We study the dynamics of a generalization of quantum coin walk on the line which is a natural model for a diffusion modified by quantum or interference effects. In particular, our results provide surprisingly simple explanations to phenomena observed by Bouwmeester et al. (Phys. Rev. A, 61, 13410 (1999)) in their optical Galton board experiment, and a description of a stroboscopic quantum walks given by Buershaper and Burnett (quant-ph/0406039) through numerical simulations. We also provide heuristic explanations for the behavior of our model which show, in particular, that its dynamics can be viewed as a discrete version of Bloch oscillations.
We realize quasi-periodic dynamics of a quantum walker over 2.5 quasi-periods by realizing the walker as a single photon passing through a quantum-walk optical-interferometer network. We introduce fully controllable polarization-independent phase shifters in each optical path to realize arbitrary site-dependent phase shifts, and we employ large clear-aperture beam displacers, while maintaining high-visibility interference, to enable reaching 10 quantum-walk steps. By varying the half-wave-plate setting, we control the quantum-coin bias thereby observing a transition from quasi-periodic dynamic…
Quantum walk is a key operation in quantum computing, simulation, communication and information. Here, we report for the first time the demonstration of quantum walks and localized quantum walks in a new type of optical fibers having a ring of cores constructed with both periodic and quasiperiodic Fibonacci sequences, respectively. Good agreement between theoretical and experimental results have been achieved. The new multicore ring fibers provide a new platform for experiments of quantum effects in low-loss optical fibers which is critical for scalability of real applications with large-size …
We generalize the quantum random walk protocol for a particle in a one-dimensional chain, by using several types of biased quantum coins, arranged in aperiodic sequences, in a manner that leads to a rich variety of possible wave function evolutions. Quasiperiodic sequences, following the Fibonacci prescription, are of particular interest, leading to a sub-ballistic wavefunction spreading. In contrast, random sequences leads to diffusive spreading, similar to the classical random walk behaviour. We also describe how to experimentally implement these aperiodic sequences.
We introduce quantum walks with a time-dependent coin, and show how they include, as a particular case, the generalized quantum walk recently studied by Wojcik et al. {[}Phys. Rev. Lett. \textbf{93}, 180601(2004){]} which exhibits interesting dynamical localization and quasiperiodic dynamics. Our proposal allows for a much easier implementation of this particular rich dynamics than the original one. Moreover, it allows for an additional control on the walk, which can be used to compensate for phases appearing due to external interactions. To illustrate its feasibility, we discuss an example us…
We propose a new theory on a relation between diffusive and coherent nature in one dimensional wave mechanics based on a quantum walk. It is known that the quantum walk in homogeneous matrices provides the coherent property of wave mechanics. Using the recent result of a localization phenomenon in a one-dimensional quantum walk (Konno, Quantum Inf. Proc. (2010) 9, 405-418), we numerically show that the randomized localized matrices suppress the coherence and give diffusive nature.
Quantum walks obey unitary dynamics: they form closed quantum systems. The system becomes open if the walk suffers from imperfections represented as missing links on the underlying basic graph structure, described by dynamical percolation. Openness of the system's dynamics creates decoherence, leading to strong mixing. We present a method to analytically solve the asymptotic dynamics of coined, percolated quantum walks for a general graph structure. For the case of a circle and a linear graph we derive the explicit form of the asymptotic states. We find that a rich variety of asymptotic evolut…
The quantum walk is a dynamical protocol which describes the motion of spinful particles on a lattice. Also, it has been demonstrated to be a powerful platform to explore topological quantum matter. Recently, the quantum walk in coherent state space has been proposed theoretically and realized experimentally. However, due to the inherent characteristics of coherent states, it is challenging to control the number of photons when we need the coherent space to be a nearly orthogonal space in practice. Here, we demonstrate that the nonorthogonality of coherent sates, on the one hand can be cancell…
We address the performance of a coin-biased quantum walk as a generator for non-classical position states of the walker. We exploit a phenomenon of coherent localisation in the position space --- resulting from the choice of small values of the coin parameter and assisted by post-selection --- to engineer large-size coherent superpositions of distinguishable position states of the walker. The protocol that we design appears to be remarkably robust against both the actual value taken by the coin parameter and strong dephasing-like noise acting on the spatial degree of freedom. We finally illust…
We study the coherent exciton transport on Apollonian networks generated by simple iterative rules. The coherent exciton dynamics is modeled by continuous-time quantum walks and we calculate the transition probabilities between two nodes of the networks. We find that the transport depends on the initial nodes of the excitation. For networks less than the second generation the coherent transport shows perfect revivals when the initial excitation starts at the central node. For networks of higher generation, the transport only shows partial revivals. Moreover, we find that the excitation is most…
