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arXiv:0911.4898

The effect of decoherence on mixing time in continuous-time quantum walks on one-dimension regular networks

S. Salimi, Roya Radgohar

Published Nov 25, 2009• 0 citations
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Abstract

In this paper, we study decoherence in continuous-time quantum walks (CTQWs) on one-dimension regular networks. For this purpose, we assume that every node is represented by a quantum dot continuously monitored by an individual point contact(Gurvitz's model). This measuring process induces decoherence. We focus on small rates of decoherence and then obtain the mixing time bound of the CTQWs on one-dimension regular network which its distance parameter is $l\geq 2$. Our results show that the mixing time is inversely proportional to rate of decoherence which is in agreement with the mentioned results for cycles in \cite{FST,VKR}. Also, the same result is provided in \cite{SSRR} for long-range interacting cycles. Moreover, we find that this quantity is independent of distance parameter $l(l\geq 2)$ and that the small values of decoherence make short the mixing time on these networks.

DOI
10.48550/arxiv.0911.4898

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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.
In this paper, we study mixing and large decoherence in continuous-time quantum walks on one dimensional regular networks, which are constructed by connecting each node to its 2l2l2l nearest neighbors(lll on either side). In our investigation, the nodes of network are represented by a set of identical tunnel-coupled quantum dots in which decoherence is induced by continuous monitoring of each quantum dot with nearby point contact detector. To formulate the decoherent CTQWs, we use Gurvitz model and then calculate probability distribution and the bounds of instantaneous and average mixing times. We show that the mixing times are linearly proportional to the decoherence rate. Moreover, adding links to cycle network, in appearance of large decoherence, decreases the mixing times.
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.
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