2007/12/19 by Franklin de Lima Marquezino, F. L. Marquezino, R. Portugal +3 · 2 citations
Computer Science · Physics and Astronomy · #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum-Dot Cellular Automata #quant-ph
paper · pdf · doi:10.1103/physreva.77.042312
published as Phys. Rev. A 77, 042312 (2008) · REVTeX, 9 pages, with small corrections
arxiv created 2007/12/19 · openalex publication_date 2008/04/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(n log n). We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally nonuniform) distribution \ensuremathπ(x) in time O(n), and the stationary distribution is determined by the initial state of the walk. An explicit expression for \ensuremathπ(x) is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate p\ensuremath≈0.1. A similar effect was previously reported for a quantum walk on an N-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps in obtaining---and preserving---a uniform distribution over the 2n sites of the hypercube in the shortest possible time.