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

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.