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arXiv:1106.2036 · The European Physical Journal D

Quantum walk with jumps

Hynek Lavička, Václav Potoček, T. Kiss, E. Lutz, I. Jex

Published Jul 27, 2011• 18 citations
arXivHTMLPDF
DOI
10.1140/epjd/e2011-20138-8

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We define the hitting time for a model of continuous-time open quantum walks in terms of quantum jumps. Our starting point is a master equation in Lindblad form, which can be taken as the quantum analogue of the rate equation for a classical continuous-time Markov chain. The quantum jump method is well known in the quantum optics community and has also been applied to simulate open quantum walks in discrete time. This method however, is well-suited to continuous-time problems. It is shown here that a continuous-time hitting problem is amenable to analysis via quantum jumps: The hitting time can be defined as the time of the first jump. Using this fact, we derive the distribution of hitting times and explicit exressions for its statistical moments. Simple examples are considered to illustrate the final results. We then show that the hitting statistics obtained via quantum jumps is consistent with a previous definition for a measured walk in discrete time [Phys. Rev. A 73, 032341 (2006)] (when generalised to allow for non-unitary evolution and in the limit of small time steps). A caveat of the quantum-jump approach is that it relies on the final state (the state which we want to hit) to share only incoherent edges with other vertices in the graph. We propose a simple remedy to restore the applicability of quantum jumps when this is not the case and show that the hitting-time statistics will again converge to that obtained from the measured discrete walk in appropriate limits.

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We propose a definition of the hitting time for a model of continuous-time quantum walks based on quantum jumps and derive explicit expressions for its hitting statistics. This method works naturally for open quantum walks where incoherent transitions occur but can also be modified to cover the unitary case when incoherent transitions are absent.

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Compendium of Quantum Physics•Jan 1, 2009•1 citation
We analyze a set of discrete-time quantum walks for which the displacements on a chain follow binary aperiodic jumps according to three paradigmatic sequences: Fibonacci, Thue-Morse, and Rudin-Shapiro. We use a generalized Hadamard coin, C[over ̂]_{H}, as well as a generalized Fourier coin, C[over ̂]_{K}. We verify the QW experiences a slowdown of the wave packet spreading, σ^{2}(t)∼t^{α}, by the aperiodic jumps whose exponent, α, depends on the type of aperiodicity. Additional aperiodicity-induced effects also emerge, namely, (1) while the superdiffusive regime (1<α<2) is predominant, α displays an unusual sensibility with the type of coin operator where the more pronounced differences emerge for the Rudin-Shapiro and random protocols and (2) even though the angle θ of the coin operator is homogeneous in space and time, there is a nonmonotonic dependence of α with θ. Fingerprints of the aperiodicity in the hoppings are also found when distributional measures such as the Shannon and von Neumann entropies, the Inverse Participation Ratio, the Jensen-Shannon dissimilarity, and the kurtosis are computed, which allow assessing informational and delocalization features arising from these protocols and understanding the impact of linear and nonlinear correlations of the jump sequence in a quantum walk as well. Finally, we argue the spin-lattice entanglement is enhanced by aperiodic jumps.
We define the hitting time for a model of continuous-time open quantum walks in terms of quantum jumps. Our starting point is a master equation in Lindblad form, which can be taken as the quantum analogue of the rate equation for a classical continuous-time Markov chain. The quantum jump method is well known in the quantum optics community and has also been applied to simulate open quantum walks in discrete time. This method however, is well-suited to continuous-time problems. It is shown here that a continuous-time hitting problem is amenable to analysis via quantum jumps: The hitting time can be defined as the time of the first jump. Using this fact, we derive the distribution of hitting times and explicit exressions for its statistical moments. Simple examples are considered to illustrate the final results. We then show that the hitting statistics obtained via quantum jumps is consistent with a previous definition for a measured walk in discrete time [Phys. Rev. A 73, 032341 (2006)] (when generalised to allow for non-unitary evolution and in the limit of small time steps). A caveat of the quantum-jump approach is that it relies on the final state (the state which we want to hit) to share only incoherent edges with other vertices in the graph. We propose a simple remedy to restore the applicability of quantum jumps when this is not the case and show that the hitting-time statistics will again converge to that obtained from the measured discrete walk in appropriate limits.