Molchanov's Formula and Quantum Walks: A Probabilistic Approach

TL;DR

This paper introduces a probabilistic representation for quantum walks, adapting Molchanov's formula to both continuous and discrete time, enabling classical stochastic analysis of quantum dynamics.

Contribution

It extends Molchanov's formula to quantum walks, providing a new probabilistic framework for analyzing quantum systems through classical processes.

Findings

High fidelity in representing Hadamard walk evolution

Probabilistic approach offers new analytical tools for quantum systems

Framework applicable to multidimensional quantum walks

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.