This paper seeks to expand the scope of the alkaline seepage site hydrothermal mound scenario of Russell et al, by appealing to its wider canvas for hypothesizing self-assembly of gregite clusters via interplay of forces within the gel phase of FeS membranes: directed heat transport a la Rayleigh-Benard convection for dissociation, vs oriented attachment (small clusters) and magnetic forces (large clusters) for association, the latter assisted by magnetic mound constituents. The directed movement of tiny clusters through cluster layers are reminiscent of processes like budding, molecular motor…
The coherent superposition of position states in a quantum walk (QW) can be precisely engineered towards the desired distributions to meet the need of quantum information applications. The coherent distribution can make full use of quantum parallel in computation and simulation. Particularly, the uniform superposition provides the robust non-locality, which has wide applications such as the generation of genuine multi-bit random numbers without post-processing. We experimentally demonstrate that the rich dynamics featured with arbitrary coherent distributions can be obtained by introducing dif…
Multi-photon quantum walks in integrated optics are an attractive controlled quantum system, that can mimic less readily accessible quantum systems and exhibit behavior that cannot in general be accurately replicated by classical light without an exponential overhead in resources. The ability to observe time evolution of such systems is important for characterising multi-particle quantum dynamics---notably this includes the effects of boundary conditions for walks in spaces of finite size. Here we demonstrate the coherent evolution of quantum walks of two indistinguishable photons using planar…
We apply the quantum-walk approach recently proposed in arXiv:quant-ph-1506.04213 to a radical-pair reaction where realistic estimates for the intermediate transition rates are available. The well-known average hitting time from quantum walks can be adopted as a measure of how quickly the reaction occurs and we calculate this for varying degrees of dephasing in the radical pair. The time for the radical pair to react to a product is found to be independent of the amount of dephasing introduced, even in the limit of no dephasing where the transient population dynamics exhibit strong coherent os…
Many disordered systems show a superdiffusive dynamics, intermediate between the diffusive one, typical of a classical stochastic process, and the so called ballistic behaviour, which is generally expected for the spreading in a quantum process. We have experimentally investigated the superdiffusive behaviour of a quantum walk (QW), whose dynamics can be related to energy transport phenomena, with a resolution which is high enough to clearly distinguish between different disorder regimes. By our experimental setup, the region between ballistic and diffusive spreading can be effectively scanned…
Quantum walks in atomic systems, owing to their continuous nature, are especially well-suited for the simulation of many-body physics and can potentially offer an exponential speedup in solving certain black box problems. Photonics offers an alternate route to simulating such nonclassical behavior in a more robust platform. However, in photonic implementations to date, an increase to the depth of a continuous quantum walk requires modifying the footprint of the system. Here we report continuous walks of a two-photon quantum frequency comb with entanglement across multiple dimensions. The coupl…
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 walk research has mainly focused on evolutions due to repeated applications of time-independent unitary coin operators. However, the idea of controlling the single particle evolution using time-dependent unitary coins has still been a subject of multiple studies as it not only hosts interesting possibilities for quantum information processing but also opens a much richer array of phenomena including static and dynamic localizations. So far, such studies have been performed only for single quantum walkers. In case of multi-walker systems, time-dependent coins may generate measurable phe…
In this paper, we study coherent exciton transport of continuous-time quantum walks on star graph. Exact analytical results of the transition probabilities are obtained by means of the Gram-Schmidt orthonormalization of the eigenstates. Our results show that the coherent exciton transport displays perfect revivals and strong localization on the initial node. When the initial excitation starts at the central node, the transport on star graph is equivalent to the transport on a complete graph of the same size.
In this article we investigate the effects of shifting position decoherence, arisen from the tunneling effect in the experimental realization of the quantum walk, on the one-dimensional discreet time quantum walk. We show that in the regime of this type of noise the quantum behavior of the walker does not fade, in contrary to the coin decoherence for which the walker undergos the quantum-to-classical transition even for weak noise. Particularly, we show that the quadratic dependency of the variance on the time and also the coin-position entanglement, i.e. two important quantum aspects of the c…
The appearance of topological effects in systems exhibiting a non-trivial topological band structure strongly relies on the coherent wave nature of the equations of motion. Here, we reveal topological dynamics in a classical stochastic random walk version of the Su-Schrieffer-Heeger model with no relation to coherent wave dynamics. We explain that the commonly used topological invariant in the momentum space translates into an invariant in a counting-field space. This invariant gives rise to clear signatures of the topological phase in an associated escape time distribution.
