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

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.

Spectral Analysis of Quantum Bernoulli Noise

“…be quantum Bernoulli noise. In a recent paper [7] , it was shown that there exists a close link between operators of form…”

Multiparticle quantum walk–based error correction algorithm with two-lattice Bose–Hubbard model

“…As early as 2003, Shapira et al used computer simulation to study the influence of unitary noise on onedimensional quantum walk and showed the variation of probability distribution of quantum walk under the influence of noise [19]. Also, specific noise types are studied, such as the non-Markov continuous time quantum walk algorithm with dynamic noise [20] and the quantum Bernoulli central limit theorem in the quantum walk algorithm [21]. In 2015, Ambainis et al improved the potential barrier using the Grover algorithm, improved the amplitude of quantum walking coin state, and reduced the impact of noise during particle quantum walking [21,23].…”

“…Also, specific noise types are studied, such as the non-Markov continuous time quantum walk algorithm with dynamic noise [20] and the quantum Bernoulli central limit theorem in the quantum walk algorithm [21]. In 2015, Ambainis et al improved the potential barrier using the Grover algorithm, improved the amplitude of quantum walking coin state, and reduced the impact of noise during particle quantum walking [21,23]. In 2016, Wang et al proposed to construct time-varying quantum walking with infinite degrees of freedom by using quantum Bernoulli noise [24].…”