We devise a protocol to build 1D time-dependent quantum walks in 1D maximizing the spatial spread throughout the procedure. We allow only one of the physical parameters of the coin-tossing operator to vary, i.e. the angle $θ$, such that for $θ=0$ we have the $\hatσ_z$, while for $θ=π/4$ we obtain the Hadamard gate. The optimal $θ$ sequences present non-trivial patterns, with mostly $θ\approx 0$ alternated with $θ\approx π/4$ values after increasingly long periods. We provide an analysis of the entanglement properties, quasi-energy spectrum and survival probability, providing a full physical picture.
We investigate the impact of decoherence and static disorder on the dynamics of quantum particles moving in a periodic lattice. Our experiment relies on the photonic implementation of a one-dimensional quantum walk. The pure quantum evolution is characterized by a ballistic spread of a photon's wave packet along 28 steps. By applying controlled time-dependent operations we simulate three different environmental influences on the system, resulting in a fast ballistic spread, a diffusive classical walk and the first Anderson localization in a discrete quantum walk architecture.
We generalize the discrete quantum walk on the line using a time dependent unitary coin operator. We find an analytical relation between the long-time behaviors of the standard deviation and the coin operator. Selecting the coin time sequence allows to obtain a variety of predetermined asymptotic wave-function spreadings: ballistic, sub-ballistic, diffusive, sub-diffusive and localized.
We present a generalized version of the discrete time quantum walk, using the SU(2) operation as the quantum coin. By varying the coin parameters, the quantum walk can be optimized for maximum variance subject to the functional form $σ^2 \approx N^2$ and the probability distribution in the position space can be biased. We also discuss the variation in measurement entropy with the variation of the parameters in the SU(2) coin. Exploiting this we show how quantum walk can be optimized for improving mixing time in an $n$-cycle and for quantum walk search.
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…
In a recent work by Novo et al. (Sci. Rep. 5, 13304, 2015), the invariant subspace method was applied to the study of continuous-time quantum walk (CTQW). The method helps to reduce a graph into a simpler version that allows more transparent analyses of the quantum walk model. In this work, we adopt the aforementioned method to investigate the optimality of a quantum walk search of a marked element on a complete multi-partite graph. We formulate the eigenbasis that would facilitate the transport between the two lowest energy eigenstates and demonstrate how to set the appropriate coupling facto…
We study a 2-D disordered time-discrete quantum walk based on 1-D `generalized elephant quantum walk' where an entangling coin operator is assumed and which paves the way to a new set of properties. We show that considering a given disorder in one direction, it is possible to control the degree of spreading and entanglement in the other direction. This observation helps assert that the random quantum walks of this ilk serve as a controllable decoherence channel with the degree of randomness being the tunable parameter and highlight the role of dimensionality in quantum systems regarding inform…
This paper examines the stability of the quantum random walk search algorithm, when the walk coin is constructed by generalized Householder reflection and additional phase shift, against inaccuracies in the phases used to construct the coin. The optimization of the algorithm is done by numerical methods - Monte Carlo, neural networks, and supervised machine learning. The results of numerical simulations show that, with such a construction of the Householder reflection, the algorithm is more stable to inaccuracies in the specific values of these phases, as long as it is possible to control the …
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 investigate the ballistic spreading behavior of the one-dimensional discrete time quantum walks whose time evolution is driven by any balanced quantum coin. We obtain closed-form expressions for the long-time variance of position of quantum walks starting from any initial qubit (spin-$1/2$ particle) and position states following a delta-like (local), Gaussian and uniform probability distributions. By averaging over all spin states, we find out that the average variance of a quantum walk starting from a local state is independent of the quantum coin, while from Gaussian and uniform states it…
Infinite reduction in absorbing time in quantum walks over classical ones
Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute（哈里什-昌德拉研究所，霍米·巴哈国家学院下属机构）；National Institute of Technology Agartala（阿加塔拉国立技术学院）
We introduce the Peierls substitution to a two-dimensional discrete-time quantum walk on a square lattice to examine the spreading dynamics and the coin-position entanglement in the presence of an artificial gauge field. We use the ratio of the magnetic flux through the unit cell to the flux quantum as a control parameter. For a given flux ratio, we obtain faster spreading for a small number of steps and the walker tends to be highly localized around the origin. Moreover, the spreading of the walk can be suppressed and decreased within a limited time interval for specific rational values of fl…
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 show that the standard quantum-walk quantum-to-classical transition, characterized by ballistic-to-diffusive spreading of the walker's position, can be controlled by externally modulating the coin state. We illustrate by showing an oscillation between classical diffusive and quantum ballistic spreading using numerical and asymptotically exact closed-form solutions, and we prove that the walker is in a controllable incoherent mixture of classical and quantum walks with a reversible quantum-to-classical transition.
