Asymptotic Dynamics of Coined Quantum Walks on Percolation Graphs

Quantum walks obey unitary dynamics: they form closed quantum systems. The system becomes open if the walk suffers from imperfections represented as missing links on the underlying basic graph structure, described by dynamical percolation. Openness of the system’s dynamics creates decoherence, leading to strong mixing. We present a method to analytically solve the asymptotic dynamics of coined, percolated quantum walks for a general graph structure. For the case of a circle and a linear graph we derive the explicit form of the asymptotic states. We find that a rich variety of asymptotic evolutions occur: not only the fully mixed state, but other stationary states; stable periodic and quasiperiodic oscillations can emerge, depending on the coin operator, the initial state, and the topology of the underlying graph.

Figure 1

Distance of the joint coin-position distributions from the uniform distribution $\mathcal{M}(p,r)\equiv \sum _{}^{}|p_i-r_i|$ for percolated quantum walks in dependence on the number of iterations. Initial states are parametrized according to Eq. (16). (a) Hadamard walk on the 7 cycle with $\theta =\pi /2$, $\varphi =-\pi /2$. Inset: quantum fidelity with respect to the totally mixed state. (b) The same walk as in (a) on the 8 cycle. The steady state here is not totally mixed. (c) 7-linear graph with $\theta =-\pi /2$, $\varphi =-\pi$. An asymptotic limit cycle occurs. (d) Special walk on the 8 cycle using the coin $C(\pi /2,\sqrt{2})$ [see Eq. (11)] with the same initial state as in (c). The asymptotic behavior becomes quasiperiodic. In all cases $p=0.5$; the points are connected for visual aid.

Figure 2

Position mixing time (circles) for a percolated Hadamard walk on the 7 cycle as a function of distance threshold $ϵ$. Initially, the walker was localized on a vertex with a totally mixed coin state. The line represents our analytic estimation Eq. (18).