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.
In recent years, open systems with balanced loss and gain, that are invariant under the combined parity and time-reversal ($\mathcal{PT}$) operations, have been studied via asymmetries of their solutions. They represent systems as diverse as coupled optical waveguides and electrical or mechanical oscillators. We numerically investigate the asymmetries of incompressible viscous flow in two and three dimensions with "balanced" inflow-outflow ($\mathcal{PT}$-symmetric) configurations. By introducing configuration-dependent classes of asymmetry functions in velocity, kinetic energy density, and vo…
In this paper we study continuous-time quantum walks on Cayley graphs of the symmetric group, and prove various facts concerning such walks that demonstrate significant differences from their classical analogues. In particular, we show that for several natural choices for generating sets, these quantum walks do not have uniform limiting distributions, and are effectively blind to large areas of the graphs due to destructive interference.
We study theoretically how loss impacts the amplification and squeezing performance of a generic quantum travelling wave parametric amplifier. Unlike previous studies, we analyze how having different levels of loss at signal and idler frequencies can dramatically alter properties compared to the case of frequency-independent loss. We find that loss asymmetries increase the amplifier's added noise in comparison to the symmetric loss case. More surprisingly, even small levels of loss asymmetry can completely destroy any quantum squeezing of symmetric collective output quadratures, while nonethel…
The theory of random walks on finite graphs is well developed with numerous applications. In quantum walks, the propagation is governed by quantum mechanical rules; generalizing random walks to the quantum setting. They have been successfully applied in the development of quantum algorithms. In particular, to solve problems that can be mapped to searching or property testing on some specific graph. In this paper we investigate the discrete time coined quantum walk (DTCQW) model using tools from non-commutative Fourier analysis. Specifically, we are interested in characterizing the DTCQW on Cay…
Perfect state transfer between two marked vertices of a graph by means of discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that for the coined quantum walks on star graph and complete graph with self-loops perfect state transfer between the sender and receiver vertex is achieved for arbitrary number of vertices $N$ in $O(\sqrt{N})$ steps of the walk. Finally, we show that Szegedy's walk with queries on complete graph allows for state transfer with unit fidelity in the limit o…
We show that the hitting time of the discrete quantum walk on a symmetric Cayley graph over $\Z_2^n $ from a vertex to its antipodal is polynomial in degree of the graph. We prove that returning time of quantum walk on a symmetric Cayley graph over $\Z_2^n $ is polynomial and the probability to hit is almost one. To prove it, we give a new estimation of Kravchuk coefficients. We give an example of a probabilistic polynomial algorithm that finds an antipodal vertex in symmetric Cayley graphs.
The quantum walk differs fundamentally from the classical random walk in a number of ways, including its linear spreading and initial condition dependent asymmetries. Using stationary phase approximations, precise asymptotics have been derived for one-dimensional two-state quantum walks, one-dimensional three-state Grover walks, and two-dimensional four-state Grover walks. Other papers have investigated asymptotic behavior of a much larger set of two-dimensional quantum walks and it has been shown that in special cases the regions of polynomial decay can be parameterized. In this paper, we sho…
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…
Non-Hermitian topological edge states have many intriguing properties, but have so far mainly been discussed in terms of bulk-boundary correspondence. Here we propose to use a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient is found to show unique features with the topological phase transitions driven by paritytime( PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite artificial boundaries are not constructed by inhomogeneous quantum walk. For a Hermitian system, a turnin…
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 …
$\mathcal{PT}$ 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 ent…
Various quantum-walk based algorithms have been proposed to analyse and rank the centrality of graph vertices. However, issues arise when working with directed graphs --- the resulting non-Hermitian Hamiltonian leads to non-unitary dynamics, and the total probability of the quantum walker is no longer conserved. In this paper, we discuss a method for simulating directed graphs using PT-symmetric quantum walks, allowing probability conserving non-unitary evolution. This method is equivalent to mapping the directed graph to an undirected, yet weighted, complete graph over the same vertex set, an…
The visibility of the quantum interference "dip" seen in the Hong-Ou-Mandel experiment is optimized when a symmetric 50/50 beamsplitter is used in the interferometer. Here we show that the reduction in visibility caused by an asymmetric beamsplitter can be compensated by manipulating the polarization states of the two input photons. We experimentally demonstrate this by using a highly asymmetric 10/90 beamsplitter, and converting an initial dip visibility of 22% to a compensated value of 99%.
We model a quantum walk of identical particles that can change their exchange statistics by hopping across a domain wall in a 1D lattice. Such a "statistical boundary" is transparent to single particles and affects the dynamics only by swapping multiple particles arriving together. We find that the two-particle interference is dramatically altered by reflections of these bunched waves at the interface, producing strong measurable asymmetries. Depending on the phases on the two sides, a bunched wavepacket can get completely reflected or split into a superposition of a reflected wave and an anti…
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, an…
We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an $M$-partite graph…
Quantum walks often provide telling insights about the structure of the system on which they are performed. In PT-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 PT symmetric dimer lattic…
We present an open-system master equation study of the coherent and incoherent resonance fluorescence spectrum from a two-level quantum system under coherent pulsed excitation. Several pronounced features which differ from the fluorescence under a constant drive are highlighted, including a multi-peak structure and a pronounced off-resonant spectral asymmetry, in stark contrast to the conventional symmetrical Mollow triplet. We also study semiconductor quantum dot systems using a polaron master equation, and show how the key features of dynamic resonance fluorescence change with electron--acou…
We analyze the probability distributions of the quantum walks induced from Markov chains by Szegedy (2004). The first part of this paper is devoted to the quantum walks induced from finite state Markov chains. It is shown that the probability distribution on the states of the underlying Markov chain is always convergent in the Cesaro sense. In particular, we deduce that the limiting distribution is uniform if the transition matrix is symmetric. In the cases of non-symmetric Markov chain, we exemplify that the limiting distribution of the quantum walk is not necessarily identical with the stati…
Classical random walks on finite graphs have an underrated property: a walk from any vertex can reach every other vertex in finite time, provided they are connected. Discrete-time quantum walks on finite connected graphs however, can have infinite hitting times. This phenomenon is related to graph symmetry, as previously characterized by the group of direction-preserving graph automorphisms that trivially affect the coin Hilbert space. If a graph is symmetric enough (in a particular sense) then the associated quantum walk unitary will contain eigenvectors that do not overlap a set of target ve…
Quantum walks are expected to provide useful algorithmic tools for quantum computation. This paper introduces absorbing probability and time of quantum walks and gives both numerical simulation results and theoretical analyses on Hadamard walks on the line and symmetric walks on the hypercube from the viewpoint of absorbing probability and time.
We provide a classification of translation invariant one-dimensional quantum walks with respect to continuous deformations preserving unitarity, locality, translation invariance, a gap condition, and some symmetry of the tenfold way. The classification largely matches the one recently obtained (arXiv:1611.04439) for a similar setting leaving out translation invariance. However, the translation invariant case has some finer distinctions, because some walks may be connected only by breaking translation invariance along the way, retaining only invariance by an even number of sites. Similarly, if …
Topological phases and the associated multiple edge states are studied for parity and time-reversal $(\mathcal{PT})$ symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk-edge correspondence. Therefore, the bulk-edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge s…
We clarify that coined quantum walk is determined by only the choice of local quantum coins. To do so, we characterize coined quantum walks on graph by disjoint Euler circles with respect to symmetric arcs. In this paper, we introduce a new class of coined quantum walk by a special choice of quantum coins determined by corresponding quantum graph, called quantum graph walk. We show that a stationary state of quantum graph walk describes the eigenfunction of the quantum graph.
