Quantum walks on embedded hypercubes: Nonsymmetric and nonlocal cases

TL;DR

This paper demonstrates that exponential quantum speed-up in hitting times persists even when hypercubes are embedded in nonsymmetric or nonlocal graphs, and introduces a mapping to analyze high-dimensional quantum walks efficiently.

Contribution

It shows that quantum speed-up is robust against symmetry and locality disruptions and presents a new mapping technique for efficient analysis of high-dimensional quantum walks.

Findings

Exponential speed-up survives nonsymmetric embeddings.

Speed-up persists under nonlocal embeddings and edge removal.

A new mapping reduces Hilbert space size for analysis.

Abstract

The expected hitting time of discrete quantum walks on a hypercube (HC) is numerically known to be exponentially shorter than that of their classical analogs in terms of the scaling with the HC dimension. Recent numerical analyses illustrated that this scaling exists not only on the bare HC, but also when the HC graph is symmetrically and locally embedded into larger graphs. The present work investigates the necessity of symmetry and locality for the speed-up by considering embeddings that are nonsymmetric or nonlocal. We provide numerical evidence that the exponential speed-up survives also in these cases. Furthermore, our numerical simulations demonstrate that removing a single edge from the HC also does not destroy the exponential speed-up. In the nonlocal embedding of the HC we encounter dark states, which we analyze. We provide a general and detailed presentation of the mapping that reduces the exponentially large Hilbert space of the quantum walk to an effective subspace of polynomial scaling. This mapping is our essential tool to numerically study quantum walks in such high-dimensional structures.