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arXiv:1204.1287 · Journal of Physics A Mathematical and Theoretical

Decoherence in two-dimensional quantum walks using four- and two-state particles

C. M. Chandrashekar, Thomas Busch

Published Feb 21, 2013• 14 citations
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Abstract

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.

DOI
10.1088/1751-8113/46/10/105306

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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.
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 present a new scheme for a discrete-time quantum walk on two- and three-dimensional lattices using a two-state particle. We use different Pauli basis as translational eigestates for different axis and show that the coin operation, which is necessary for one-dimensional walk is not a necessary requirement for two- and three- dimensional walk but can serve as an additional resource. Using this scheme, the probability distribution from Grover walk using four-state particle and other equivalent schemes on a square lattice using coin operation is reproduced. We also present the Hamiltonian form of evolution which can serve as a general framework to simulate, control, and study the dynamics in different physical systems.
We theoretically analyze the case of noisy Quantum walks (QWs) by introducing four qubit decoherence models into the coin degree of freedom of linear and cyclic QWs. These models include flipping channels (bit flip, phase flip and bit-phase flip), depolarizing channel, phase damping channel and generalized amplitude damping channel. Explicit expressions for the probability distribution of QWs on a line and on a cyclic path are derived under localized and delocalized initial states. We show that QWs which begin from a delocalized state generate mixture probability distributions, which could give rise to useful algorithmic applications related to data encoding schemes. Specifically, we show how the combination of delocalzed initial states and decoherence can be used for computing the binomial transform of a given set of numbers. However, the sensitivity of QWs to noisy environments may negatively affect various other applications based on QWs.
The classicalization of a decoherent discrete-time quantum walk on a line or an nnn-cycle can be demonstrated in various ways that do not necessarily provide a geometry-independent description. For example, the position probability distribution becomes increasingly Gaussian, with a concomitant fall in the standard deviation, in the former case, but not in the latter. As another example, each step of the quantum walk on a line may be subjected to an arbitrary phase gate, without affecting the position probability distribution, no matter whether the walk is noiseless or noisy. This symmetry, which is absent in the case of noiseless cyclic walk, but is restored in the presence of sufficient noise, serves as an indicator of classicalization, but only in the cyclic case. Here we show that the degree of quantum correlations between the coin and position degrees of freedom, quantified by a measure based on the disturbance induced by local measurements [Luo, Phys. Rev. A 77, 022301 (2008)], provides a suitable measure of classicalization across both type of walks. Applying this measure to compare the two walks, we find that cyclic quantum walks tend to classicalize faster than quantum walks on a line because of more efficient phase randomization due to the self-interference of the two counter-rotating waves. We model noise as acting on the coin, and given by the squeezed generalized amplitude damping (SGAD) channel, which generalizes the generalized amplitude damping channel.
One of the proposals for the exploitation of two-dimensional quantum walks has been the efficient generation of entanglement. Unfortunately, the technological effort required for the experimental realization of standard two-dimensional quantum walks is significantly demanding. In this respect, an alternative scheme with less challenging conditions has been recently studied, particularly in terms of spatial-entanglement generation [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. Here, we extend the investigation to a scenario where a measurement is performed on the coin degree of freedom after the evolution, allowing a further comparison with the standard two-dimensional Grover walk.
Decoherence and disorder are two major difficulties limiting the performance of quantum systems in practical settings. Yet they can potentially counteract each other to partially restore the systems' quantum signatures. We adopt the particle swarm optimization method to find the optimal disorder for mitigating the effects of decoherence in one- and two-dimensional quantum random walks, achieving substantial increase in the mean walking distance for a wide range of decoherence strength. This result suggests a viable approach to constructing practical quantum systems robust against decoherence and disorder.