We study the state transfer through quantum walks placed on a bounded one-dimensional path. We first consider continuous-time quantum walks from a Gaussian state. We find such a state when superposing centered on the starting and antipodal positions preserves a high fidelity for a long time and when sent on large circular graphs. Furthermore, it spreads with a null group velocity. We also explore discrete-time quantum walks to evaluate the qubit fidelity throughout the walk. In this case, the initial state is a product of states between a qubit and a Gaussian superposition of position states. Then, we add two $σ_x$ gates to confine this delocalized qubit. We also find that this bounded system dynamically enables periodic recovery of the initial separable state. We outline some applications of our results in dynamic graphs and propose quantum circuits to implement them based on the available literature.
We study the entanglement between the internal (spin) and external (position) degrees of freedom of the one-dimensional discrete time quantum walk starting from local and delocalized initial states whose time evolution is driven by Hadamard and Fourier coins. We obtain the dependence of the asymptotic entanglement with the initial dispersion of the state and establish a way to connect the asymptotic entanglement between local and delocalized states. We find out that the delocalization of the state increases the number of initial spin states which achieves maximal entanglement from two states (…
Traditional continuous variable teleportation can only approach unit fidelity in the limit of an infinite (and unphysical) amount of squeezing. We describe a new method for continuous variable teleportation that approaches unit fidelity with finite resources. The protocol is not based on squeezed states as in traditional teleportation but on an ensemble of single photon entangled states. We characterize the teleportation scheme with coherent states, Schrodinger cat states and two-mode squeezed state and we find several situations in which near-unity teleportation fidelity can be obtained with …
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…
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…
Cat states are coherent quantum superpositions of macroscopically distinct states and are useful for understanding the boundary between the classical and the quantum world. Due to their macroscopic nature, cat states are difficult to prepare in physical systems. We propose a method to create cat states in one-dimensional quantum walks using delocalized initial states of the walker. Since the quantum walks can be performed on any quantum system, our proposal enables a platform-independent realization of the cat states. We further show that the linear dispersion relation of the effective quantum…
A unit evolution step of discrete-time quantum walks is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the quantum walks starting with a localized initial state have been derived. In the present paper, we compute limit distributions of a 2-state quantum walk with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip oper…
We investigate the transport properties and entanglement between spin and position of one-dimensional quantum walks starting from a qubit over position states following a delta-like (local state) and Gaussian (delocalized state) distributions. We find out that if the initial state is delocalized enough and a NOT gate reflects this state backwards, then the interference pattern extinguishes the position dispersion without preventing the propagation of the state. This effect allows the creation of a Trojan wave packet, a non-spreading and non-stationary double-peak quantum state.
We propose a novel implementation of discrete time quantum walks for a neutral atom in an array of optical microtraps or an optical lattice. We analyze a one-dimensional walk in position space, with the coin, the additional qubit degree of freedom that controls the displacement of the quantum walker, implemented as a spatially delocalized qubit, i.e., the coin is also encoded in position space. We analyze the dependence of the quantum walk on temperature and experimental imperfections as shaking in the trap positions. Finally, combining a spatially delocalized qubit and a hyperfine qubit, we a…
We study the existence and charaterization of self-trapping phenomena in discrete-time quantum walks. By considering a Kerr-like nonlinearity, we associate an acquisition of the intensity-dependent phase to the walker while it propagates on the lattice. Adjusting the nonlinear parameter ($χ$) and the quantum gates ($θ$), we will show the existence of different quantum walking regimes, including those with travelling soliton-like structures or localized by self-trapping. This latter scenario is absent for quantum gates close enough to Pauli-X. It appears for intermediate configurations and beco…
We have studied how decoherence affects a quantum walk on the line. As expected, it is highly sensitive, consisting as it does of an extremely delocalized particle. We obtain an expression for the rate at which the standard deviation falls from the quantum value as decoherence increases and show that it is proportional to the number of decoherence "events" occuring during the walk.
