Complete Characterization of Mixing Time for the Continuous Quantum Walk on the Hypercube with Markovian Decoherence Model

TL;DR

This paper thoroughly analyzes how Markovian decoherence affects the mixing times of quantum random walks on the hypercube, revealing conditions for classical and instantaneous mixing and emphasizing the importance of environmental interactions.

Contribution

It provides a complete characterization of the mixing behavior of hypercube quantum walks under a realistic decoherence model, including conditions for instantaneous mixing.

Findings

Classical mixing always occurs under the considered decoherence model.

Instantaneous mixing exists under specific conditions related to environmental interaction.

Mixing times are heavily influenced by the strength and nature of decoherence.

Abstract

The n-dimensional hypercube quantum random walk (QRW) is a particularily appealing example of a quantum walk because it has a natural implementation on a register on $n$ qubits. However, any real implementation will encounter decoherence effects due to interactions with uncontrollable degrees of freedom. We present a complete characterization of the mixing properties of the hypercube QRW under a physically relevant Markovian decoherence model. In the local decoherence model considered the non-unitary dynamics are modeled as a sum of projections on individual qubits to an arbitrary direction on the Bloch sphere. We prove that there is always classical (asymptotic) mixing in this model and specify the conditions under which instantaneous mixing \textit{always} exists. And we show that the latter mixing property, as well as the classical mixing time, depend heavily on the exact environmental interaction and its strength. Therefore, algorithmic applications of the QRW on the hypercube, if they intend to employ mixing properties, need to consider both the walk dynamics and the precise decoherence model.