We analyze the quantum walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier transform arise as natural extensions of the quantum coin toss in the one-dimensional walk simulation, and other illustrative transformations are also investigated. We find that entanglement between the dimensions serves to reduce the rate of spread of the quantum walk. The classical limit is obtained by introducing a random phase variable.
Quantum walk acts obviously different from its classical counterpart, but decoherence will lessen and close the gap between them. To understand this process, it is necessary to investigate the evolution of quantum walk under different situation of decoherence. In this article, we study a non-Markovian decoherent quantum walk on a line. In the short time regime, the behavior of the walk deviates from both idea quantum walks and classical random walks. The position variance as a measure of quantum walk starts oscillating from the first several steps and tends to be linear on time and showing a d…
In this paper, we consider the quantum walk on $\mathbb{Z}$ with attachment of one-length path periodically. This small modification to $\mathbb{Z}$ provides localization of the quantum walk. The eigenspace causing this localization is generated by finite length round trip paths. We find that the localization is due to the eigenvalues of an underlying random walk. Moreover we find that the transience of the underlying random walk provides a slow down of the pseudo velocity of the induced quantum walk and a different limit distribution from the Konno distribution.
We study the dynamics of continuous-time quantum walks (CTQW) on networks with highly degenerate eigenvalue spectra of the corresponding connectivity matrices. In particular, we consider the two cases of a star graph and of a complete graph, both having one highly degenerate eigenvalue, while displaying different topologies. While the CTQW spreading over the network - in terms of the average probability to return or to stay at an initially excited node - is in both cases very slow, also when compared to the corresponding classical continuous-time random walk (CTRW), we show how the spreading i…
Quantum walks, both discrete (coined) and continuous time, form the basis of several quantum algorithms and have been used to model processes such as transport in spin chains and quantum chemistry. The enhanced spreading and mixing properties of quantum walks compared with their classical counterparts have been well-studied on regular structures and also shown to be sensitive to defects and imperfections in the lattice. As a simple example of a disordered system, we consider percolation lattices, in which edges or sites are randomly missing, interrupting the progress of the quantum walk. We us…
We quantitatively differentiate between the spreads of discrete-time quantum and classical random walks on a cyclic graph. Due to the closed nature of any cyclic graph, there is additional "collision"- like interference in the quantum random walk along with the usual interference in any such walk on any graph, closed or otherwise. We find that the quantum walker remains localized in comparison to the classical one, even in the absence of disorder, a phenomenon that is potentially attributable to the additional interference in the quantum case. This is to be contrasted with the situation on ope…
Quantum walks are promising for information processing tasks because on regular graphs they spread quadratically faster than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, whereby the spread of the walker stays within a finite region even in the infinite time limit. It is therefore important to understand when we can expect a quantum walk to be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a generic one-dimensional qua…
Transport phenomena play a crucial role in modern physics and applied sciences. Examples include the dissipation of energy across a large system, the distribution of quantum information in optical networks, and the timely modeling of spreading diseases. In this work, we experimentally prove the feasibility of disordered quantum walks to realize a quantum simulator that is able to model general subdiffusive phenomena, exhibiting a sublinear spreading in space over time. Our experiment simulates such phenomena by means of a finely controlled insertion of various levels of disorder during the evo…
Quantum walks are known to propagate quadratically faster than their classical counterparts and are used to model dynamics in various quantum systems. The spread of the quantum walk in position space shows anomalous diffusion behavior. By controlling the action of quantum coin operation on the corresponding coin degree of freedom of the walker, one can demonstrate control over the diffusion behavior. In this work, we report different forms of coin operations on quantum walks exhibiting anomalous diffusion behavior. Homogeneous and accelerated quantum walks display superdiffusive behavior, wher…
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…
We study the spreading of a quantum particle placed in a single site of a lattice or binary tree with the Hamiltonian permitting particle number changes. We show that the particle number-changing interactions accelerate the spreading beyond the ballistic expansion limit by inducing off-resonant Rabi oscillations between states of different numbers of particles. We consider the effect of perturbative number-changing couplings on Anderson localization in one-dimensional disordered lattices and show that they lead to decrease of localization. The effect of these couplings is shown to be larger at…
We study maximal coin-position entanglement generation via a discrete-time quantum walk, in which the coin operation is randomly selected from one of two coin operators set at each step. We solve maximal entanglement generation as an optimization problem with quantum process fidelity as the cost function. Then we determine the maximal entanglement that can be rigorously generated for any step beyond the second regardless of initial condition with appropriate coin sequences. The simplest coin sequence comprising Hadamard and identity operations is equivalent to the generalized elephant quantum …
The evolution of a walker in standard "Discrete-time Quantum Walk (DTQW)" is determined by coin and shift unitary operators. The conditional shift operator shifts the position of the walker to right or left by unit step size while the direction of motion is specified by the coin operator. This scenario can be generalized by choosing the step size randomly at each step in some specific interval. For example, the value of the roll of a dice can be used to specify the step size after throwing the coin. Let us call such a quantum walk "Discrete-time Random Step Quantum Walk (DTRSQW)". A completely…
By pursuing the deep relation between the one-dimensional Dirac equation and quantum walks, the physical role of quantum interference in the latter is explained. It is shown that the time evolution of the probability density of a quantum walker, initially localized on a lattice, is directly analogous to relativistic wave-packet spreading. Analytic wave-packet solutions reveal a striking connection between the discrete and continuous time quantum walks.
