We introduce and solve from first principles a continuous-time quantum walk with absorption generated by a Lindblad boundary sink of arbitrary strength. Projecting onto the surviving walker sector maps the problem to a non-Hermitian tight-binding Hamiltonian with a rank-one imaginary defect on the semi-infinite line. We obtain closed-form expressions for the exact propagator, from which the first-passage statistics follow. Weak coupling limits absorption through inefficient transfer into the sink, whereas for strong dissipation, occupation at the boundary is suppressed by Zeno-like reflection, accompanied by the emergence of a localized non-Hermitian mode. Despite the different physical origins of these suppression mechanisms, their respective absorption probabilities exhibit an asymptotic duality. The evolution is conveniently visualized in phase space, where the localized mode has an exponentially confined Wigner representation near the edge site.
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.
In this paper, we study Grover walks on a line with one and two absorbing boundaries. In particular, we present some results for the absorbing probabilities both in a semi-finite and finite line. Analytical expressions for these absorbing probabilities are presented by using the combinatorial approach. These results are perfectly matched with numerical simulations. We show that the behavior of Grover walks on a line with absorbing boundaries is strikingly different from that of classical walks and that of Hadamard walks.
Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Q_n where n is the lattice size. The conditional probability d…
We consider a two-state quantum walk on a line where after the first step an absorbing sink is placed at the origin. The probability of finding the walker at position $j$, conditioned on that it has not returned to the origin, is investigated in the asymptotic limit. We prove a limit theorem for the conditional probability distribution and show that it is given by the Konno's density function modified by a pre-factor ensuring that the distribution vanishes at the origin. In addition, we discuss the relation to the problem of recurrence of a quantum walk and determine the Polya number. Our appr…
In this paper we analyze the behavior of quantum random walks. In particular we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the one-dimensional case. We compute these probabilites both by employing generating functions and by use of an eigenfunction approach. The generating function method is used to determine some simple properties of the walks we consider, but appears to have limitations. The eigenfunction approach works by relating the problem of absorption to a unitary problem that has identical dynamics inside a certain…
Szegedy's quantum walk is a quantization of a classical random walk or Markov chain, where the walk occurs on the edges of the bipartite double cover of the original graph. To search, one can simply quantize a Markov chain with absorbing vertices. Recently, Santos proposed two alternative search algorithms that instead utilize the sign-flip oracle in Grover's algorithm rather than absorbing vertices. In this paper, we show that these two algorithms are exactly equivalent to two algorithms involving coined quantum walks, which are walks on the vertices of the original graph with an internal deg…
Discrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a noisy environment may destroy these phases. We investigate the behavior of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian is $\mathc…
Quantum walks are known to have nontrivial interaction with absorbing boundaries. In particular, Ambainis et.\ al.\ \cite{ambainis01} showed that in the $(\Z ,C_1,H)$ quantum walk (one-dimensional Hadamard walk) an absorbing boundary partially reflects information. These authors also conjectured that the left absorption probabilities $P_n^{(1)}(1,0)$ related to the finite absorbing Hadamard walks $(\Z ,C_1,H,\{ 0,n\} )$ satisfy a linear fractional recurrence in $n$ (here $P_n(1,0)$ is the probability that a Hadamard walk particle initialized in $|1\rangle |R\rangle$ is eventually absorbed at $…
We extend the non-Hermitian one-dimensional quantum walk model [Phys. Rev. Lett. 102, 065703 (2009)] by taking the dephasing effect into account. We prove that the feature of topological transition does not change even when dephasing between the sites within units is present. The potential experimental observation of our theoretical results in the circuit QED system consisting of superconducting qubit coupled to a superconducting resonator mode is discussed and numerically simulated. The results clearly show a topological transition in quantum walk and display the robustness of such a system t…
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…
Dipartimento Interuniversitario di Fisica, Università di Bari（巴里大学跨学科物理系）；INFN, Sezione di Bari（意大利国家核物理研究所巴里分部）；Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague（布拉格捷克理工大学核科学与物理工程学院物理系）
When searching for a marked vertex in a graph, Szegedy's usual search operator is defined by using the transition probability matrix of the random walk with absorbing barriers at the marked vertices. Instead of using this operator, we analyze searching with Szegedy's quantum walk by using reflections around the marked vertices, that is, the standard form of quantum query. We show we can boost the probability to 1 of finding a marked vertex in the complete graph. Numerical simulations suggests that the success probability can be improved for other graphs, like the two-dimensional grid. We also …
Finite-size Effects on The Edge Loss Probability in Non-Hermitian Quantum Walks
Hunan Key Laboratory for Super-Microstructure and Ultrafast Process, School of Physics, Central South University（中南大学物理学院）；School of Microelectronics and Physics, Hunan University of Technology and Business（湖南工业大学微电子与物理学院）；Hunan Provincial Key Laboratory of Flexible Electronic Materials Genome Engineering, School of Physics and Electronic Sciences, Changsha University of Science and Technology（长沙理工大学物理与电子科学学院）
We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix, and clarify explicit formulas of their supports of the square. In addition, we give tables by computer on the identification of digraphs by their eigenvalues.