Classical chemical kinetics use rate-equation models to describe how a reaction proceeds in time. Such models are sufficient for describing state transitions in a reaction where coherences between different states do not arise, or in other words, a reaction which contain only incoherent transitions. A prominent example reaction containing coherent transitions is the radical-pair model. The kinetics of such reactions is defined by the so-called reaction operator which determines the radical-pair state as a function of intermediate transition rates. We argue that the well-known concept of quantu…
The study of quantum walks has been shown to have a wide range of applications in areas such as artificial intelligence, the study of biological processes, and quantum transport. The quantum stochastic walk, which allows for incoherent movement of the walker, and therefore, directionality, is a generalization on the fully coherent quantum walk. While a quantum stochastic walk can always be described in Lindblad formalism, this does not mean that it can be microscopically derived in the standard weak-coupling limit under the Born-Markov approximation. This restricts the class of quantum stochas…
Controlled Zeno-Induced Localization of Free Fermions in a Quasiperiodic Chain
Indian Institute of Science Education and Research, Bhopal（印度科学教育研究所博帕尔分校）；Nanyang Technological University（南洋理工大学）；National Institute of Science Education and Research（国家科学教育研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；University of Oxford（牛津大学）；University College London（伦敦大学学院）
Quantum walks are central to a wide range of applications such as quantum search, quantum information processing, and entanglement transport. Gaining control over the duration and the direction of quantum walks (QWs) is crucial to implementing dedicated processing. However, in current systems, it is cumbersome to achieve in a scalable format. High-dimensional quantum states, encoded in the photons' frequency degree of freedom in on-chip devices are great assets for the scalable generation and reliable manipulation of large-scale complex quantum systems. These states, viz. quantum frequency com…
We show that quantum walks interpolate between a coherent `wave walk' and a random walk depending on how strongly the walker's coin state is measured; i.e., the quantum walk exhibits the quintessentially quantum property of complementarity, which is manifested as a trade-off between knowledge of which path the walker takes vs the sharpness of the interference pattern. A physical implementation of a quantum walk (the quantum quincunx) should thus have an identifiable walker and the capacity to demonstrate the interpolation between wave walk and random walk depending on the strength of measureme…
The coin-position entanglement generated by the evolution operator of a discrete--time quantum walk converges, in the long time limit, to a well defined value which depends on the initial state. We also discuss the asymptotic bi-partite entanglement generated by several non-separable coin operations for two coherent quantum walkers, for which the entropy of entanglement increases logarithmically.
We model coherent exciton transport in dendrimers by continuous-time quantum walks (CTQWs). For dendrimers up to the second generation the coherent transport shows perfect recurrences, when the initial excitation starts at the central node. For larger dendrimers, the recurrence ceases to be perfect, a fact which resembles results for discrete quantum carpets. Moreover, depending on the initial excitation site we find that the coherent transport to certain nodes of the dendrimer has a very low probability. When the initial excitation starts from the central node, the problem can be mapped onto …
This paper reviews recent advances in continuous-time quantum walks (CTQW) and their application to transport in various systems. The introduction gives a brief survey of the historical background of CTQW. After a short outline of the theoretical ideas behind CTQW and of its relation to classical continuous-time random walks (CTRW) in Sec.~2, implications for the efficiency of the transport are presented in Sec.~3. The fourth section gives an overview of different types of networks on which CTQW have been studied so far. Extensions of CTQW to systems with long-range interactions and with stati…
We propose a scheme to implement the quantum walk for SU(1,1) in the phase space, which generalizes those associated with the Heisenberg-Weyl group. The movement of the walker described by the SU(1,1) coherent states can be visualized on the hyperboloid or the Poincaré disk. In both one-mode and two-mode realizations, we introduce the corresponding coin-flip and conditional-shift operators for the SU(1,1) group, whose relations with those for Heisenberg-Weyl group are analyzed. The probability distribution, standard deviation and the von Neumann entropy are employed to describe the walking pro…
Computational advantages gained by quantum algorithms rely largely on the coherence of quantum devices and are generally compromised by decoherence. As an exception, we present a quantum algorithm for graph isomorphism testing whose performance is optimal when operating in the partially coherent regime, as opposed to the extremes of fully coherent or classical regimes. The algorithm builds on continuous-time quantum stochastic walks (QSWs) on graphs and the algorithmic performance is quantified by the distinguishing power (DIP) between non-isomorphic graphs. The QSW explores the entire graph a…
