Research has shown that quantum walks can accelerate certain quantum algorithms and act as a universal paradigm for quantum processing. The discrete-time quantum walk (DTQW) model, owing to its discrete nature, stands out as one of the most suitable choices for circuit implementation. Nevertheless, most current implementations are characterized by extensive, multi-layered quantum circuits, leading to higher computational expenses and a notable decrease in the number of confidently executable time steps on current quantum computers. Since quantum computers are not scalable enough in this NISQ era, we also must confine ourselves to the ancilla-free frontier zone. Therefore, in this paper, we have successfully cut down the circuit cost concerning gate count and circuit depth by half through our proposed methodology in qubit systems as compared to the state-of-the-art increment-decrement approach. Furthermore, for the engineering excellence of our proposed approach, we implement DTQW in any finite-dimensional quantum system with akin efficiency. To ensure an efficient implementation of quantum walks without requiring ancilla, we have incorporated an intermediate qudit technique for decomposing multi-qubit gates. Experimental outcomes hold significance far beyond the realm of just a few time steps, laying the groundwork for dependable implementation and utilization on quantum computers.
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…
We investigate a space-inhomogeneous discrete-time quantum walk in one dimension. We show that the walk exhibits localization by a path counting method.
Discrete-time quantum walk in one-dimension is studied from a path-integral perspective. This enables derivation of a closed-form expression for amplitudes corresponding to any coin-position basis of the state vector of the quantum walker at an arbitrary step of the walk. This provides a new approach to the foundations and applications of quantum walks.
A proof that continuous time quantum walks are universal for quantum computation, using unweighted graphs of low degree, has recently been presented by Childs [PRL 102 180501 (2009)]. We present a version based instead on the discrete time quantum walk. We show the discrete time quantum walk is able to implement the same universal gate set and thus both discrete and continuous time quantum walks are computational primitives. Additionally we give a set of components on which the discrete time quantum walk provides perfect state transfer.
We consider discrete-time nearest-neighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk.
We show that a quantum state transfer, previously studied as a continuous time process in networks of interacting spins, can be achieved within the model of discrete time quantum walks with position dependent coin. We argue that due to additional degrees of freedom, discrete time quantum walks allow to observe effects which cannot be observed in the corresponding continuous time case. First, we study a discrete time version of the engineered coupling protocol due to Christandl et. al. [Phys. Rev. Lett. 92, 187902 (2004)] and then discuss the general idea of conversion between continuous time q…
We analyze final-time dependent discrete-time quantum walks in one dimension. We compute asymptotics of the return probability of the quantum walk by a path counting approach. Moreover, we discuss a relation between the quantum walk and the corresponding final-time dependent classical random walk.
Higher-order topological insulators not only exhibit exotic bulk-boundary correspondence principle, but also have an important application in quantum computing. However, they have never been achieved in quantum walk. In this paper, we construct a two-dimensional coinless discrete-time quantum walk to simulate second-order topological insulator with zero-dimensional corner states. We show that both of the corner and edge states can be observed through the probability distribution of the walker after multi-step discrete-time quantum walks. Furthermore, we demonstrate the robustness of the topolo…
The quantum random walk is a possible approach to construct new quantum algorithms. Several groups have investigated the quantum random walk and experimental schemes were proposed. In this paper we present the experimental implementation of the quantum random walk algorithm on a nuclear magnetic resonance quantum computer. We observe that the quantum walk is in sharp contrast to its classical counterpart. In particular, the properties of the quantum walk strongly depends on the quantum entanglement.
In this paper we obtain some properties of orthogonal polynomials given by a weight function which is a limit density of a rescaled discrete-time quantum walk on the line.
I obtain the dynamics of the continuous time quantum walk on a $d$-dimensional lattice, with periodic boundary conditions, as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of arXiv:quant-ph/0606050 which proved this limit for the infinite line. By highlighting the main features of the limiting procedure, I then extend it to general graphs. For a given discrete time quantum walk on a general graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics.