We study the effects of braiding interactions on single anyon dynamics using a quantum walk model on a quasi-1-dimensional ladder filled with stationary anyons. The model includes loss of information of the coin and nonlocal fusion degrees of freedom on every second time step, such that the entanglement between the position states and the exponentially growing auxiliary degrees of freedom is lost. The computational complexity of numerical calculations reduces drastically from the fully coherent anyonic quantum walk model, allowing for relatively long simulations for anyons which are spin-1/2 i…
We show how multi-walker quantum walks can be implemented in a quantum quincunx created via cavity quantum electrodynamics. The implementation of a quantum walk with a multi-walker opens up the interesting possibility to introduce entanglement and more advanced walks. With different coin tosses and initial states the multi-walker quantum walk shows different probability distributions which deviate strongly from the classical random walks with quadratic enhanced spreadings and localization effects. By introducing decoherence, the transition from quantum walks to the classical versions is observ…
Quantum Stochastic Walks (QSW) allow for a generalization of both quantum and classical random walks by describing the dynamic evolution of an open quantum system on a network, with nodes corresponding to quantum states of a fixed basis. We consider the problem of quantum state discrimination on such a system, and we solve it by optimizing the network topology weights. Finally, we test it on different quantum network topologies and compare it with optimal theoretical bounds.
We perform a finite-time scaling analysis over the detrapping point of a three-state quantum walk on the line. The coin operator is parameterized by $ρ$ that controls the wavepacket spreading velocity. The input state prepared at the origin is set as symmetric linear combination of two eigenstates of the coin operator with a characteristic mixing angle $θ$, one of them being the component that results in full spreading occurring at $θ_c(ρ)$ for which no fraction of the wavepacket remains trapped near the initial position. We show that relevant quantities such as the survival probability and th…
It is well known that many real world networks have the power-law degree distribution (scale-free property). However there are no rigorous results for continuous-time quantum walks on such realistic graphs. In this paper, we analyze space-time behaviors of continuous-time quantum walks and random walks on the threshold network model which is a reasonable candidate model having scale-free property. We show that the quantum walker exhibits localization at the starting point, although the random walker tends to spread uniformly.
We attempt to extract a homological structure of two kinds of graphs by the Grover walk. The first one consists of a cycle and two semi-infinite lines and the second one is assembled by a periodic embedding of the cycles in $\mathbb{Z}$. We show that both of them have essentially the same eigenvalues induced by the existence of cycles in the infinite graphs. The eigenspace of the homological structure appears as so called {\it localization} in the Grover walks, in that the walk is partially trapped by the homological structure. On the other hand, the difference of the absolutely continuous par…
We present the first robust implementation of a coined quantum walk over five steps using only passive optical elements. By employing a fiber network loop we keep the amount of required resources constant as the walker's position Hilbert space is increased. We observed a non-Gaussian distribution of the walker's final position, thus characterizing a faster spread of the photon wave-packet in comparison to the classical random walk. The walk is realized for many different coin settings and initial states, which opens the way for the implementation of a quantum walk-based search algorithm.
We show analytically that particle trapping appears in a quantum process called "quantum walk", in which the particle moves macroscopically correlating to the inner states. It has been well known that a particle in the ``Hadamard walk" with two inner states spreads away quickly on a line. In contrast, we find one-dimensional quantum walk with multi-state in which a particle stays at the starting point entirely with high positive probability. This striking difference is explained from difference between degeneration of eigenvalues of the time-evolution matrices.