Quantum random walks use interference to obtain faster state space exploration, which can be used for algorithmic purposes. Photonic technologies provide a natural platform for many recent experimental demonstrations. Here we analyze quantum random walks implemented by coherent light propagation in series-coupled photonic ring resonators. We propose a family of graphs modeling these devices and compare quantum and classical random walks on these structures, calculating steady-state and time-dependent solutions. We obtain conditions for quantum advantage in this setting and show how to recover …
This work focuses on the study of quantum stochastic walks, which are a generalization of coherent, i. e. unitary quantum walks. Our main goal is to present a measure of a coherence of the walk. To this end, we utilize the asymptotic scaling exponent of the second moment of the walk i. e. of the mean squared distance covered by a walk. As the quantum stochastic walk model encompasses both classical random walks and quantum walks, we are interested how the continuous change from one regime to the other influences the asymptotic scaling exponent. Moreover this model allows for behavior which is …
We design a quantum probing protocol using Quantum Walks to investigate the Quantum Information spreading pattern. We employ Quantum Fisher Information, as a figure of merit, to quantify extractable information about an unknown parameter encoded within the Quantum Walk evolution. Although the approach is universal, we focus on the coherent static and dynamic disorder to investigate anomalous and classical transport as well as Anderson localization. Our results show that a Quantum Walk can be considered as a readout device of information about defects and perturbations occurring in complex netw…
Multi-dimensional quantum walks can exhibit highly non-trivial topological structure, providing a powerful tool for simulating quantum information and transport systems. We present a flexible implementation of a 2D optical quantum walk on a lattice, demonstrating a scalable quantum walk on a non-trivial graph structure. We realized a coherent quantum walk over 12 steps and 169 positions using an optical fiber network. With our broad spectrum of quantum coins we were able to simulate the creation of entanglement in bipartite systems with conditioned interactions. Introducing dynamic control all…
A scheme for implementing the discrete-time quantum walk on the Bloch sphere is proposed, which is closely related to the SU(2) group. A spin cluster serves as the walker, whereas its location on the Bloch sphere is described by the spin coherent state. An additional spin that interacts with the spin cluster plays the role of a coin, whose state determines the rotation of the spin cluster. The Wigner function is calculated to visualize the movement of the walker on the Bloch sphere, with which the probability distribution and the standard deviation are also achieved. The quadratic enhancement …
Quantum walks are a well-established model for the study of coherent transport phenomena and provide a universal platform in quantum information theory. Dynamically influencing the walker's evolution gives a high degree of flexibility for studying various applications. Here, we present time-multiplexed finite quantum walks of variable size, the preparation of non-localized input states and their dynamical evolution. As a further application, we implement a state transfer scheme for an arbitrary input state to two different output modes. The presented experiments rely on the full dynamical cont…
We consider the coherent exciton transport, modeled by continuous-time quantum walks, on Erdös-Rény graphs in the presence of a random distribution of traps. The role of trap concentration and of the substrate dilution is deepened showing that, at long times and for intermediate degree of dilution, the survival probability typically decays exponentially with a (average) decay rate which depends non monotonically on the graph connectivity; when the degree of dilution is either very low or very high, stationary states, not affected by traps, get more likely giving rise to a survival probability …
We present a Mathematica package, QSWalk, to simulate the time evaluation of Quantum Stochastic Walks (QSWs) on arbitrary directed and weighted graphs. QSWs are a generalization of continuous time quantum walks that incorporate both coherent and incoherent dynamics and as such, include both quantum walks and classical random walks as special cases. The incoherent component allows for quantum walks along directed graph edges. The dynamics of QSWs are expressed using the Lindblad formalism, originally developed for open quantum systems, which frames the problem in the language of density matrice…
Symmetrically evolving discrete quantum walk results in dynamic localization with zero mean displacement when the standard evolution operations are replaced by a temporal disorder evolution operation. In this work we show that the quantum ratchet action, that is, a directed transport in standard or disordered discrete-time quantum walk can be realized by introducing a pawl like effect realized by using a fixed coin operation at marked positions that is, different from the ones used for evolution at other positions. We also show that the combination of standard and disordered evolution operatio…
We consider a quantum particle (walker) on a line who coherently chooses to jump to the left or right depending on the result of toss of a quantum coin. The lengths of the jumps are considered to be independent and identically distributed quenched Poisson random variables. We find that the spread of the walker is significantly inhibited, whereby it resides in the near-origin region, with respect to the case when there is no disorder. The scaling exponent of the quenched-averaged dispersion of the walker is sub-ballistic but super-diffusive. We also show that the features are universal to a cla…
