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However, as the dimensionality and complexity of the target distribution increase, the convergence of these methods can slow considerably, demanding substantial computational resources. In this context, quantum computing emerges as a promising framework to address this limitation, with different quantum circuit constructions serving as quantum MCMC methods. In this work, we present a quantum circuit based on the discrete quantum walk (DQW) algorithm that implements the algorithmic steps of the Metropolis\u2013Hastings method, encoding the target distribution in a position quantum register. The circuit incorporates auxiliary registers to compute and discretize the acceptance probability, and uses controlled operations to modulate the walker\u2019s evolution accordingly. Simulation results show that the marginal probability distribution encoded in the position register converges asymptotically toward the discretized target distribution for both single and multimodal distributions. We further analyze the effect of circuit parameters on convergence behavior and approximation quality, refining the construction by enabling the evaluation of multiple candidate moves per iteration, therefore reducing the number of iterations required for convergence within our implementation. Finally, we provide a qualitative comparison of our approach with related quantum MCMC methods, namely Szegedy\u2019s quantum walk and QAOA, and discuss structural differences as well as the potential incorporation of adaptive proposal techniques within the proposed circuit framework.<\/jats:p>","DOI":"10.1007\/s11128-026-05158-5","type":"journal-article","created":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T06:15:21Z","timestamp":1777270521000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the convergence of Markov chain distribution within quantum walk circuit subspace"],"prefix":"10.1007","volume":"25","author":[{"given":"Aingeru","family":"Ramos","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jose A.","family":"Pascual","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Javier","family":"Navaridas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ivan","family":"Coluzza","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,27]]},"reference":[{"key":"5158_CR1","volume-title":"Understanding Molecular Simulation: From Algorithms to Applications","author":"D Frenkel","year":"1996","unstructured":"Frenkel, D., Smit, B.: Understanding Molecular Simulation: From Algorithms to Applications, 1st edn. 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