Mixing times in quantum walks on the hypercube

Abstract

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 non-uniform) distribution π(x) in time O(n) and the stationary distribution is determined by the initial state of the walk. An explicit expression for π(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 ≈ 0.1. A similar effect was previously reported for the QW on the N -cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps to obtain-and preserve-a uniform distribution over the 2 n sites of the hypercube in the shortest possible time.

Physical Implementation of Quantum Walks

“…It can be shown that for an unbiased Hadamard walk on the line, the walk is almost uniformly distributed over the interval h t= p 2; t= p 2 i after t time steps (Nayak and Vishwan 2000). Extending this study, Marquezino and Portugal (2008) showed that the asymptotic distribution of a discrete-time quantum walk on the hypercube is in general not uniform, and characterized the average mixing time to this distribution. Moore and Russell (2002) analyzed both continuousand discrete-time quantum walks on the hypercube and showed that in both cases, the walk has an instantaneous mixing time at n =4.…”

“…More specifically, they observed a highly uniform distribution on the line, a very fast mixing time on the cycle, and more reliable hitting times across the hypercube. Likewise Marquezino and Portugal (2008) established that for a walk on a hypercube a controlled amount of decoherence, due to randomly breaking links between connected sites, helps in obtaining and preserving a uniform distribution in the shortest possible time. Maloyer and Kendon (2007) showed that the mixing time of a discrete-time quantum walk on an N -cycle may be reduced by allowing for some decoherence from repeated measurements, provided that those measurements affect the position of the walk.…”

Quantum Walks on Sierpinski Gaskets

“…We note that it decays approximately as 1=T not only for the Sierpinski gasket but also for twodimensional lattices 11 and hypercubes. 12 This numerical result can be explained analytically. Expanding the quantum walk state in the eigenbasis that diagonalizes the evolution operator, one can obtain an explicit expression for jj" pðT ; x; yÞ À ðx; yÞjj for any initial condition.…”

“…[6][7][8][9] Quantum walks have been analyzed on two-dimensional square lattices, 10,11 and on the hypercube. 12 A spatial search using the discrete-time quantum walk model has been undertaken on the Sierpinski gasket, 13 and on the Hanoi network of degree 3. 14 A quantum walk on the dual Sierpinski gasket using the continuous-time quantum walk model has been analyzed by Agliari et al 15 In this paper, we focus our attention on discrete-time quantum walks on the Sierpinski gasket.…”

Incoherent tunneling effects in a one-dimensional quantum walk

“…Many aspects of QW have been studied theoretically, for example some researches focus on the network over which the walks takes place. In this area, the QW on a line has been well studied [11][12][13][14][15][16], but other topologies such as cycles [17,18], two-dimensional lattices [19][20][21][22], or n-dimensional hypercubes [23,24] have also been investigated.…”