We analyze the quantum walk on a cycle using discrete Wigner functions as a way to represent the states and the evolution of the walker. The method provides some insight on the nature of the interference effects that make quantum and classical walks different. We also study the behavior of the system when the quantum coin carried by the walker interacts with an environment. We show that for this system quantum coherence is robust for initially delocalized states of the walker. The use of phase-space representation enables us to develop an intuitive description of the nature of the decoherence …
We study the localization properties, energy spectra and coin-position entanglement of the aperiodic discrete-time quantum walks. The aperiodicity is described by spatially dependent quantum coins distributed on the lattice, whose distribution is neither periodic (Bloch-like) nor random (Anderson-like). Within transport properties we identified delocalized/localized quantum walks mediated by a proper adjusting of aperiodic parameter. Both scenarios are studied by exploring typical quantities (inverse participation ratio and survival probability), as well as the energy spectra of an associated …
We investigate the relation between transport properties and entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional discrete time quantum walks. We obtain closed-form expressions for the long-time position variance and asymptotic entanglement of quantum walks whose time evolution is given by any balanced quantum coin, starting from any initial qubit and position states following $δ$-like (local) and Gaussian distributions. We find out that the knowledge of the limit velocity of the walker together with the polar angle of the initial qubit provide…
The notion of fidelity susceptibility, introduced within the context of quantum metric tensor, has been an important quantity to characterize the criticality near quantum phase transitions. We demonstrate that for topological phase transitions in Dirac models, provided the momentum space is treated as the manifold of the quantum metric, the fidelity susceptibility coincides with the curvature function whose integration gives the topological invariant. Thus the quantum criticality of the curvature function near a topological phase transition also describes the criticality of the fidelity suscep…
We perform the fidelity analysis for Boltzmann-Gibbs-like states in order to investigate whether the topological order of 1D fermionic systems at zero temperature is maintained at finite temperatures. We use quantum walk protocols that are known to simulate topological phases and the respective quantum phase transitions for chiral symmetric Hamiltonians. Using the standard approaches of the fidelity analysis and the study of edge states, we conclude that no thermal-like phase transitions occur as temperature increases, i.e., the topological behaviour is washed out gradually. We also show that …
We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker's position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.
We investigate coined quantum walk search and state transfer algorithms, focusing on the complete $M$-partite graph with $N$ vertices in each partition. First, it is shown that by adding a loop to each vertex the search algorithm finds the marked vertex with unit probability in the limit of a large graph. Next, we employ the evolution operator of the search with two marked vertices to perform a state transfer between the sender and the receiver. We show that when the sender and the receiver are in different partitions the algorithm succeeds with fidelity approaching unity for a large graph. Ho…
High-fidelity quantum state transfer is critical for quantum communication and scalable quantum computation. Current quantum state transfer algorithms on the complete bipartite graph, which are based on discrete-time quantum walk search algorithms, suffer from low fidelity in some cases. To solve this problem, in this paper we propose a two-stage quantum state transfer algorithm on the complete bipartite graph. The algorithm is achieved by the generalized Grover walk with one marked vertex. The generalized Grover walk's coin operators and the query oracles are both parametric unitary matrices,…
Phases of matter with non-trivial topological order are predicted to exhibit a variety of exotic phenomena, such as the existence robust localized bound states in 1D systems, and edge states in 2D systems, which are expected to display spin-helicity, immunity to back-scattering, and weak anti-localization. In this Letter, we present an experimental observation of topological structures generated via the controlled implementation of two consecutive non-commuting rotations in photonic discrete-time quantum walks. The second rotation introduces valley-like Dirac points in the system, allowing to …
Systems with purely off-diagonal disorder have peculiar features such as the localization-delocalization transition and long-range correlations in their wavefunctions. To motivate possible experimental studies of the physics of off-diagonal disorder, we study in detail disordered discrete-time quantum walk in a finite chain, where the diagonal disorder can be set to zero by construction. Starting from a transfer matrix approach, we show, both theoretically and computationally, that the dynamics of the quantum walk with disorder manifests all the main features of off-diagonal disorder. We also …
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…
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…
