Molchanov's Formula and Quantum Walks: A Probabilistic Approach
Hoang Vu
TL;DR
The paper addresses the problem of connecting quantum walk dynamics with classical stochastic processes by deriving probabilistic representations for both continuous-time and discrete-time quantum walks. It introduces Molchanov's probabilistic framework for CTQW on $\mathbb{Z}^d$ and, for DTQW on $\mathbb{Z}$, constructs Poisson-driven representations for coins such as $C=e^{i\lambda \sigma_2}$ and $C=e^{i\lambda_1\sigma_3}e^{i\lambda_2\sigma_2}e^{i\lambda_3\sigma_3}$. Key contributions include explicit formulas for the amplitudes, a unified probabilistic decomposition across coin types, and an efficient Monte Carlo simulation algorithm validated against the Hadamard walk. This probabilistic lens enables scalable simulation of high-dimensional quantum walks and opens analytical avenues to study quantum systems via classical stochastic methods.
Abstract
This paper establishes a robust link between quantum dynamics and classical ones by deriving probabilistic representation for both continuous time and discrete time quantum walks. We first adapt Molchanov formula, originally employed in the study of Schrodinger operators on multidimensional integer lattice, to characterize the evolution of continuous time quantum walks. Extending this framework, we develop a probabilistic method to represent discrete time quantum walks on an infinite integer line, bypassing the locality constraints that typically inhibit direct application of Molchanov formula. The validity of our representation is empirically confirmed through a benchmark analysis of the Hadamard walk, demonstrating high fidelity with traditional unitary evolution. Our results suggest that this probabilistic lens offer a powerful alternative for learning multidimensional quantum walks and provides new analytical pathways for investigating quantum systems via classical stochastic processes.