Symmetric Quantum Walks Research Articles

Disorder in parity–time symmetric quantum walks

We experimentally investigate the impact of static disorder and dynamic disorder on the non-unitary dynamics of parity–time (PT)-symmetric quantum walks. Via temporally alternating photon losses in an interferometric network, we realize the passive PT-symmetric quantum dynamics for single photons. Controllable coin operations allow us to simulate different environmental influences, which result in three different behaviors of quantum walkers: a standard ballistic spread, a diffusive behavior, and a localization, respectively, in a PT-symmetric quantum walk architecture.

Chiral Floquet Systems and Quantum Walks at Half-Period

We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators in terms of five integer indices. On the basis of these indices it can be decided whether the half-step operator can be obtained from a continuous Hamiltonian driving, or not. The half-step operator determines two Floquet operators, obtained by starting the driving at zero or at half period, respectively. These are called timeframes and are chiral symmetric quantum walks. Conversely, we show under which conditions two chiral symmetric walks determine a common half-step operator. Moreover, we clarify the connection between the classification of half-step operators and the corresponding quantum walks. Within this theory we prove bulk-edge correspondence and show that a second timeframe allows to distinguish between symmetry protected edge states at $+1$ and $-1$ which is not possible for a single timeframe.

Experimental Parity-Time Symmetric Quantum Walks for Centrality Ranking on Directed Graphs.

Using quantum walks (QWs) to rank the centrality of nodes in networks, represented by graphs, is advantageous compared to certain widely used classical algorithms. However, it is challenging to implement a directed graph via QW, since it corresponds to a non-Hermitian Hamiltonian and thus cannot be accomplished by conventional QW. Here we report the realizations of centrality rankings of a three-, a four-, and a nine-vertex directed graph with parity-time (PT) symmetric quantum walks by using high-dimensional photonic quantum states, multiple concatenated interferometers, and dimension dependent loss to achieve these. We demonstrate the advantage of the QW approach experimentally by breaking the vertex rank degeneracy in a four-vertex graph. Furthermore, we extend our experiment from single-photon to two-photon Fock states as inputs and realize the centrality ranking of a nine-vertex graph. Our work shows that a PT symmetric multiphoton quantum walk paves the way for realizing advanced algorithms.

Topological quantum walk with discrete time-glide symmetry

Discrete quantum walks are periodically driven systems with discrete time evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian exists, and the one-period time evolution is given directly by a series of unitary operators. Regarding each constituent unitary operator as a discrete time step, we formulate discrete space-time symmetry in quantum walks and evaluate the corresponding symmetry protected topological phases. In particular, we study chiral and/or time-glide symmetric topological quantum walks in this formalism. Due to discrete nature of time evolution,the topological classification is found to be different from that in conventional Floquet systems. As a concrete example, we study a two-dimensional quantum walk having both chiral and time-glide symmetries, and identify the anomalous edge states protected by these symmetries.

The Witten index for 1D supersymmetric quantum walks with anisotropic coins

Chirally symmetric discrete-time quantum walks possess supersymmetry, and their Witten indices can be naturally defined. The Witten index gives a lower bound for the number of topologically protected bound states. The purpose of this paper is to give a complete classification of the Witten index associated with a one-dimensional split-step quantum walk. It turns out that the Witten index of this model exhibits striking similarity to the one associated with a Dirac particle in supersymmetric quantum mechanics.

Supersymmetry for chiral symmetric quantum walks

Quantum walks have attracted attention as a promising platform realizing topological phenomena, and many physicists have introduced various types of indices to characterize topologically protected bound states that are robust against perturbations. In this paper, we introduce an index from a supersymmetric point of view. This allows us to define indices for all chiral symmetric quantum walks such as multi-dimensional split-step quantum walks and quantum walks on graphs, for which there has been no index theory. Moreover, the index gives a lower bound on the number of bound states robust against compact perturbations. We also calculate the index for several concrete examples including the unitary transformation that appears in Grover’s search algorithm.

Fragile aspects of topological transition in lossy and parity-time symmetric quantum walks

Quantum walks often provide telling insights about the structure of the system on which they are performed. In {mathscr{P}}{mathscr{T}}-symmetric and lossy dimer lattices, the topological properties of the band structure manifest themselves in the quantization of the mean displacement of such a walker. We investigate the fragile aspects of a topological transition in these two dimer models. We find that the transition is sensitive to the initial state of the walker on the Bloch sphere, and the resultant mean displacement has a robust topological component and a quasiclassical component. In {mathscr{P}}{mathscr{T}} symmetric dimer lattices, we also show that the transition is smeared by nonlinear effects that become important in the {mathscr{P}}{mathscr{T}}-symmetry broken region. By carrying out consistency checks via analytical calculations, tight-binding results, and beam-propagation-method simulations, we show that our predictions are easily testable in today’s experimental systems.

