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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum-classical crossover in quantum walks mixing time

Leonid Evgenevich Fedichkin, Fedor Pavlovich Meshchaninov

Published Dec 30, 2016• 5 citations
Publisher

Abstract

Continuous-time quantum walks on a cycle graph are analyzed in the presence of decoherence. The electron evolution in the nanostructure composed of ring of tunnel-coupled semiconductor quantum dots is considered as realistic model for quantum walks on a cycle graph. Quantum-classical crossover from fast quantum dynamics to classical diffusive behavior is observed as decoherence rate parameter grows. The dependence of transition decoherence rate and mixing time versus size of the graph are studied both analytically and numerically.

DOI
10.1117/12.2267246

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Continuous-time quantum walks on a cycle graph are analyzed in the presence of decoherence. The electron evolution in the nanostructure composed of ring of tunnel-coupled semiconductor quantum dots is considered as realistic model for quantum walks on a cycle graph. Quantum-classical crossover from fast quantum dynamics to classical diffusive behavior is observed as decoherence rate parameter grows. The dependence of transition decoherence rate and mixing time versus size of the graph are studied both analytically and numerically.
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We prove analytical results showing that decoherence can be useful for mixing time in a continuous-time quantum walk on finite cycles. This complements the numerical observations by Kendon and Tregenna (Physical Review A 67 (2003), 042315) of a similar phenomenon for discrete-time quantum walks. Our analytical treatment of continuous-time quantum walks includes a continuous monitoring of all vertices that induces the decoherence process. We identify the dynamics of the probability distribution and observe how mixing times undergo the transition from quantum to classical behavior as our decoherence parameter grows from zero to infinity. Our results show that, for small rates of decoherence, the mixing time improves linearly with decoherence, whereas for large rates of decoherence, the mixing time deteriorates linearly towards the classical limit. In the middle region of decoherence rates, our numerical data confirms the existence of a unique optimal rate for which the mixing time is minimized.
In this paper, we consider decoherence in continuous-time quantum walks on long-range interacting cycles (LRICs), which are the extensions of the cycle graphs. For this purpose, we use Gurvitz's model and assume that every node is monitored by the corresponding point-contact induced by the decoherence process. Then, we focus on large rates of decoherence and calculate the probability distribution analytically and obtain the lower and upper bounds of the mixing time. Our results prove that the mixing time is proportional to the rate of decoherence and the inverse of the square of the distance parameter ( m ). This shows that the mixing time decreases with increasing range of interaction. Also, what we obtain for m = 0 is in agreement with Fedichkin, Solenov and Tamon's results [48] for cycle, and we see that the mixing time of CTQWs on cycle improves with adding interacting edges.
We study the effect of small decoherence in continuous-time quantum walks on long-range interacting cycles, which are constructed by connecting all the two nodes of distance m on the cycle graph. In our investigation, each node is continuously monitored by an individual point contact, which induces the decoherence process. We obtain the analytical probability distribution and the mixing time upper bound. Our results show that, for small rates of decoherence, the mixing time upper bound is independent of distance parameter m and is proportional to inverse of decoherence rate.