We introduce the quantum quincunx, which physically demonstrates the quantum walk and is analogous to Galton's quincunx for demonstrating the random walk. In contradistinction to the theoretical studies of quantum walks over orthogonal lattice states, we introduce quantum walks over nonorthogonal lattice states (specifically, coherent states on a circle) to demonstrate that the key features of a quantum walk are observable albeit for strict parameter ranges. A quantum quincunx may be realized with current cavity quantum electrodynamics capabilities, and precise control over decoherence in such…
The presence of disorder and inhomogeneities in quantum networks has often been unexpectedly beneficial for both quantum and classical resources. Here, we experimentally realize a controllable inhomogenous Quantum Walk dynamics, which can be exploited to investigate the effect of coherent disorder on the quantum correlations between two indistinguishable photons. Through the imposition of suitable disorder configurations, we observe two photon states which exhibit an enhancement in the quantum correlations between two modes of the network, compared to the case of an ordered Quantum Walk. Diffe…
In the quest for signatures of coherent transport we consider exciton trapping in the continuous-time quantum walk framework. The survival probability displays different decay domains, related to distinct regions of the spectrum of the Hamiltonian. For linear systems and at intermediate times the decay obeys a power-law, in contrast to the corresponding exponential decay found in incoherent continuous-time random walk situations. To differentiate between the coherent and incoherent mechanisms, we present an experimental protocol based on a frozen Rydberg gas structured by optical dipole traps.
We make and generalize the observation that summing of probability amplitudes of a discrete-time quantum walk over partitions of the walking graph consistent with the step operator results in a unitary evolution on the reduced graph which is also a quantum walk. Since the effective walking graph of the projected walk is not necessarily simpler than the original, this may bring new insights into the dynamics of some kinds of quantum walks using known results from thoroughly studied cases like Euclidean lattices. We use abstract treatment of the walking space and walker displacements in aim for …
We put forward a new, versatile and highly-scalable experimental setup for the realization of discrete two-dimensional quantum random walks with a single-qubit coin and tunable degree of decoherence. The proposed scheme makes use of a small number of simple optical components arranged in a multi-path Mach-Zehnder-like configuration, where a weak coherent state with average photon number of 1 is injected. Environmental effects (decoherence) are generated by a spatial light modulator, which introduces pure dephasing in the transverse spatial plane, perpendicular to the direction of propagation o…
Coherent evolution governs the behaviour of all quantum systems, but in nature it is often subjected to influence of a classical environment. For analysing quantum transport phenomena quantum walks emerge as suitable model systems. In particular, quantum walks on percolation structures constitute an attractive platform for studying open system dynamics of random media. Here, we present an implementation of quantum walks differing from the previous experiments by achieving dynamical control of the underlying graph structure. We demonstrate the evolution of an optical time-multiplexed quantum wa…
We propose a scheme to implement the simplest and best-studied version of quantum random walk, the discrete Hadamard walk, in one dimension using coherent macroscopic sample of ultracold atoms, Bose-Einstein condensate (BEC). Implementation of quantum walk using BEC gives access to the familiar quantum phenomena on a macroscopic scale. This paper uses rf pulse to implement Hadamard operation (rotation) and stimulated Raman transition technique as unitary shift operator. The scheme suggests implementation of Hadamard operation and unitary shift operator while the BEC is trapped in long Rayleigh…
A quantum computing system is typically represented by a set of non-interacting (local) two-state systems - qubits. Many physical systems can naturally have more accessible states, both local and non-local. We show that the resulting non-local network of states connecting qubits can be efficiently addressed via continuous time quantum random walks, leading to substantial speed-up of multiqubit entanglement manipulations. We discuss a three-qubit Toffoli gate and a system of superconducting qubits as an illustration.
Integrated optics provides an ideal test bed for the emulation of quantum systems via continuous-time quantum walks. Here we study the evolution of two-photon states in an elliptic array of waveguides. We characterise the photonic chip via coherent-light tomography and use the results to predict distinct differences between temporally indistinguishable and distinguishable two-photon inputs which we then compare with experimental observations. Our work highlights the feasibility for emulation of coherent quantum phenomena in three-dimensional waveguide structures.
Measurements on a quantum particle unavoidably affect its state, since the otherwise unitary evolution of the system is interrupted by a non-unitary projection operation. In order to probe measurement-induced effects in the state dynamics using a quantum simulator, the challenge is to implement controlled measurements on a small subspace of the system and continue the evolution from the complementary subspace. A powerful platform for versatile quantum evolution is represented by photonic quantum walks due to their high control over all relevant parameters. However, measurement-induced dynamics…
