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Modern Physics Letters A

Decoherence in two-dimensional quantum walks with two- and four-state coins

Yu‐Guang Yang, Xixi Wang, Jian Li, Dan Li, Yi‐Hua Zhou, Wei‐Min Shi

Published Sep 28, 2021• 2 citations
Publisher

Abstract

Two-dimensional quantum walks using a two-state coin have simpler experimental implementation than two-dimensional quantum walks using a four-state coin. However, decoherence occurs inevitably during the evolution of quantum walks due to the coupling between the quantum systems and their environment. Thus, it is interesting to investigate the robustness against decoherence for two- and four-state two-dimensional quantum walks. Here, we investigate the effects of the decoherence on two- and four-state two-dimensional quantum walks produced by the broken-link-type noise and compare their robustness against the broken-link-type noise. Specifically, we analyze the quantum correlation between the two spatial dimensions x and y by using measurement-induced disturbance for the two-state quantum walks, i.e. the alternate walk and the Pauli walk, and the four-state quantum walks, i.e. the Grover, Hadamard and Fourier walks, respectively. Our analysis shows that the two-state walks are more robust against the broken-link-type noise than the four-state walks.

DOI
10.1142/s0217732321502102

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Two-dimensional quantum walks using a two-state coin have simpler experimental implementation than two-dimensional quantum walks using a four-state coin. However, decoherence occurs inevitably during the evolution of quantum walks due to the coupling between the quantum systems and their environment. Thus, it is interesting to investigate the robustness against decoherence for two- and four-state two-dimensional quantum walks. Here, we investigate the effects of the decoherence on two- and four-state two-dimensional quantum walks produced by the broken-link-type noise and compare their robustness against the broken-link-type noise. Specifically, we analyze the quantum correlation between the two spatial dimensions x and y by using measurement-induced disturbance for the two-state quantum walks, i.e. the alternate walk and the Pauli walk, and the four-state quantum walks, i.e. the Grover, Hadamard and Fourier walks, respectively. Our analysis shows that the two-state walks are more robust against the broken-link-type noise than the four-state walks.
We study the decoherence effects originating from state flipping and depolarization for two-dimensional discrete-time quantum walks using four- and two-state particles. By comparing the quantum correlations between the two spatial ( x − y ) degrees of freedom using measurement-induced disturbance, we show that the two schemes using a two-state particle are more robust against decoherence than the Grover walk, which uses a four-state particle. We also show that the symmetries which hold for two-state quantum walks break down for the Grover walk, adding to the various other advantages of using two-state rather than four-state particles.
We analyze the decoherence in quantum walks in two-dimensional lattices generated by broken-link-type noise. In this type of decoherence, the links of the lattice are randomly broken with some given constant probability. We obtain the evolution equation for a quantum walker moving on two-dimensional (2D) lattices subject to this noise, and we point out how to generalize for lattices in more dimensions. In the nonsymmetric case, when the probability of breaking links in one direction is different from the probability in the perpendicular direction, we have obtained a nontrivial result. If one fixes the link-breaking probability in one direction, and gradually increases the probability in the other direction from 0 to 1, the decoherence initially increases until it reaches a maximum value, and then it decreases. This means that, in some cases, one can increase the noise level and still obtain more coherence. Physically, this can be explained as a transition from a decoherent 2D walk to a coherent 1D walk.
We investigate the evolution of the [Formula: see text] norm coherence in two-dimensional quantum walk on lattices for Fourier, Grover and Hadamard coins. Our study shows that the coherence of both the whole system and the position subsystem increases quadratically with the number of steps. We also analyze the influence of three kinds of noises, i.e. broken links, lattice congestion and lattice traps, on the two-dimensional quantum walk. The result shows that the increase of coherence with the number of steps gradually inclines to be linear as the probabilities of broken links and lattice congestion increase. In the quantum walk on trapped lattices, the coherence increases with the increasing time step before the crossover time [Formula: see text] and then gradually decreases after [Formula: see text].
We reveal the quantum and classical behaviors of the two-dimensional (2D) alternative quantum walk (AQW) in the presence of decoherence. For different kinds of decoherence, the analytic expressions for the moments of position distribution of the AQW are obtained. Taking the broken line noise and coin decoherence as examples of decoherence, we find that when decoherence emerges in only one direction, the anisotropic position distribution pattern appears, and not all the motions of the walker exhibit the transition from quantum to classical behaviors. Considering the effect of decoherence, we reveal the anisotropic correlations between the x(y)x\phantom{\rule{0.28em}{0ex}}(y)x(y) position of the 2D walker and the state of the coin in 2D AQWs.
Quantum walk (QW) provides a versatile platform for the realization of quantum algorithms. Due to the existence of the inevitable noises in the walk, the different quantum algorithms accommodating to different noises are demanded. Thus, the success of the algorithms based on the QW requires us to sense different noises in the walk. Until now, the way to distinguish different noises in the walk has been discussed rarely. Here, we propose an efficient way to sense the noises in the one and two dimensional QWs. The populations of the coin in the walk with or without decoherence are presented. By only detecting the populations of the coin in the QW, we can determine whether there exists the decoherence in the total QW system. Moreover, the non-Markovianity of the coin in the one and two dimensional QWs is revealed, in which the coin is taken as an open quantum system, and the other components of the QW system is taken as the large environment. With the measured value of the non-Markovianity for the coin, we can conjecture which kinds of noise emerges in the one and two dimensional QWs.