \(\mathcal {O}(\varvec{\sigma },\varvec{\sigma }')\) with the current spin configuration \(\varvec{\sigma }\) and proposed one \(\varvec{\sigma }'\). We focus on the cumulative probability of Hamming distance between two spin configurations and the absolute energy difference \(|\Delta E|(\varvec{\sigma },\varvec{\sigma }')=|{\langle {\varvec{\sigma }}|}\hat{H}_0{|{\varvec{\sigma }}\rangle }-{\langle {\varvec{\sigma }'}|}\hat{H}_0{|{\varvec{\sigma }'}\rangle }|\). The cumulative probability is calculated from \(n_{\textrm{all}}=10^5\) spin configurations as

$$\begin{aligned} P_{\textrm{cum}}(d)&=\sum _{x=0}^{d} P_{\textrm{ham}}(x),\hspace{1mm} P_{\textrm{ham}}(x)=\frac{\# d(\varvec{\sigma },\varvec{\sigma }')=x}{n_{\textrm{all}}},\end{aligned}$$
(12)
$$\begin{aligned} P_{\textrm{cum}}(|\Delta E|)&=\int _{0}^{|\Delta E|} P_{|\Delta E|}(x)dx,\hspace{1mm} P_{|\Delta E|}(x)=\frac{\# |\Delta E|(\varvec{\sigma },\varvec{\sigma }')=x}{n_{\textrm{all}}}, \end{aligned}$$
(13)

where \(\# \mathcal {O}(\varvec{\sigma },\varvec{\sigma }')=x\) denotes the number of samples satisfied with \(\mathcal {O}(\varvec{\sigma },\varvec{\sigma }')=x\). For the uniform and QA updates, we utilize the data with \(\tau _{\textrm{MCS}}=10^5\) in Fig. 5. Figure 6 (a) shows the dependence of \(P_{\textrm{cum}}(d)\) on Hamming distance. Each marker denotes the average over 100 different instances used in Figs. 2 and 5. The error bar and the shaded region denote the standard deviation. As defined, the local update proposes the next state with \(d=1\), while the proposal distribution of the uniform update concentrates around \(d=5\), which is the highest number of states. This result is consistent with the result presented in the previous study23. The QA update gives the samples with a smaller Hamming distance than the uniform update. The main feature of the QA update is that we can obtain the samples with a large Hamming distance, which is far from the current state. Figure 6 (b) exhibits the dependence on the absolute difference of the energy between the current and proposed states represented as \(|\Delta E|\). Each curve denotes the results for different instances. In many instances, the QA update gives samples with a smaller \(|\Delta E|\) than the uniform and local updates. As shown in Fig. 6 (c), the small \(|\Delta E|\) leads to a high acceptance probability. Therefore, the QA update can yield the samples with large d and small \(|\Delta E|\). We can interpret the QA update, which enables us to transition to the different local minima.

Discussion and conclusion

In this paper, we proposed QAEMCMC, a hybrid algorithm in which QA was employed to generate proposal states within the MCMC subroutine. To evaluate its performance, we benchmarked QAEMCMC against conventional MCMC methods using the SK model. Our results demonstrated that QAEMCMC yielded a significantly larger absolute spectral gap than classical update schemes, particularly in the low-temperature regime. This enhancement translated into faster convergence toward the Gibbs–Boltzmann distribution. Indeed, the scaling exponent of the spectral gap exhibited more than a twofold improvement compared to the classical baseline. Furthermore, we analyzed the time evolution of key observables, including the average energy and the total variation distance between the empirical and target distributions. QAEMCMC achieved more accurate estimates in a smaller Monte Carlo step, highlighting the effectiveness of QA as a global proposal generator. The resulting sample characteristics indicated that QA could traverse broader regions of configuration space and overcome trapping in local minima more efficiently. These findings suggest that the QA update is a valuable component for enhancing other sampling frameworks, such as NMCMC algorithms53.

It is noted that QA exhibits biased sampling over degenerate ground states, particularly due to the transverse field term in the driver Hamiltonian, which tends to suppress isolated solutions in the long annealing time limit59,60. The biased sampling can be mitigated by introducing higher-order driver terms or by adjusting the annealing schedule appropriately. In the intermediate annealing time region, the probability of reaching isolated ground states remains a finite value, suggesting that the unfair sampling can be alleviated by carefully choosing the annealing time. This unfair sampling problem does not preclude QA applications from being included in the MCMC subroutine.

In addition, recent work61 has explored time-dependent Hamiltonian evolution by using reverse annealing62 as an alternative method to improve QEMCMC. Whereas reverse annealing focuses on exploring the vicinity of a given initial configuration, our approach generates global proposals that are independent of the current state. A detailed study is needed to investigate how the selection of the algorithms affects the convergence of MCMC.

Moreover, it is worth discussing the practical aspects of QAEMCMC implementation. All numerical simulations in this work were performed using classical emulation of QA, and direct access to the actual output distribution of QA for large-scale systems remains infeasible due to the difficulty of simulating quantum dynamics. A pragmatic alternative is to empirically approximate the output distribution by sampling directly from the quantum annealer. For instance, in Ref45., a Gaussian approximation was constructed using empirical moments of samples obtained from the quantum annealer. Assessing the accuracy of such approximations is crucial for validating QAEMCMC in real-device settings. In our simulation, we assumed a closed quantum system. Since the quantum annealer is subject to device-specific imperfections, such as thermal effects, noise, and decoherence44, which introduce additional sampling bias, the empirical distribution deviates from the ideal one. Further studies are needed to quantify how these effects alter the sampling behavior and convergence properties of QAEMCMC.

Finally, the expected runtime of QAEMCMC in practical settings warrants further investigation. While a single annealing run on a D-Wave quantum annealer can yield a low-energy configuration in approximately 20 microseconds, the overall runtime of QAEMCMC depends on how accurately the output distribution is estimated, particularly in computing the M-H acceptance probability. A realistic runtime evaluation, including the cost of empirical distribution estimation under noisy sampling and limited readout, remains an open challenge. Additionally, the performance of QAEMCMC depends on the annealing time, which affects the shape of the output distribution. In our simulations, the annealing time was optimized to maximize the spectral gap from the transition probability. Since diagonalizing the transition probability is infeasible in practical situations, we need to explore the optimization criteria to decide the hyperparameters, such as the acceptance probability25. These considerations outline key challenges for practical deployment and guide future research.