Recently, quantized versions of random walks have been explored as effective elements for quantum algorithms. In the simplest case of one dimension, the theory has remained divided into the discrete-time quantum walk and the continuous-time quantum walk. Though the properties of these two walks have shown similarities, it has remained an open problem to find the exact relation between the two. The precise connection of these two processes, both quantally and classically, is presented. Extension to higher dimensions is also discussed.
Quantum walks have emerged as an interesting alternative to the usual circuit model for quantum computing. While still universal for quantum computing, the quantum walk model has very different physical requirements, which lends itself more naturally to some physical implementations, such as linear optics. Numerous authors have considered walks with one or two walkers, on one dimensional graphs, and several experimental demonstrations have been performed. In this paper we discuss generalizing the model of discrete time quantum walks to the case of an arbitrary number of walkers acting on arbit…
In this work we explain the necessity for consistently labeled rotation maps for efficiently computing coined discrete time quantum walks on regular graphs.
We extend the idea of a discrete-time quantum walk on a graph by placing a qubit on each vertex, and allowing the walker to interact with the qubit at its current position. We show that allowing for a controlled-Z interaction at each time step between the walker and the vertex qubits generates multipartite entanglement between the qubits. We demonstrate that for particular coin operators the system generates a highly entangled cluster state, of use in measurement based quantum computation.
We introduce the driven discrete time quantum walk, where walkers are added during the walk instead of only at the beginning. This leads to interference in walker number and very different dynamics when compared to the original quantum walk. These dynamics have two regimes, which we illustrate using the one-dimensional line. Then, we explore a search application which has certain advantages over current search protocols, namely that it does not require a complicated initial state nor a specific measurement time to observe the marked state. Finally, we describe a potential experimental implemen…
In discrete time, coined quantum walks, the coin degrees of freedom offer the potential for a wider range of controls over the evolution of the walk than are available in the continuous time quantum walk. This paper explores some of the possibilities on regular graphs, and also reports periodic behaviour on small cyclic graphs.
A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in $2$D curved space-time is revisited. Usual continuous covariant derivatives and spin-connections are generalized into discrete covariant derivatives along the lattice coordinates and discrete connections. The concepts of metrics and $2$-beins are also extended to the discrete realm. Two slightly different Riemann curvatures are then defined on the space-time lattice as the curvatures of the discrete spin connection. These two curvatures are closely related and one of them tends at the continuous limit …
Berry and Wang [Phys. Rev. A {\bf 83}, 042317 (2011)] show numerically that a discrete-time quantum random walk of two noninteracting particles is able to distinguish some non-isomorphic strongly regular graphs from the same family. Here we analytically demonstrate how it is possible for these walks to distinguish such graphs, while continuous-time quantum walks of two noninteracting particles cannot. We show analytically and numerically that even single-particle discrete-time quantum random walks can distinguish some strongly regular graphs, though not as many as two-particle noninteracting d…
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 consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph $X$. Given that $X$ has diameter $d$ and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on $X$ is a product of at most $d$ commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of $X$. We also obtain a similar factorization for any graph $X$ in a Bose Mesner algebra.
Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete-time q…
We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. [1] showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle [2], expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al. [3].
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…
We propose a quantum-electrodynamics scheme for implementing the discrete-time, coined quantum walk with the walker corresponding to the phase degree of freedom for a quasi-magnon field realized in an ensemble of nitrogen-vacancy centres in diamond. The coin is realized as a superconducting flux qubit. Our scheme improves on an existing proposal for implementing quantum walks in cavity quantum electrodynamics by removing the cumbersome requirement of varying drive-pulse durations according to mean quasiparticle number. Our improvement is relevant to all indirect-coin-flip cavity quantum-electr…
We show that probability is locally conserved in discrete time quantum walks, corresponding to a particle evolving in discrete space and time. In particular, for a spatial structure represented by an arbitrary directed graph, and any unitary evolution of a particle which respects that locality structure, we can define probability currents which also respect the locality structure and which yield the correct final probability distribution.
The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time random walk have recently been attracted from various and interdisciplinary fields. In this review, the weak limit theorem, that is, the asymptotic behavior, of the one-dimensional discrete time quantum walk is analytically shown. From the limit distribution of the discrete time quantum walk, the discrete time quantum walk can be taken as the quantum dynamical simulator of some physical systems.
