analiza aiqbench
This research establishes a rigorous mathematical link between p-adic Schrödinger equations and continuous-time quantum Markov chains, demonstrating that their limiting distributions significantly outperform classical counterparts. By leveraging unitary evolution at the probability amplitude level rather than probabilities alone, these models offer novel pathways for advancing fundamental quantum algorithms without relying on standard graph adjacency matrices. The study provides essential theoretical comparisons between p-adic heat equations and Schrödinger dynamics through both analytical solutions and numerical simulations of limiting distributions.
Continuous-Time Quantum Markov Chains And Discretizations Of p-Adic Schr\"odinger Equations: Comparisons And Simulations
arXiv:2508.06712v5 Announce Type: replace Abstract: The continuous-time quantum walks (CTQWs) are a fundamental tool in the development of quantum algorithms. Recently, it was shown that discretizations of p-adic Schr\"odinger equations give rise to continuous-time quantum Markov chains (CTQMCs); this type of Markov chain includes the CTQWs constructed using adjacency matrices of graphs as a particular case. Unlike classical continuous-time Markov chains, the CTQMCs studied here do not satisfy the classical Chapman-Kolmogorov equation; this is a genuine quantum effect, due to the fact that the underlying unitary evolution composes at the level of probability amplitudes rather than probabilities, and we make precise the sense in which the term ``quantum Markov chain'' is used. In this paper, we study a large class of p-adic Schr\"odinger equations and the associated CTQMCs by comparing them with p-adic heat equations and the associated continuous-time Markov chains (CTMCs). The comparison is done by a mathematical study of the mentioned equations, which requires, for instance, solving the initial value problems attached to the mentioned equations, and through numerical simulations. We conducted multiple simulations, including numerical approximations of the limiting distribution. Our simulations show that the limiting distribution of quantum Markov chains is greater than the stationary probability of their classical counterparts, for a large class of CTQMCs.
Abstract:The continuous-time quantum walks (CTQWs) are a fundamental tool in the development of quantum algorithms. Recently, it was shown that discretizations of p-adic Schrödinger equations give rise to continuous-time quantum Markov chains (CTQMCs); this type of Markov chain includes the CTQWs constructed using adjacency matrices of graphs as a particular case. Unlike classical continuous-time Markov chains, the CTQMCs studied here do not satisfy the classical Chapman-Kolmogorov equation; this is a genuine quantum effect, due to the fact that the underlying unitary evolution composes at the level of probability amplitudes rather than probabilities, and we make precise the sense in which the term ``quantum Markov chain'' is used. In this paper, we study a large class of p-adic Schrödinger equations and the associated CTQMCs by comparing them with p-adic heat equations and the associated continuous-time Markov chains (CTMCs). The comparison is done by a mathematical study of the mentioned equations, which requires, for instance, solving the initial value problems attached to the mentioned equations, and through numerical simulations. We conducted multiple simulations, including numerical approximations of the limiting distribution. Our simulations show that the limiting distribution of quantum Markov chains is greater than the stationary probability of their classical counterparts, for a large class of CTQMCs.
arXiv:2508.06712 [quant-ph]
(or arXiv:2508.06712v5 [quant-ph] for this version)
From: W. A. Zuniga-Galindo [view email] [v1] Fri, 8 Aug 2025 21:28:18 UTC (635 KB) [v2] Fri, 16 Jan 2026 01:35:16 UTC (641 KB) [v3] Wed, 25 Feb 2026 18:04:43 UTC (642 KB) [v4] Wed, 12 Aug 2026 21:59:23 UTC (646 KB) [v5] Tue, 18 Aug 2026 21:28:06 UTC (646 KB)
Text integral sub licența deschisă a publicației · 264 cuvinte
Subiecte conexe
Cost-augmented Schr\"odinger bridges on graphs are exactly solvable: a Feynman-Kac tilt replaces learned control
arXiv · 2 oct. 2026
Top Photonic Quantum Computing Companies
Quantum Zeitgeist · 27 mai 2026
p-Adic Dirac Equations, Continuous-Time Quantum Walks, and Quantum Networks
arXiv · 7 sept. 2026
On the quantum computational complexity of classical linear dynamics with geometrically local interactions: Dequantization and universality
Quantum Journal · 3 aug. 2026