We construct a two-dimensional, discrete-time quantum walk exhibiting non-Hermitian skin effects under open-boundary conditions. As a confirmation of the non-Hermitian bulk-boundary correspondence, we show that the emergence of topological edge states are consistent with Floquet winding numbers calculated using a non-Bloch band theory invoking time-dependent generalized Billouin zones. Further, the non-Bloch topological invariants associated with quasienergy bands are captured by a non-Hermitian local Chern marker in real space, defined through local biorthogonal eigen wave functions of the no…
We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set $M$ consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time $HT(P,M)$ of any reversible random walk $P$ on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity $HT^+(\mathit{P,M})$ which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolatio…
In this paper we present a model exhibiting a new type of continuous-time quantum walk (as a quantum mechanical transport process) on networks, which is described by a non-Hermitian Hamiltonian possessing a real spectrum. We call it pseudo-Hermitian continuous-time quantum walk. We introduce a method to obtain the probability distribution of walk on any vertex and then study a specific system. We observe that the probability distribution on certain vertices increases compared to that of the Hermitian case. This formalism makes the transport process faster and can be useful for search algorithm…
Discrete-time photonic quantum walks on a synthetic lattice, where both spatial and temporal evolution of light is discretized, have provided recently a fascinating platform for the observation of a wealth of non-Hermitian physical phenomena and for the control of light scattering in complex media. A rather open question is whether invisible potentials, analogous to the ones known for continuous optical media, do exist in such discretized systems. Here it is shown that, under certain conditions, slowly-drifting Kramers-Kronig potentials behave as invisible potentials in discrete-time photonic …
Recently, the study of non-Hermitian physics has attracted considerable attention. The modified bulk-boundary correspondence has been proposed to understand topological edge states in non-Hermitian static systems. Here we report a new experimental observation of edge states in non-Hermitian periodically driven systems. Some unconventional edge states are found not to be satisfied with the bulk-boundary correspondence when the system belongs to the broken parity-time (PT) symmetric phase. The experiments are performed in our constructed non-Hermitian light quantum walk platform with left and ri…
We examine the coherent propagation of the one-dimensional Frenkel exciton (correlated electron-hole pair system) based on a model of a quantum walker in multi-dimensional Hilbert space. The walk is governed by a non-Hermitian coin flip operation coupled to a generalized shift process. The dissipative coin flip operation is associated with amplitude leakages at occupied sites, typical of processes which occur when an exciton is transferred along dimer sites in photosynthetic protein complexes. We analyze the characteristics probability distribution of the one-dimensional quantum walk for vario…
Delocalization transition is numerically found in a non-Hermitian extension of a discrete-time quantum walk on a one-dimensional random medium. At the transition, an eigenvector gets delocalized and at the same time the corresponding energy eigenvalue (the imaginary unit times the phase of the eigenvalue of the time-evolution operator) becomes complex. This is in accordance with a non-Hermitian extension of the random Anderson model in one dimension, called, the Hatano-Nelson model. We thereby numerically find that all eigenstates of the Hermitian quantum walk share a common localization lengt…
A hallmark of topological band theory in periodic media is that bulk properties are not affected by boundary conditions. Remarkably, in certain non-Hermitian lattices the bulk properties are largely affected by boundaries, leading to such major effects as the non-Hermitian skin effect and violation of the bulk-boundary correspondence. Here we unveil that non-unitary discrete-time quantum walks of photons in systems involving gain and loss show rather generally non-Bloch parity-time ($\mathcal{PT}$) symmetry breaking phase transitions, and suggest a bulk probing method to detect such boundary-d…
We study the influence of particle interaction on a quantum walk on a bipartite one-dimensional lattice with decay from every second site. The corresponding non-interacting (linear) system has been shown to have a topological transition described by the average displacement before decay. Here we use this topological quantity to distinguish coherent quantum dynamics from incoherent classical dynamics caused by a breaking of the translational symmetry. We furthermore analyze the behavior by means of a rate equation providing a quantitative description of the incoherent nonlinear dynamics.
