Quantum Walk in Terms of Quantum Bernoulli Noise and Quantum Central Limit Theorem for Quantum Bernoulli Noise

Caishi Wang,Suling Ren,Ce Wang,Yuling Tang

doi:10.1155/2018/2507265

Caishi Wang, Suling Ren + Show 2 more

Open Access

https://doi.org/10.1155/2018/2507265

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Abstract

As a unitary quantum walk with infinitely many internal degrees of freedom, the quantum walk in terms of quantum Bernoulli noise (recently introduced by Wang and Ye) shows a rather classical asymptotic behavior, which is quite different from the case of the usual quantum walks with a finite number of internal degrees of freedom. In this paper, we further examine the structure of the walk. By using the Fourier transform on the state space of the walk, we obtain a formula that links the moments of the walk’s probability distributions directly with annihilation and creation operators on Bernoulli functionals. We also prove some other results on the structure of the walk. Finally, as an application of these results, we establish a quantum central limit theorem for the annihilation and creation operators themselves.

Highlights

Quantum walks are quantum analogs of the classical random walk, which have found wide application in quantum information, quantum computing, and many other fields [1,2,3]
Similar properties have been found for continuous-time quantum walks with a finite number of internal degrees of freedom [11]
Based walk has a Gaussian limit distribution with scaling speed √n, which is in striking contrast with the case of the usual discrete-time quantum walks with a finite number of internal degrees of freedom

Summary

Introduction

Quantum walks are quantum analogs of the classical random walk, which have found wide application in quantum information, quantum computing, and many other fields [1,2,3]. Konno [10] proved that, for localized initial states, a discrete-time quantum walk on the line with a finite number of internal degrees of freedom usually has a limit distribution with scaling speed n, which is far from being Gaussian. Based walk has a Gaussian limit distribution with scaling speed √n, which is in striking contrast with the case of the usual discrete-time quantum walks with a finite number of internal degrees of freedom. Machida [6] has found that, for a very particular nonlocalized initial state, a discrete-time quantum walk on the line with 2 internal degrees of freedom can generate a Gaussian limit distribution with scaling speed n. Letters like j, k, and n stand for nonnegative integers, namely, elements of N

Quantum Bernoulli Noise

QBN-Based Walk

Structure Property of QBN-Based Walk

Quantum Central Limit Theorem

Conclusion