On the Convergence of Markov Chain Distribution within Quantum Walk Circuit Subspace

TL;DR

This work designs a Quantum MCMC (QMCMC) framework by embedding Metropolis–Hastings acceptance into a Discrete Quantum Walk circuit. By introducing dedicated gates (TRIAL, DISC, C GROUP, SHIFT) and registers for actions, positions, trials, acceptance, and coin flips, the circuit enables sampling from a target distribution $f$ via quantum superposition, with convergence demonstrated for Gaussian and Gaussian‑mixture targets. The authors show that increasing the move space per iteration via a larger action register $|\mathcal{H}_a|$ can substantially speed up convergence, at the cost of more qubits, and compare QMCMC to classical MH, noting faithful target reproduction but without a universal superiority proof. The work highlights both potential quantum advantages in sampling and the practical challenges of hardware‑level implementation, pointing to future theoretical convergence analyses, optimal resource tradeoffs, and gate‑level realizations as key directions.

Abstract

Markov Chain Monte Carlo (MCMC) methods are algorithms for sampling probability distributions, commonly applied to the Boltzmann distribution in physical and chemical models such as protein folding and the Ising model. These methods enable exploration of such systems by sampling their most probable states. However, sampling multidimensional and multimodal distributions with MCMC requires substantial computational resources, leading to the development of techniques aimed at improving sampling efficiency. In this context, quantum computing, with its potential to accelerate classical methods, emerges as a promising solution to the sampling problem. In this work, we present the design of a new circuit based on the Discrete Quantum Walk (DQW) algorithm to perform MCMC sampling over a desired distributions. Simulation results show convergence behavior in the superposition of the quantum register that encodes the target distribution. This design is further refined to increase convergence speed and, consequently, the scalability of the algorithm.

Figures (7)

• Figure 1: Left: sample chains generated using different acceptance functions. Right: the resulting distributions for each chain, compared against the true distribution ${\mathcal{N}}(0,1)$. The figure illustrates how ergodicity affects convergence to the target distribution.
• Figure 2: Comparison of the probability distributions over the state space $\Sigma$ after 100 iterations of a classical Random Walk (red dashed line) and a Discrete Quantum Walk (blue solid line), both starting from the initial position $x_0 = 0$ with equal probability ($p = 0.5$) of moving left or right. The quantum version exhibits faster spatial spread and higher variance than its classical counterpart.
• Figure 3: Quantum circuit for the proposed QMCMC algorithm. The circuit is composed of four main operations: TRIAL, DISC, C GROUP (a collection of controlled C gates) and SHIFT. Gates marked with the symbol $\dagger$ represent the inverses of their respective operations and are included to enable the reversibility and iterability of the process.
• Figure 4: Simulation results illustrating the circuit's ability to approximate various target distributions. Each subplot contains: a heatmap representing the evolution of the quantum state's probability distribution over successive iterations (columns), where yellow indicates low probability and blue indicates high probability and a plot comparing the expected discretized distribution (dashed line) with the obtained distribution from the quantum circuit (solid line). The function being approximated and the size of the position register $\ket{x}$ are specified in each subplot caption. It can be observed that the circuit successfully approximates the shape of the target function $f$, which in these experiments corresponds to individual and mixtures of gaussians. Increasing the size of the register $\ket{x}$ enhances the resolution of the approximation, although it also leads to a higher number of iterations required to reach convergence.
• Figure 5: Impact of the size of the $\ket{\text{acc}}$ register (which encodes the acceptance probability) on the circuit's ability to approximate the target distribution $\mathcal{N}(0; 1)$. Subfigure (a) uses a coarse discretization (4 intervals), (b) uses 8 intervals, and (c) presents a theoretical non-discretized case. The plots show how increasing the resolution of acceptance probability improves fidelity to the expected distribution.