In all existing quantum walk models, the assumption about a pre-existing fixed background causal structure is always made and has been taken for granted. Nevertheless, in this work we will get rid of this tacit assumption especially by introducing indefinite causal order ways of coin tossing and investigate this modern scenario. We find that an ideal-shape and fast-spreading uniform distribution can be prepared with our new model. First we will show how an always-symmetrical instantaneous distribution appears in an indefinite causal order quantum walk, which then paves the way for deriving con…
We experimentally investigate a discrete time quantum walk in a system of coupled fiber loops and observe typical phenomena known from the wave propagation in periodic structures as ballistic spreading or an oscillation between two internal quantum states similar to Zitterbewegung (trembling motion). If a position-dependent phase gradient is applied we find localization and Bloch oscillations of the field for moderate as well as Landau-Zener tunneling for strong phase gradients.
Recently, it was introduced a generalization of a nonstandard step operator named the elephant quantum walk (EQW). With proper statistical distribution for the steps, that generalized EQW (gEQW) can be tuned to exhibit a myriad of dynamical scaling behavior ranging from standard diffusion to %and superdiffusion to ballistic and hyperballistic spreading. In this work, we study the influence of the statistics of the step size and the delocalization of the initial states on the entanglement entropy of the coin. Our results show that the gEQW generates maximally entangled states for almost all ini…
We show that a quantum walk process can be used to construct and secure quantum memory. More precisely, we show that a localized quantum walk with temporal disorder can be engineered to store the information of a single, unknown qubit on a compact position space and faithfully recover it on demand. Since the localization occurss with a finite spread in position space, the stored information of the qubit will be naturally secured from the simple eavesdropper. Our protocol can be adopted to any quantum system for which experimental control over quantum walk dynamics can be achieved.
Quantum walks (QWs) exhibit different properties compared with classical random walks (RWs), most notably by linear spreading and localization. In the meantime, random walks that replicate quantum walks, which we refer to as quantum-walk-replicating random walks (QWRWs), have been studied in the literature where the eventual properties of QWRW coincide with those of QWs. However, we consider that the unique attributes of QWRWs have not been fully utilized in the former studies to obtain deeper or new insights into QWs. In this paper, we highlight the directivity of one-dimensional discrete qua…
Quantum walks are well-known for their ballistic dispersion, traveling $Θ(t)$ away in $t$ steps, which is quadratically faster than a classical random walk's diffusive spreading. In physical implementations of the walk, however, the particle may need to tunnel through a potential barrier to hop, and a naive calculation suggests this could eliminate the ballistic transport. We show by explicit calculation, however, that such a loss does not occur. Rather, the $Θ(t)$ dispersion is retained, with only the coefficient changing, which additionally gives a way to detect and quantify the hopping erro…
Operator spreading, often characterized by out-of-time-order correlators (OTOCs), is one of the central concepts in quantum many-body physics. However, measuring OTOCs is experimentally challenging due to the requirement of reversing the time evolution of systems. Here we apply Floquet engineering to investigate operator spreading in a superconducting 10-qubit chain. Floquet engineering provides an effective way to tune the coupling strength between nearby qubits, which is used to demonstrate quantum walks with tunable couplings, reversed time evolution, and the measurement of OTOCs. A clear l…
We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us to answer an open question on how the graph topology ultimately bounds their mixing performance, and that of any stochastic local evolution. The results highlight that speedups in mixing and transport phenomena are not necessarily diagnostic of quantum effects, although superdiffusive spreading is more prominent with quantum walks.
In this paper, we study the properties of lackadaisical quantum walks on a line. This model is first proposed in~\cite{wong2015grover} as a quantum analogue of lazy random walks where each vertex is attached $τ$ self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate rigorously the wave function of the walker starting from the origin and obtain a long time approximation for the entire probability density function. As an application of the density function, we …
Quantum versions of random walks on the line and cycle show a quadratic improvement in their spreading rate and mixing times respectively. The addition of decoherence to the quantum walk produces a more uniform distribution on the line, and even faster mixing on the cycle by removing the need for time-averaging to obtain a uniform distribution. We calculate numerically the entanglement between the coin and the position of the quantum walker and show that the optimal decoherence rates are such that all the entanglement is just removed by the time the final measurement is made.
We investigate numerically and theoretically the effect of spatial disorder on two-dimensional split-step discrete-time quantum walks with two internal "coin" states. Spatial disorder can lead to Anderson localization, inhibiting the spread of quantum walks, putting them at a disadvantage against their diffusively spreading classical counterparts. We find that spatial disorder of the most general type, i.e., position-dependent Haar random coin operators, does not lead to Anderson localization but to a diffusive spread instead. This is a delocalization, which happens because disorder places the…