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 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 discuss how the coined quantum walk on the line or on the circle can be implemented using optical waves. We propose several optical cavity configurations for these implementations.
Estimation of the coin parameter(s) is an important part of the problem of implementing more robust schemes for quantum simulation using quantum walks. We present the estimation of the quantum coin parameter used for one-dimensional discrete-time quantum walk evolution using machine learning algorithms on their probability distributions. We show that the models we have implemented are able to estimate these evolution parameters to a good accuracy level. We also implement a deep learning model that is able to predict multiple parameters simultaneously. Since discrete-time quantum walks can be u…
Quantum random walks are shown to have non-intuitive dynamics which makes them an attractive area of study for devising quantum algorithms for long-standing open problems as well as those arising in the field of quantum computing. In the case of continuous-time quantum random walks, such peculiar dynamics can arise from simple evolution operators closely resembling the quantum free-wave propagator. We investigate the divergence of quantum walk dynamics from the free-wave evolution and show that in order for continuous-time quantum walks to display their characteristic propagation, the state sp…
We perform a systematic study of the discrete time Quantum Walk on one dimension using Wigner functions, which are generalized to include the chirality (or coin) degree of freedom. In particular, we analyze the evolution of the negative volume in phase space, as a function of time, for different initial states. This negativity can be used to quantify the degree of departure of the system from a classical state. We also relate this quantity to the entanglement between the coin and walker subspaces.
The theory of random walks on finite graphs is well developed with numerous applications. In quantum walks, the propagation is governed by quantum mechanical rules; generalizing random walks to the quantum setting. They have been successfully applied in the development of quantum algorithms. In particular, to solve problems that can be mapped to searching or property testing on some specific graph. In this paper we investigate the discrete time coined quantum walk (DTCQW) model using tools from non-commutative Fourier analysis. Specifically, we are interested in characterizing the DTCQW on Cay…
We formalize a notion of discrete Lorentz transforms for Quantum Walks (QW) and Quantum Cellular Automata (QCA), in (1 + 1)-dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit equivalence rules. Within this framework, we show the first-order-only covariance of the Dirac QW. We then introduce the Clock QW and the Clock QCA, and prove that they are exactly discrete Lorentz covariant. The theory also allows for non-homogeneous Lorentz transforms, between non-inertial frames.
We address the question of symmetries of an important type of quantum walks. We introduce all the necessary definitions and provide a rigorous formulation of the problem. Using a thorough analysis, we reach the complete answer by presenting a constructive method of finding all solutions of the problem with minimal additional assumptions. We apply the results on an example of a quantum walk on a line to demonstrate the practical significance of the theory.
This note introduces some examples of quantum random walks in d-dimensional Eucilidean space and proves the weak convergence of their rescaled n-step densities. One of the examples is called the Plancherel quantum walk because the "quantum coin flip" is the Fourier Integral (or Plancherel) Transform. The other examples are the Birkhoff quantum walks, so named because the coin flips are effected by means of measure preserving transformations to which the Birkhoff's Ergodic Theorem is applied.
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.
We propose a scheme for perfect transfer of an unknown qubit state via the discrete-time quantum walk on a line or a circle. For this purpose, we introduce an additional coin operator which is applied at the end of the walk. This operator does not depend on the state to be transferred. We show that perfect state transfer over an arbitrary distance can be achieved only if the walk is driven by an identity or a flip coin operator. Other biased coin operators and Hadamard coin allow perfect state transfer over finite distances only. Furthermore, we show that quantum walks ending with a perfect st…
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 identify a key difference between quantum search by discrete- and continuous-time quantum walks: a discrete-time walk typically performs one walk step per oracle query, whereas a continuous-time walk can effectively perform multiple walk steps per query while only counting query time. As a result, we show that continuous-time quantum walks can outperform their discrete-time counterparts, even though both achieve quadratic speedups over their corresponding classical random walks. To provide greater equity, we allow the discrete-time quantum walk to also take multiple walk steps per oracle qu…
We theoretically analyze the dynamical evolution of photonic quantum walks on Möbius strips and other exotic structures in 3D integrated photonics. Our flexible design allows discrete observations of continuous time quantum walks of photons in a variety of waveguide arrays. Furthermore, our design allows one to inject photons during the evolution, allowing the possibility of interacting with the photons as they are 'walking'. We find that non-trivial array topologies introduce novel time-dependent symmetries of the two-photon correlations. These properties allow a large degree of control for q…
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.