In non-Hermitian quasicrystals, mobility edges (ME) separating localized and extended states in complex energy plane can arise as a result of non-Hermitian terms in the Hamiltonian. Such ME are of topological nature, i.e. the energies of localized and extended states exhibit distinct topological structures in the complex energy plane. However, depending on the origin of non-Hermiticity, i.e. asymmetry of hopping amplitudes or complexification of the incommensurate potential phase, different winding numbers are introduced, corresponding to different transport features in the bulk of the lattice…
Spectral singularities are predicted to occur in a non-Hermitian extension of the Friedrichs-Fano-Anderson model describing the decay of a discrete state $|a >$ coupled to a continuum of modes. A physical realization of the model, based on electronic or photonic transport in a semi-infinite tight-binding lattice with an imaginary impurity site at the lattice boundary, is proposed. The occurrence of the spectral singularities is shown to correspond either to a diverging reflection probability (for an amplifying impurity) or to a vanishing reflection probability (for an absorbing impurity) from …
Non-Hermiticity significantly enriches the properties of topological models, leading to exotic features such as the non-Hermitian skin effects and non-Bloch bulk-boundary correspondence that have no counterparts in Hermitian settings. Its impact is particularly illustrating in non-Hermitian quasicrystals where the interplay between non-Hermiticity and quasiperiodicity results in the concurrence of the delocalization-localization transition, the parity-time (PT)-symmetry breaking, and the onset of the non-Hermitian skin effects. Here we experimentally simulate non-Hermitian quasicrystals using …
Quantum walks have a host of applications, ranging from quantum computing to the simulation of biological systems. We present an intrinsically stable, deterministic implementation of discrete quantum walks with single photons in space. The number of optical elements required scales linearly with the number of steps. We measure walks with up to 6 steps and explore the quantum-to-classical transition by introducing tunable decoherence. Finally, we also investigate the effect of absorbing boundaries and show that decoherence significantly affects the probability of absorption.
A review of discrete quantum walk with two particle is given. The use of different states encountered in identical particle, and the idea of entanglement and superposition is explored to explored the interesting dynamics of two particle quantum walk. Boundary conditions can specify certain dynamics and so a survey of periodic boundary condition (circle) is presented. A simulation for a Hadamard walk for different periods of a circle is considered and results are compared for various periods and for a case with absorbing boundaries.
We present a novel application of the HHL (Harrow-Hassidim-Lloyd) algorithm -- a quantum algorithm solving systems of linear equations -- in solving an open problem about quantum random walks, namely computing hitting (or absorption) probabilities of a general (not only Hadamard) one-dimensional quantum random walks with two absorbing boundaries. This is achieved by a simple observation that the problem of computing hitting probabilities of quantum random walks can be reduced to inverting a matrix. Then a quantum algorithm with the HHL algorithm as a subroutine is developed for solving the pro…
We define the hitting (or absorbing) time for the case of continuous quantum walks by measuring the walk at random times, according to a Poisson process with measurement rate $λ$. From this definition we derive an explicit formula for the hitting time, and explore its dependence on the measurement rate. As the measurement rate goes to either 0 or infinity the hitting time diverges; the first divergence reflects the weakness of the measurement, while the second limit results from the Quantum Zeno effect. Continuous-time quantum walks, like discrete-time quantum walks but unlike classical random…
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 extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modelling the propagation of electromagnetic waves travelling in a wave guide half and half filed with gain and absorbing media, we give a perturbative construction of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. I…
We formalize the treatment of directed (or chiral) quantum walks using Hermitian adjacency matrices, bridging two developing fields of research in quantum information and spectral graph theory. We display results and simulations which highlight the conceptual differences between having directions encoded in the Hamiltonians or not. This leads to a construction of a new type of quantum phenomenon: zero transfer between pairs of sites in a connected coupled network, which is only possible in the directed model we study. Our main result is a description of several families of directed cycles that…
The dynamics of the survival probability of quantum walkers on a one-dimensional lattice with random distribution of absorbing immobile traps are investigated. The survival probability of quantum walkers is compared with that of classical walkers. It is shown that the time dependence of survival probability of quantum walkers has a piecewise stretched exponential character depending on the density of traps in numerical and analytical observations. The crossover between the quantum analogs of the Rosenstock and Donsker-Varadhan behaviors is identified.
