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arXiv:1706.06257

On the spreading of quantum walks starting from local and delocalized states

Alexandre C. Orthey, Edgard P. M. Amorim

Published Jun 20, 2017• 4 citations
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Abstract

We investigate the ballistic spreading behavior of the one-dimensional discrete time quantum walks whose time evolution is driven by any balanced quantum coin. We obtain closed-form expressions for the long-time variance of position of quantum walks starting from any initial qubit (spin-$1/2$ particle) and position states following a delta-like (local), Gaussian and uniform probability distributions. By averaging over all spin states, we find out that the average variance of a quantum walk starting from a local state is independent of the quantum coin, while from Gaussian and uniform states it depends on the sum of relative phases between spin states given by the quantum coin, being non-dispersive for a Fourier walk and large initial dispersion. We also perform numerical simulations of the average probability distribution and variance along the time to compare them with our analytical results.

DOI
10.48550/arxiv.1706.06257

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We investigate the ballistic spreading behavior of the one-dimensional discrete time quantum walks whose time evolution is driven by any balanced quantum coin. We obtain closed-form expressions for the long-time variance of position of quantum walks starting from any initial qubit (spin-1/21/21/2 particle) and position states following a delta-like (local), Gaussian and uniform probability distributions. By averaging over all spin states, we find out that the average variance of a quantum walk starting from a local state is independent of the quantum coin, while from Gaussian and uniform states it depends on the sum of relative phases between spin states given by the quantum coin, being non-dispersive for a Fourier walk and large initial dispersion. We also perform numerical simulations of the average probability distribution and variance along the time to compare them with our analytical results.
We investigate the spreading behavior of the one-dimensional discrete time quantum walk starting from local and Gaussian states. In order to compare both cases, we obtain analytically the average variance of the position by averaging over all spin states. We find out that the average variance for the local case is independent on the quantum coin. However, when the quantum walk starts from a Gaussian state, its average variance depends on the sum of relative phases between spin states given by the quantum coin, and also, on the initial dispersion of the state. We perform numerical simulations of the average probability and variance along the time to confront them with our analytical results.
We investigate the relation between the ballistic spreading and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following a delta-like (local) and Gaussian distributions. We find out that the knowledge of long-time expected position together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while the expected position with the quantum coin phases give it for highly delocalized states.
We investigate the relation between transport properties and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following $\ensuremath{\delta}$-like (local) and Gaussian distributions. We find out that the knowledge of the limit velocity of the walker together with the polar angle of the initial qubit provide the asymptotic entanglement for local states, while this velocity with the quantum coin phases give it for highly delocalized states.
We investigate the evolution dynamics of inhomogeneous discrete-time one-dimensional quantum walks displaying long-range correlations in both space and time. The associated quantum coin operators of internal states are built to exhibit random inhomogeneity distribution of long-range correlations embedded in the time evolution protocol through a fractional Brownian motion with spectrum following a power-law behavior, S(k)∼1/k^{ν}. From extensive numerical simulations with averages over a large number of independent realizations of the phases of quantum coins, the power-law correlated disorder encoded in the coin phases is shown to give rise to a wide variety of spreading patterns of the qubit states, from localized to subdiffusive, diffusive, and superdiffusive (including ballistic) behavior, depending on the relative strength of the parameters driving the correlation degree. Dispersion control is possible in one-dimensional discrete-time quantum walks by tuning the long-range correlation properties assigned to the inhomogeneous quantum coin operator.
A unit evolution step of discrete-time quantum walks (QWs) is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the QWs starting with a localized initial state have been derived. In this paper, we compute limit distributions of a 2-state QW with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the (α, β) delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.
For a discrete time quantum walk (QW) on the N-cycle, allowing for decoherence on the coin, we derive a number of results, including an explicit formula for the position probability distribution. For a QW of this type, we show that the mixing behavior tends, in the long run, to a uniform distribution regardless of the initial state of the system and irrespective of the parity of the number of nodes N. These results confirm the findings of previous authors who arrived at similar conclusions through extensive numerical simulations. In particular, we infer that the mixing time M(epsilon) for the time-averaged probability distribution is of order no greater than O(N2/).