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Bounds on the dynamics and entanglement in a periodic quantum walks

N. Pradeep Kumar, Radhakrishnan Balu, Raymond Laflamme, C. M. Chandrashekar

Published Nov 16, 2017• 1 citation
OpenAlex

Abstract

We study the dynamics of discrete-time quantum walk using quantum coin operations, $\hat{C}(\theta_1)$ and $\hat{C}(\theta_2)$ in time dependent periodic sequence. For two-period quantum walk with the parameters $\theta_1$ and $\theta_2$ in the coin operations we show that the standard deviation ($\sigma_{\theta_1, \theta_2} (t)$) is same as the minimum of standard deviation obtained from one of the one-period quantum walk with coin operations $\theta_1$ or $\theta_2$, $\sigma_{\theta_1, \theta_2}(t) = \min \{\sigma_{\theta_1}(t), \sigma_{\theta_2}(t) \}$. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walk, we present the bounds on the dynamics of three- and higher period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using $\theta_1$ and $\theta_2$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $\theta=0$, here we show the recovery of massless Dirac equation with non-zero $\theta$ parameters contributing to the intriguing interference in the dynamics in a totally non-relativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.

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We study the dynamics of discrete-time quantum walk using quantum coin operations, C^(θ1)\hat{C}(\theta_1)C^(θ1​) and C^(θ2)\hat{C}(\theta_2)C^(θ2​) in time dependent periodic sequence. For two-period quantum walk with the parameters θ1\theta_1θ1​ and θ2\theta_2θ2​ in the coin operations we show that the standard deviation (σθ1,θ2(t)\sigma_{\theta_1, \theta_2} (t)σθ1​,θ2​​(t)) is same as the minimum of standard deviation obtained from one of the one-period quantum walk with coin operations θ1\theta_1θ1​ or θ2\theta_2θ2​, σθ1,θ2(t)=min⁡{σθ1(t),σθ2(t)}\sigma_{\theta_1, \theta_2}(t) = \min \{\sigma_{\theta_1}(t), \sigma_{\theta_2}(t) \}σθ1​,θ2​​(t)=min{σθ1​​(t),σθ2​​(t)}. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walk, we present the bounds on the dynamics of three- and higher period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using θ1\theta_1θ1​ and θ2\theta_2θ2​. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter θ=0\theta=0θ=0, here we show the recovery of massless Dirac equation with non-zero θ\thetaθ parameters contributing to the intriguing interference in the dynamics in a totally non-relativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.
We study the dynamics of discrete-time quantum walk using quantum coin operations, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}({\ensuremath{\theta}}_{1})$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}({\ensuremath{\theta}}_{2})$, in time-dependent periodic sequence. For the two-period quantum walk with the parameters ${\ensuremath{\theta}}_{1}$ and ${\ensuremath{\theta}}_{2}$ in the coin operations we show that the standard deviation $[{\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{1},{\ensuremath{\theta}}_{2}}(t)]$ is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations ${\ensuremath{\theta}}_{1}$ or ${\ensuremath{\theta}}_{2}$, ${\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{1},{\ensuremath{\theta}}_{2}}(t)=\mathrm{min}{{\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{1}}(t),{\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{2}}(t)}$. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walks, we present the bounds on the dynamics of three- and higher-period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using ${\ensuremath{\theta}}_{1}$ and ${\ensuremath{\theta}}_{2}$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $\ensuremath{\theta}=0$, here we show the recovery of the massless Dirac equation with nonzero $\ensuremath{\theta}$ parameters contributing to the intriguing interference in the dynamics in a totally nonrelativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.
Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC , pp 59–68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.
Properties of one-dimensional discrete-time quantum walks (DTQWs) are sensitive to the presence of inhomogeneities in the substrate, which can be generated by defining position-dependent coin operators. Deterministic aperiodic sequences of two or more symbols provide ideal environments where these properties can be explored in a controlled way. Based on an exhaustive numerical study, this work discusses a two-coin model resulting from the construction rules that lead to the usual fractal Cantor set. Although the fraction of the less frequent coin [Formula: see text] as the size of the chain is increased, it leaves peculiar properties in the walker dynamics. They are characterized by the wave function, from which results for the probability distribution and its variance, as well as the entanglement entropy, were obtained. A number of results for different choices of the two coins are presented. The entanglement entropy has shown to be very sensitive to uncovering subtle quantum effects present in the model.
In this work, we extend the scattering quantum walk (SQW) framework to a lattice of energy-dependent point interactions. This yields, within the coined quantum walk (CQW) formalism, a coin operator that is directly related to the scattering matrix of zero-range potentials. The model thus provides a discrete-time quantum-walk (DTQW) analog of a periodic array of point interactions of the Kronig-Penney type, where the walker's wavenumber serves as a continuous, physically transparent control parameter for the coin operation. We analyze the spectra and the dynamics of position probability and entanglement, yielding distinct results for specific energies and point interactions. We relate the spectral structure to the spatial probability distribution and explicitly characterize the long-time entanglement behavior for each point interaction. The transmission modulus determines the quasienergy gap, bandwidth, and maximum group velocity, while also controlling the long-time coin-position entanglement for the initial state considered. The four families of one-dimensional point interactions (δδδ, δ′δ'δ′, crossed and asymmetric) realize qualitatively distinct transmission profiles and span the full range of behavior, including enhanced or strongly suppressed spreading and oscillatory entanglement.
Quantum Walk (QW) has very different transport properties to its classical counterpart due to interference effects. Here we study the discrete-time quantum walk (DTQW) with on-site static/dynamic phase disorder following either binary or uniform distribution in both one and two dimensions. For one dimension, we consider the Hadamard coin; for two dimensions, we consider either a 2-level Hadamard coin (Hadamard walk) or a 4-level Grover coin (Grover walk) for the rotation in coin-space. We study the transport properties e.g. inverse participation ratio (IPR) and the standard deviation of the density function (σ) as well as the coin-position entanglement entropy (EE), due to the two types of phase disorders and the two types of coins. Our numerical simulations show that the dimensionality, the type of coins, and whether the disorder is static or dynamic play a pivotal role and lead to interesting behaviors of the DTQW. The distribution of the phase disorder has very minor effects on the quantum walk.
Discrete-time quantum walk evolve by a unitary operator which involves two operators a conditional shift in position space and a coin operator. This operator entangles the coin and position degrees of freedom of the walker. In this paper, we investigate the asymptotic behavior of the coin position entanglement (CPE) for an inhomogeneous quantum walk which determined by two orthogonal matrices in one-dimensional lattice. Free parameters of coin operator together provide many conditions under which a measurement perform on the coin state yield the value of entanglement on the resulting position quantum state. We study the problem analytically for all values that two free parameters of coin operator can take and the conditions under which entanglement becomes maximal are sought.