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arXiv:1204.6149 · Physical Review Letters

Asymptotic Dynamics of Coined Quantum Walks on Percolation Graphs

Bálint Kollár, T. Kiss, Jaroslav Novotný, Igor Jex

Published Jun 5, 2012• 54 citations
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Abstract

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.

DOI
10.1103/physrevlett.108.230505

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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.
Randomly breaking connections in a graph alters its transport properties, a model used to describe percolation. In the case of quantum walks, dynamic percolation graphs represent a special type of imperfections, where the connections appear and disappear randomly in each step during the time evolution. The resulting open system dynamics is hard to treat numerically in general. We shortly review the literature on this problem. We then present our method to solve the evolution on finite percolation graphs in the long time limit, applying the asymptotic methods concerning random unitary maps. We work out the case of one-dimensional chains in detail and provide a concrete, step-by-step numerical example in order to give more insight into the possible asymptotic behavior. The results about the case of the two-dimensional integer lattice are summarized, focusing on the Grover-type coin operator.
Quantum walks on graphs can model physical processes and serve as efficient tools in quantum information theory. Once we admit random variations in the connectivity of the underlying graph, we arrive at the problem of percolation, where the long-time behaviour appears untreatable with direct numerical methods. We develop novel analytic methods based on the theory of random unitary operations which help us to determine explicitly the asymptotic dynamics of quantum walks on two-dimensional finite integer lattices with percolation. Based on this theory, we find new unexpected features of percolated walks like asymptotic position inhomogeneity or special directional symmetry breaking.
Quantum walks, both discrete (coined) and continuous time, form the basis of several quantum algorithms and have been used to model processes such as transport in spin chains and quantum chemistry. The enhanced spreading and mixing properties of quantum walks compared with their classical counterparts have been well studied on regular structures and also shown to be sensitive to defects and imperfections in the lattice. As a simple example of a disordered system, we consider percolation lattices, in which edges or sites are randomly missing, interrupting the progress of the quantum walk. We use numerical simulation to study the properties of coined quantum walks on these percolation lattices in one and two dimensions. In one dimension (the line), we introduce a simple notion of quantum tunnelling and determine how this affects the properties of the quantum walk as it spreads. On two-dimensional percolation lattices, we show how the spreading rate varies from linear in the number of steps down to zero as the percolation probability decreases towards the critical point. This provides an example of fractional scaling in quantum-walk dynamics.
We present an alternative definition of discrete-time coined quantum walks convenient for capturing a rather broad spectrum of a 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 nonstationary 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.
Abstract Asymptotic dynamics of a Hadamard walk of two non-interacting quantum particles on a dynamically percolated finite line or a circle is investigated. We construct a basis of the attractor space of the corresponding random-unitary dynamics and prove the completeness of our solution. In comparison to the one-particle case, the structure of the attractor space is much more complex, resulting in intriguing asymptotic dynamics. General results are illustrated on two examples. First, for circles of length not divisible by 4 the boundary conditions reduces the number of attractors considerably, allowing for fully analytic solution. Second, we investigate line of length 4 and determine the asymptotic cycle of reduced coin states and position distributions, focusing on the correlations between the two particles. Our results show that a random unitary evolution, which is a combination of quantum dynamics and a classical stochasticity, leads to correlations between initially uncorrelated particles. This is not possible for purely unitary evolution of non-interacting quantum particles. The shared dynamically percolated graph can thus be considered as a weak form of interaction.
Open quantum walks often lead to a classical asymptotic behavior. Here we look for a simple open quantum walk whose asymptotic behavior can be nonclassical. We consider a discrete-time quantum walk on an nnn-cycle subject to a random coin-dependent phase shift at a single position. This finite system, whose evolution is described by only two Kraus operators, can exhibit all kinds of asymptotic behavior observable in quantum Markov chains: It either evolves towards a maximally mixed state, or partially mixed state, or tends to an oscillatory motion on an asymptotic orbit. We find that the asymptotic orbits do not have a product structure; therefore, the corresponding states can manifest entanglement between the position and the coin degrees of freedom, even if the system started in a product state.