Recent work has shown that a simple chain of interacting spins can be used as a medium for high-fidelity quantum communication. We describe a scheme for quantum communication using a spin system that conserves z-spin, but otherwise is arbitrary. The sender and receiver are assumed to directly control several spins each, with the sender encoding the message state onto the larger state-space of her control spins. We show how to find the encoding that maximises the fidelity of communication, using a simple method based on the singular-value decomposition. Also, we show that this solution can be u…
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 demonstrate a platform for implementing quantum walks that overcomes many of the barriers associated with photonic implementations. We use coupled fiber-optic cavities to implement time-bin encoded walks in an integrated system. We show that this platform can achieve very low losses combined with high-fidelity operations, enabling an unprecedented large number of steps in a passive system, as required for scenarios with multiple walkers. Furthermore the platform is reconfigurable, enabling variation of the coin, and readily extends to multidimensional lattices. We demonstrate variation of t…
Stability and instability of quantum evolution are studied in the interaction between a two-level atom with photon recoil and a quantized field mode in an ideal cavity, the basic model of cavity quantum electrodynamics (QED). It is shown that the Jaynes-Cummings dynamics can be unstable in the regime of chaotic walking of the atomic center-of-mass in the quantized field of a standing wave in the absence of any kind of interaction with environment. This kind of quantum instability manifests itself in strong variations of reduced quantum purity and entropy, correlating with the respective classi…
Quantum walk (QW) in presence of lattice disorders leads to a multitude of interesting phenomena, such as Anderson localization. While QW has been realized in various optical and atomic systems, its implementation with superconducting qubits still remains pending. The major challenge in simulating QW with superconducting qubits emerges from the fact that on-chip superconducting qubits cannot hop between two adjacent lattice sites. Here we overcome this barrier and develop a scheme to realize the discrete time QW by placing a pair of superconducting qubits on each site of a 1D lattice and treat…
An ideal quantum walk transitions from one vertex to another with perfect fidelity, but in physical systems, the particle may be hindered by potential energy barriers. Then the particle has some amplitude of tunneling through the barriers, and some amplitude of staying put. We investigate the algorithmic consequence of such barriers for the quantum walk formulation of Grover's algorithm. We prove that the failure amplitude must scale as $O(1/\sqrt{N})$ for search to retain its quantum $O(\sqrt{N})$ runtime; otherwise, it searches in classical $O(N)$ time. Thus searching larger "databases" requ…
We investigate the counterparts of random walk in universal quantum computing and their implementation using standard quantum circuits. Quantum walk have been recently well investigated for traversing graphs with certain oracles. We focus our study on traversing a 1-D graph, namely a circle, and show how to implement discrete circular quantum walk in quantum circuits built with universal CNOT and single quit gates. We review elementary quantum gates and circuit decomposition and propose a a generalized version of the all CNOT based quantum discrete circular walk. We simulated these circuits on…
Quantum state transfer between different sites is a significant problem for quantum networks and quantum computers. By selecting quantum walks with two coins as the basic model and two coin spaces as the communication carriers, we successfully implement quantum state transfer on various graphs (EPL, \textbf{124} (2018) 60009) \cite{Shang_2019}. Here, we demonstrate the experimental implementation of this scheme using IBM quantum experience platform. In particular, we show the transfer of Bell state, GHZ state and W state on complete graph on the quantum device. Also, we observe that our protoc…
The development of universal quantum computers has achieved remarkable success in recent years, culminating with the quantum supremacy reported by Google. Now is possible to implement short-depth quantum circuits with dozens of qubits and to obtain results with significant fidelity. Quantum walks are good candidates to be implemented on the available quantum computers. In this work, we implement discrete-time quantum walks with one and two interacting walkers on cycles, two-dimensional lattices, and complete graphs on IBM quantum computers. We are able to obtain meaningful results using the cy…
Quantum walks are an analog of classical random walks in quantum systems. Quantum walks have smaller hitting times compared to classical random walks on certain types of graphs, leading to a quantum advantage of quantum-walks-based algorithms. An important feature of quantum walks is that they are accompanied by the excitation transfer from one site to another, and a moment of hitting the destination site is characterized by the maximum probability amplitude of observing the excitation on this site. It is therefore prospective to consider such problems as candidates for quantum advantage demon…
We implement the proof of principle for the quantum walk of one ion in a linear ion trap. With a single-step fidelity exceeding 0.99, we perform three steps of an asymmetric walk on the line. We clearly reveal the differences to its classical counterpart if we allow the walker/ion to take all classical paths simultaneously. Quantum interferences enforce asymmetric, non-classical distributions in the highly entangled degrees of freedom (of coin and position states). We theoretically study and experimentally observe the limitation in the number of steps of our approach, that is imposed by motion…
