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arXiv:0803.4453 · Physical Review A

Symmetry-noise interplay in a quantum walk on an n -cycle

Subhashish Banerjee, R. Srikanth, C. M. Chandrashekar, Pranaw Rungta

Published Nov 10, 2008• 31 citations
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Abstract

Augmenting the unitary transformation which generates a quantum walk by a generalized phase gate $G$ is a symmetry for both noisy and noiseless quantum walk on a line, in the sense that it leaves the position probability distribution invariant. However, this symmetry breaks down in the case of a quantum walk on an $n$ cycle, and hence can be regarded as a probe of the walk topology. Noise, modeled here as phase flip and generalized amplitude damping channels, tends to restore the symmetry because it classicalizes the walk. However, symmetry restoration happens even in the regime where the walker is not entirely classical, because noise also has the effect of desensitizing the operation $G$ to the walk topology. We discuss methods for physical implementation, and talk about the wider implications to condensed matter systems.

DOI
10.1103/physreva.78.052316

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Augmenting the unitary transformation which generates a quantum walk by a generalized phase gate GGG is a symmetry for both noisy and noiseless quantum walk on a line, in the sense that it leaves the position probability distribution invariant. However, this symmetry breaks down in the case of a quantum walk on an nnn cycle, and hence can be regarded as a probe of the walk topology. Noise, modeled here as phase flip and generalized amplitude damping channels, tends to restore the symmetry because it classicalizes the walk. However, symmetry restoration happens even in the regime where the walker is not entirely classical, because noise also has the effect of desensitizing the operation GGG to the walk topology. We discuss methods for physical implementation, and talk about the wider implications to condensed matter systems.
We study some discrete symmetries of unbiased (Hadamard) and biased quantum walk on a line, which are shown to hold even when the quantum walker is subjected to environmental effects. The noise models considered in order to account for these effects are the phase flip, bit flip, and generalized amplitude damping channels. The numerical solutions are obtained by evolving the density matrix, but the persistence of the symmetries in the presence of noise is proved using the quantum trajectories approach. We also briefly extend these studies to quantum walk on a cycle. These investigations can be relevant to the implementation of quantum walks in various known physical systems. We discuss the implementation in the case of NMR quantum information processor and ultracold atoms.
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.
This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. \n \nIn the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. \n \nWe investigate symmetry operations and environmental effects on dynamics of the walk on a line and an n−n-n−cycle highlighting the interplay between noise and topology. \n \nUsing the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.
We explore static noise in a discrete quantum random walk over a homogeneous cyclic graph, focusing on the spectral and dynamical properties of the system. Using a three-parameter unitary coin, we control the spectral structure of the noiseless step operator on the unit circle in the complex plane. One parameter governs the probability amplitudes and induces two spectral bandgaps proportional to its value. The half-sum of the two phase parameters rotates the spectrum and induces twofold degeneracy under specific conditions. Degenerate spectra yield eigenstates with sinusoidal probability distributions, while non-degenerate spectra produce flat distributions. We introduce static phase noise in the sites and analyze its impact across two distinct propagation regimes. In the walk-on-the-line regime, which precedes a full traversal of the graph, we extract the spreading exponent β from the step-resolved mean squared displacement. We find that low participation ratios correlate with sub-diffusive spread, while high ratios correspond to ballistic or super-diffusive evolution. After the walker completes a cycle, finite-size effects dominate. In this walk-on-the-cycle regime, the spreading exponent no longer characterizes the behavior. We quantify localization using a convergence criterion based on the coefficient of variation of the mean squared displacement. Across both regimes, we identify a sharp crossover near static site noise strength ϕs = π/3. This transition coincides with a drop in the participation ratio, a transition from diffusive to sub-diffusive spread in the walk-on-the-line regime, and a reduction in the saturation level of the mean squared displacement in the walk-on-the-cycle regime. Our results demonstrate that the eigenstate participation ratio provides a computationally efficient spectral diagnostic that anticipates localization across both regimes, offering an alternative to full dynamical simulations.
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.
This study investigates the crossover dynamics of a one-dimensional discrete-time quantum walk (DTQW) in noisy environments using the density matrix formalism and Kraus operator representation of quantum operations. Decoherence is modeled by bit-flip, dephasing, and bit-phase noise channels, while dissipation is represented by generalized amplitude damping channels. The analysis of crossover dynamics involves examining the probability distribution, scaling exponent of variance, coherence measures, and entropy metrics such as Shannon and von Neumann entropy. The DTQW produces a symmetrical probability distribution of displacement under decoherence, with distinct features in scaling exponents of mean squared displacement or variance. Under dissipation, DTQW produces a non-symmetric probability distribution and spreads faster than classical walk under extreme decoherence. These suggest a broad class of crossover transport behavior, which can be classified as pure quantum walk, quantum-like walk, semi-classical-like walk, and classical-like walk. Remarkably, maximum Shannon entropy is achieved at lower degrees of decoherence. This study also analyzes the effect of temporal disorder in the coin operator modeled using fractional Gaussian noise (fGn). The crossover dynamic occurs in the form of weak localization, followed by enhanced entanglement and information backflow. Notably, the correlation of fGn is transferred to the coin's degree of entanglement and transpires in the two-point correlation of the von Neumann entropy fluctuation. These findings can be useful for implementing DTQW-based protocols in quantum error correction, quantum cryptography, and quantum memory on the noisy intermediate-scale quantum (NISQ) platform.