Quantum stochastic walks: A generalization of classical random walks and quantum walks

Tristan Anthony Dobrian

This is a comprehensive and self-contained introduction to the math that underlies classical random walks and quantum random walk simulations.

James Daniel Whitfield, César A. Rodríguez-Rosario, Alán Aspuru‐Guzik

We introduce the quantum stochastic walk (QSW), which determines the evolution of a generalized quantum-mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical, and quantum-stochastic transitions from a vertex as defined by its connectivity. We show how the family of possible QSWs encompasses both the classical random walk (CRW) and the quantum walk (QW) as special cases but also includes more general probability distributions. As an example, we study the QSW on a line and the glued tree of depth three to observe the behavior of the QW-to-CRW transition.

Gregor Tanner

Julia Kempe

V. S. Lamego, G. T. C. Cruz, Dennis Lima +2 more

Quantum walks (QW) offer a speed-up advantage over random walks in quantum search applications. We present an experimental study of the transition from quantum-to-classical random walk using an emulation of the decoherence process for polarization qubits that exploits maximally non-separable spin-orbit modes of an intense laser beam for the first, to the best of our knowledge, time. We are able to continuously control the input polarization mode in an all-optical quantum walk circuit to observe transitions associated with quantum, quantum stochastic, and classical random walk distributions. The results are in agreement with theoretical expectations.

Yuanbao Kang, Caishi Wang