Quantum Random Walks, which have drawn much attention over the past few decades for their distinctly non-classical behavior, is a promising subfield within Quantum Computing. Theoretical framework and applications for these walks have seen many great mathematical advances, with experimental demonstrations now catching up. In this study, we examine the viability of implementing Coin Quantum Random Walks using a Quantum Adder based Shift Operator, with quantum circuit designs specifically for superconducting qubits. We focus on the strengths and weaknesses of these walks, particularly circuit de…
We consider a quantum walk with two marked vertices, sender and receiver, and analyze its application to perfect state transfer on complete bipartite graphs. First, the situation with both the sender and the receiver vertex in the same part of the graph is considered. We show that in this case the dynamics of the quantum walk is independent of the size of the second part and reduces to the one for the star graph where perfect state transfer is achieved. Second, we consider the situation where the sender and the receiver vertex are in the opposite parts of the graph. In such a case the state tr…
We present an experimental implementation of the coined discrete time quantum walk on a square using a three qubit liquid state nuclear magnetic resonance (NMR) quantum information processor (QIP). Contrary to its classical counterpart, we observe complete interference after certain steps and a periodicity in the evolution. Complete state tomography has been performed for each of the eight steps making a full period. The results have extremely high fidelity with the expected states and show clearly the effects of quantum interference in the walk. We also show and discuss the importance of choo…
Mesoscopic integrated circuits achieving high-fidelity control of elementary quantum systems require new methodology for benchmarking. We offer circuit-level statistical description of rare-error accumulation in terms of a universal random-walk model for on-demand electron transfer. For a high-fidelity single-electron circuit, realized in the experiment as a chain of quantum dots in a GaAs/AlGaAs heterostructure, the error of the transfer operation is probed by charge counting. Error rates for extra ($P_+$) or missing ($P_-$) electrons of the electron shuttle are measured to $P_{-}=(6.92 \pm 0…
We investigate if the degradation of a quantum directional reference frame through repeated use can be modeled as a classical direction undergoing a random walk on a sphere. We demonstrate that the behaviour of the fidelity for a degrading quantum directional reference frame, defined as the average probability of correctly determining the orientation of a test system, can be fit precisely using such a model. Physically, the mechanism for the random walk is the uncontrollable back-action on the reference frame due to its use in a measurement of the direction of another system. However, we find …
The phenomenon of localization usually happens due to the existence of disorder in a medium. Nevertheless, certain quantum systems allow dynamical localization solely due to the nature of internal interactions. We study a discrete time quantum walker which exhibits disorder free localization. The quantum walker moves on a one-dimensional lattice and interacts with on-site spins by coherently rotating them around a given axis at each step. Since the spins do not have dynamics of their own, the system poses the local spin components along the rotation axis as an extensive number of conserved mom…
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 …
In this paper we consider quantum walks whose evolution converges to the Dirac equation one in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two and three space dimensions. The behaviour of particle states, defined as states smoothly peaked in some wave-vector eigenstate of the walk, is described by an approximated dispersive differential equation that for small wave-vectors gives the usual Dirac particle and antiparticle kinematics. The accuracy of the approximation is provided in terms of a lower bound on the fidelity between th…
We introduce a fidelity-based measure $\text{D}_{\text{CQ}}(t)$ to quantify the differences between the dynamics of classical (CW) and quantum (QW) walks over a graph. We provide universal, graph-independent, analytic expressions of this quantum-classical dynamical distance, showing that at short times $\text{D}_{\text{CQ}}(t)$ is proportional to the coherence of the walker, i.e. a genuine quantum feature, whereas for long times it depends only on the size of the graph. At intermediate times, $\text{D}_{\text{CQ}}(t)$ does depend on the graph topology through its algebraic connectivity. Our re…
Generation of entangled state is of paramount importance both from quantum theoretical foundation and technology applications. Entanglement swapping provides an efficient method to generate entanglement in quantum communication protocols. However, perfect Bell measurements for qudits, the key to entanglement swapping, have been proven impossible to achieve by using only linear elements and particle detectors. To avoid this bottleneck, we propose a novel scheme to generate entangled state including two-qubit entangled state, two-qudit entangled state, three-qubit GHZ state and three-qudit GHZ s…
We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles C_4, C_6 and C_8 for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of C_4 could be generalised to an arbitrary number of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for ap…
