Symmetries and noise in quantum walk

Abstract

We study some discrete symmetries of unbiased (Hadamard) and biased quantum walk on a line, which are shown to hold even when the quantum walker is subjected to environmental effects. The noise models considered in order to account for these effects are the phase flip, bit flip and generalized amplitude damping channels. The numerical solutions are obtained by evolving the density matrix, but the persistence of the symmetries in the presence of noise is proved using the quantum trajectories approach. We also briefly extend these studies to quantum walk on a cycle. These investigations can be relevant to the implementation of quantum walks in various known physical systems. We discuss the implementation in the case of NMR quantum information processor and ultra cold atoms.

Symmetry-noise interplay in a quantum walk on ann-cycle

“…Experimental implementation of QW has been reported [5], and various other schemes have been proposed for its physical realization [6,7], which has motivated us to study QW in an open quantum system [8,9]: since environmental effects (noise) will necessarily tend to destroy the coherent superposition of states central to QW, thus transforming it to CRW. This paper studies QW on an n-cycle, which is subjected to the following noise processes: the phase flip (decoherence without dissipation) [10], and generalized amplitude damping channels (decoherence with dissipation) [8,11,12,13], which are of relevance to studies in quantum optics and condensed matter systems. The latter type of noise in the QW context has been studied by us in Ref.…”

Noise-enhanced quantum transport on a closed loop using quantum walks

“…The reason for this complex behaviour can be found in the fact that the noise changes the ballistic-like probability distribution of the particle over time into a Gaussian-like distribution around the initial position [34]. For short times (t = 10) and when sink is the nearest neighbour to the initial position, this increases the fraction of the amplitude staying close to the initial position and consequently increases the transport efficiency.…”

“…However, there is one major difference in that the extrema for the transport efficiency for the dephasing channel are achieved at P = 0.5 [34].…”

“…(18) is identical to a phase flip operation at each step of the quantum walk and the resulting probability distribution is identical to the noiseless evolution. This can be seen as a symmetry of the quantum walk [34] and indicates that using a dephasing noise channel a maximally mixed stateρ(t) is obtained for P = 0.5. Note that no such symmetry exists in the presence of depolarizing noise, which means that a maximally mixed stateρ(t) in Eq.…”

“…Quantum systems are not free from the environment noise and hence, looking into the effect of noise on the quantum transport process is very important forward. In this article we present the transport process using a simple form of the discrete-time quantum walk evolution of a closed loop on n− sites [34][35][36] and study the effect of noise on the transport efficiency with one of the sites as a sink.…”