Decoherence in quantum walks on the hypercube

Abstract

We study a natural notion of decoherence on quantum random walks over the hypercube. We prove that this model possesses a decoherence threshold beneath which the essential properties of the hypercubic quantum walk, such as linear mixing times, are preserved. Beyond the threshold, we prove that the walks behave like their classical counterparts.

Decoherence in quantum walks – a review

“…At that point we can apply the usual projectors Π 0 and Π 1 to determine the probabilities of measuring 0 or 1 as a function of time. The result (see Alagić and Russell (2005) for details) is…”

“…The difference is that the continuous-time quantum walk hits exactly, while the discrete-time quantum walk hits with a probability that is less than one: see Kempe (2003b;2005) for details. For a fixed p > 4k, Alagić and Russell (2005) shows that the walk behaves much like the classical walk on the hypercube, the measurement distribution of the walk converges to the uniform distribution in time M( ) = Θ(n log n), just as in the classical case.…”

“…They augment this with numerical studies showing a minimum mixing time for intermediate decoherence rates. Alagić and Russell (2005) gives a solution for the decoherent walk on the hypercube for all rates of decoherence. These results will be discussed in separate sections below, but we will first set up a general model for decoherence in continuous-time quantum walks analogous to the basic model used for discrete-time walks.…”

“…We thus have a large number of comparisons to make between discrete and continuous-time quantum walks, and classical random walks. Alagić and Russell (2005) provides a complete solution to the dynamics of the continuous-time quantum walk on the hypercube subject to decoherence. We will sketch their method of solution, then discuss their results.…”

From Dirac to Diffusion: Decoherence in Quantum Lattice Gases

“…In general, coin decoherence produces a transition from behavior characteristic of a quantum walk in which the standard deviation of the walk grows linearly in time to behavior characteristic of a classical walk in which the standard deviation varies as the square root of time [30,[30][31][32][33][34][35]. Previous decoherence models for quantum walks considered a scattering step which is a unitary action with probability 1 − p, and a unitary action followed by a projective measurement with probability p [30][31][32][33]. Models in which measurements on the coin yield less than total information have also been considered [36].…”