We investigate quantum dynamics of a quantum walker on a finite bipartite non-Hermitian lattice, in which the particle can leak out with certain rate whenever it visits one of the two sublattices. Quantum walker initially located on one of the non-leaky sites will finally totally disappear after a length of evolution time and the distribution of decay probability on each unit cell is obtained. In one regime, the resultant distribution shows an expected decreasing behavior as the distance from the initial site increases. However, in the other regime, we find that the resultant distribution of l…
In this paper, we determine periodicity of quantum walks defined by mixed paths and mixed cycles. By the spectral mapping theorem of quantum walks, consideration of periodicity is reduced to eigenvalue analysis of $η$-Hermitian adjacency matrices. First, we investigate coefficients of the characteristic polynomials of $η$-Hermitian adjacency matrices. We show that the characteristic polynomials of mixed trees and their underlying graphs are same. We also define $n+1$ types of mixed cycles and show that every mixed cycle is switching equivalent to one of them. We use these results to discuss pe…
We analyze the scattering dynamics and spectrum of a quantum particle on a tight-binding lattice subject to a non-Hermitian (purely imaginary) local potential. The reflection, transmission and absorption coefficients are studied as a function of the strength of this absorbing potential. The system is found to have an exceptional point at a certain strength of the potential. Unusually, all (or nearly all) of the spectrum pairs up into mutually coalescing eigenstate pairs at this exceptional point. At large potential strengths, the absorption coefficient decreases and the effect of the imaginary…
Observation of Non-Hermitian Skin Dynamics in the Liouvillian Regime
The University of Hong Kong（香港大学）；HK Institute of Quantum Science and Technology（香港量子科学研究所）；State Key Laboratory of Optical Quantum Materials（光学量子材料国家重点实验室）
I review how methods from mesoscopic physics can be applied to describe the multiple wave scattering and complex wave dynamics in non-hermitian PT-symmetric resonators, where an absorbing region is coupled symmetrically to an amplifying region. Scattering theory serves as a convenient tool to classify the symmetries beyond the single-channel case and leads to effective descriptions which can be formulated in the energy domain (via Hamiltonians) and in the time domain (via time evolution operators). These models can then be used to identify the mesoscopic time and energy scales which govern the…
We model quantum transport, described by continuous-time quantum walks (CTQW), on deterministic Sierpinski fractals, differentiating between Sierpinski gaskets and Sierpinski carpets, along with their dual structures. The transport efficiencies are defined in terms of the exact and the average return probabilities, as well as by the mean survival probability when absorbing traps are present. In the case of gaskets, localization can be identified already for small networks (generations). For carpets, our numerical results indicate a trend towards localization, but only for relatively large stru…
We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be $m$-strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of…
Absolute negative mobility (ANM) in nonequilibrium systems depicts the possibility of particles propagating toward the opposite direction of an external force. We uncover in this work a phenomenon analogous to ANM regarding eigenstate localization and particle transport in non-Hermitian systems under the influence of the non-Hermitian skin effect (NHSE). A coherent coupling between two non-Hermitian chains individually possessing the same preferred direction of NHSE is shown to cause a direction reversal of NHSE for all eigenmodes. This concept is further investigated in terms of time evolutio…
We unveil an unexpected non-Hermitian phenomenon, dubbed edge burst, in non-Hermitian quantum dynamics. Specifically, in a class of non-Hermitian quantum walk in periodic lattices with open boundary condition, an exceptionally large portion of loss occurs at the system boundary. The physical origin of this edge burst is found to be an interplay between two unique non-Hermitian phenomena: non-Hermitian skin effect and imaginary gap closing. Furthermore, we establish a universal bulk-edge scaling relation underlying the non-Hermitian edge burst. Our predictions are experimentally accessible in v…