In this paper, we introduce a quantum walk whose local scattering at each vertex is denoted by a unitary circulant matrix; namely the circulant quantum walk. We also introduce another quantum walk induced by the circulant quantum walk; namely the optical quantum walk, whose underlying graph is a $2$-regular directed graph and obtained by blowing up the original graph in some way. We propose a design of an optical circuit which implements the stationary state of the optical quantum walk. We show that if the induced optical quantum walk does not have $+1$ eigenvalue, then the stationary state of…
Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the `glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum …
The discrete time quantum walk (DTQW) is a universal quantum computational model. Significant relationships between discrete and corresponding continuous quantum systems have been studied since the work of Pauli and Feynman. This work continues the study of relationships between discrete quantum models and their ostensive continuum counterparts by developing a formal transition between discrete and continuous quantum systems through a formal framework for continuum limits of the DTQW. Under this framework, we prove two constructive theorems concerning which internal discrete transitions ("coin…
In quantum computation theory, quantum random walks have been utilized by many quantum search algorithms which provide improved performance over their classical counterparts. However, due to the importance of the quantum decoherence phenomenon, decoherent quantum walks and their applications have been studied on a wide variety of structures. Recently, a unified time-inhomogeneous coin-turning random walk with rescaled limiting distributions, Bernoulli, uniform, arcsine and semi-circle laws as parameter varies have been obtained. In this paper we study the quantum analogue of these models. We o…
We show a perfect state transfer of an arbitrary unknown two-qubit state can be achieved via a discrete-time quantum walk with various settings of coin flippings, and extend this method to distribution of an arbitrary unknown multi-qubit entangled state between every pair of sites in the multi-dimensional network. Furthermore, we study the routing of quantum information on this network in a quantum walk architecture, which can be used as quantum information processors to communicate between separated qubits.
We introduce history-dependent discrete-time quantum random walk models by adding uncorrelated memory terms and also by modifying Hamiltonian of the walker to include couplings with memory-keeping agents. We next numerically study the correlation effects in these models. We also propose a correlation exponent as a relevant and promising tool for investigation of correlation or memory (hence non-Markovian) effects. Our analysis can easily be applied to more realistic models in which different regimes may emerge because of competition between different underlying physical mechanisms.
Quantum recurrence theorem holds for quantum systems with discrete energy eigenvalues and fails to hold in general for systems with continuous energy. We show that during quantum walk process dominated by interference of amplitude corresponding to different paths fail to satisfy the complete quantum recurrence theorem. Due to the revival of the fractional wave packet, a fractional recurrence characterized using quantum Pólya number can be seen.
Within a special multi-coin quantum walk scheme we analyze the effect of the entanglement of the initial coin state. For states with a special entanglement structure it is shown that this entanglement can be meausured with the mean value of the walk, which depends on the i-concurrence of the initial coin state. Further on the entanglement evolution is investigated and it is shown that the symmetry of the probability distribution is reflected by the symmetry of the entanglement distribution.
Following [Konno, arXiv:1112.4335], it is natural to ask: What is the Ito's formula for the discrete time quantum walk on a graph different than Z, the set of integers? In this paper we answer the question for the discrete time quantum walk on Z^2, the square lattice.
We show that certain types of quantum walks can be modeled as waves that propagate in a medium with phase and group velocities that are explicitly calculable. Since the group and phase velocities indicate how fast wave packets can propagate causally, we propose the use of these wave velocities in a new definition for the hitting time of quantum walks. The new definition of hitting time has the advantage that it requires neither the specification of a walker's initial condition nor of an arrival probability threshold. We give full details for the case of quantum walks on the Cayley graphs of Ab…