The Topological Classification of One-Dimensional Symmetric Quantum Walks

We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behaviour far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translationally invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at $1$ or $-1$ (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions.

Effects of Disorder on Non-unitary <i>PT</i> Symmetric Quantum Walks

$\mathcal{PT}$ symmetry, namely, a combined parity and time-reversal symmetry can make non-unitary quantum walks exhibit entirely real eigenenergy. However, it is known that the concept of $\mathcal{PT}$ symmetry can be generalized and an arbitrary anti-unitary symmetry has a possibility to substitute $\mathcal{PT}$ symmetry. The aim of the present work is to seek such non-unitary quantum walks with generalized $\mathcal{PT}$ symmetry by focusing on effects of spatially random disorder which breaks $\mathcal{PT}$ symmetry. We numerically find non-unitary quantum walks whose quasi-energy is entirely real despite $\mathcal{PT}$ symmetry is broken.

Exact solutions and symmetry analysis for the limiting probability distribution of quantum walks

In the literature, there are numerous studies of one-dimensional discrete-time quantum walks (DTQWs) using a moving shift operator. However, there is no exact solution for the limiting probability distributions of DTQWs on cycles using a general coin or swapping shift operator. In this paper, we derive exact solutions for the limiting probability distribution of quantum walks using a general coin and swapping shift operator on cycles for the first time. Based on the exact solutions, we show how to generate symmetric quantum walks and determine the condition under which a symmetric quantum walk appears. Our results suggest that choosing various coin and initial state parameters can achieve a symmetric quantum walk. By defining a quantity to measure the variation of symmetry, deviation and mixing time of symmetric quantum walks are also investigated.

Spectral stability of unitary network models

We review various unitary network models used in quantum computing, spectral analysis or condensed matter physics and establish relationships between them. We show that symmetric one-dimensional quantum walks are universal, as are CMV matrices. We prove spectral stability and propagation properties for general asymptotically uniform models by means of unitary Mourre theory.

Bulk-boundary correspondence for chiral symmetric quantum walks

Discrete-time quantum walks (DTQW) have topological phases that are richer than those of time-independent lattice Hamiltonians. Even the basic symmetries, on which the standard classification of topological insulators hinges, have not yet been properly defined for quantum walks. We introduce the key tool of timeframes, i.e., we describe a DTQW by the ensemble of time-shifted unitary timestep operators belonging to the walk. This gives us a way to consistently define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using an "inversion symmetric" pulse sequence. For one-dimensional DTQW's with CS, we identify the bulk ZxZ topological invariant that controls the number of topologically protected 0 and pi energy edge states at the interfaces between different domains, and give simple formulas for these invariants. We illustrate this bulk--boundary correspondence for DTQW's on the example of the "4-step quantum walk", where tuning CS and particle-hole symmetry realizes edge states in various symmetry classes.

Quantum Walk of Two Quantum Particles on One Dimensional System

We study two particle quantum walks on one dimensional chain. Probability distribution of two particle quantum walks is dependent on the initial state, and symmetric quantum walk or asymmetric quantum walk is analogous to that of one particle quantum walk. The quantum correlation probability is much different from classical coincidence probability. The difference reflects quantum interference between two particles.

Quantum walk of light in frequency space and its controlled dephasing

We implement, using a coherent source, a coined quantum walk of light in frequency space in a tabletop experiment utilizing a series of modified Michelson interferometers that incorporate polarization optics and acousto-optic modulators. Manipulating the phase of the radio frequency that governs the acousto-optic interaction, we achieve symmetric or asymmetric quantum walks. Introducing rapid random phase shifts electronically, while regulating the amplitude, we cause controlled dephasing and thus simulate a gradual transition from the quantum walk to the classical random walk.

Framework for discrete-time quantum walks and a symmetric walk on a binary tree

We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2, on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed form, as a function of time and the initial level $n$ in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power-law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small $n$). The quantum walk shows a polynomial algorithmic speedup in $n$ over the classical walk, which we conjecture to be of the order $2/3$, based on strong trends in data.

Asymmetries in symmetric quantum walks on two-dimensional networks

We study numerically the behavior of continuous-time quantum walks over networks which are topologically equivalent to square lattices. On short time scales, when placing the initial excitation at a corner of the network, we observe a fast, directed transport through the network to the opposite corner. This transport is not ballistic in nature, but rather produced by quantum mechanical interference. In the long time limit, certain walks show an asymmetric limiting probability distribution; this feature depends on the starting site and, remarkably, on the precise size of the network. The limiting probability distributions show patterns which are correlated with the initial condition. This might have consequences for the application of continuous time quantum walk algorithms.

Pseudo memory effects, majorization and entropy in quantum random walks

A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.