Decoherence in Quantum Walks on the Hypercube

The notion of a quantum random walk has emerged as an important element in the development of efficient quantum algorithms. In particular, it makes a dramatic appearance in the most efficient known algorithm for element distinctness [1]. The technique has also been applied to offer simple separations between quantum and classical query complexity [5]. The basic model has two natural variants, the continuous model of Childs, et al. [4], on which we will focus, and the discrete model introduced by Aharonov et al. [3]. We refer the reader to Szegedy’s [2] article for a more detailed discussion. In this continuous model, a walk on a graph G is determined by the timeevolution of the Schrödinger equation using kL as the Hamiltonian, where L is the Laplacian of the graph and k the jumping rate. In addition to being physically reasonable, this model has been successfully applied to some algorithmic problems as indicated above. As the quantum random walk is a simple, and yet still nontrivial, algorithmic ingredient, it appears to be a natural target for physical realization. An immediate question is to what extent the basic features of such quantum walks can be retained by a necessarily imperfect physical realization. In this article, we study the effects of a natural notion of decoherence on the hypercubic walk. Our notion of decoherence corresponds, roughly, to independent measurement “accidentally” taking place in each coordinate of the walk at a certain rate p. We discover that for values of p beneath a threshold depending on the energy of the system, the walk retains the basic features of the non-decohering walk; these features disappear beyond this threshold, where the behavior is analogous to the classical walk. Moore and Russell [6] analyzed both the discrete and the continuous quantum walk on a hypercube. In this article, we extend their results in the continuous case with the model of decoherence described above. In particular, we show that up to a certain rate of decoherence, both linear mixing times and hitting times still occur. Beyond the threshold, however, the walk behaves like the classical walk on the hypercube, exhibiting Θ(n log n) mixing times. As the rate of decoherence grows, mixing is retarded by the quantum Zeno effect. For a numerical analysis of the discrete case in some dimensions, see Kendon and Tregenna [7].