Percolated quantum walks with a general shift operator: From trapping to transport

Abstract

We present a generalized definition of discrete-time quantum walks convenient for capturing a rather broad spectrum of walker's behavior on arbitrary graphs. It includes and covers both: the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find, that for these walks with cyclic shift operators the existence of an edge-3-coloring of the graph allows for non-stationary asymptotic behavior of the walk. For different shift operators the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory we treat a single excitation transport on a percolated cube.

Counterintuitive role of geometry in transport by quantum walks

“…As these changes are not controlled, the resulting open dynamics involves decoherence. Second, there is a general theory allowing to construct trapped states for Grover PCQWs on a broad class of graphs [17]. Third, an analytical treatment of PCQWs provides also significant insight into transport properties of CQWs.…”

Quantum walk transport on carbon nanotube structures

“…Let us introduce a general model of walker's dynamics refined for the case of carbon nanotubes. We adopt the definition of a coined quantum walk presented in [33] as it is able to capture simultaneously the complexity of carbon nanotube structures and random disturbances of the underlying medium (percolation).…”

“…states of CQWs and PCQWs. Their general structure as well as the effect of the order of C and R is discussed in [33]. Finally, at the end of each step the part of the walker's wave function which reached the sink is taken away.…”

“…The complement p = 1 − q is the trapping probability. It is due to the presence of trapped states whose overlap with the walker's initial state determines the ATP [32,33]. The structure of trapped states is determined not only by chosen nanotube parameters (its length and chirality) but also by the choice of the sink.…”

“…All trapped states correspond to eigenvalue −1 and they form a subspace of the dimension N = 2#V − #E. A basis of this subspace can be constructed from four types of trapped states shown in . For detailed description of these states see [33]. The construction of the basis relies on faces, which are defined in planar graphs.…”