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Quantum Walks with Entangled Coins

Salvador Elías Venegas-Andraca, Jonathan L. Ball, Keith R. Burnett, Sougato Bose

Published Nov 19, 2004• 34 citations
OpenAlex

Abstract

We present a mathematical formalism for the description of unrestricted quantum walks with entangled coins and one walker. Furthermore, we examine the numerical behaviour of such walks using Bell states as initial coin states, two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins. We find the behavior of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also find that simply by changing the shift operator, we can generate widely different distributions.

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Quantum walks with entangled coins

S E Venegas-Andraca, J L Ball, K Burnett +1 more

We present a mathematical formalism for the description of un- restricted quantum walks with entangled coins and one walker. The numerical behaviour of such walks is examined when using a Bell state as the initial coin state, with two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins as well as quantum walks with coins sharing different degrees of entanglement. We illustrate that the behaviour of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also demonstrate that simply by changing the shift operator, we can generate widely different distributions. We also compare the behaviour of quantum walks with maximally entangled coins with that of quantum walks with non-entangled coins. Finally, we show that the use of different shift operators on two and three qubit coins leads to different position probability distributions in one- and two-dimensional graphs.

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arXiv:1909.08527•Sep 17, 2019•3 citations
We present a mathematical formalism for the description of unrestricted quantum walks with entangled coins and one walker. Furthermore, we examine the numerical behaviour of such walks using Bell states as initial coin states, two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins. We find the behavior of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also find that simply by changing the shift operator, we can generate widely different distributions.
We present a mathematical formalism for the description of un- restricted quantum walks with entangled coins and one walker. The numerical behaviour of such walks is examined when using a Bell state as the initial coin state, with two different coin operators, two different shift operators, and one walker. We compare and contrast the performance of these quantum walks with that of a classical random walk consisting of one walker and two maximally correlated coins as well as quantum walks with coins sharing different degrees of entanglement. We illustrate that the behaviour of our walk with entangled coins can be very different in comparison to the usual quantum walk with a single coin. We also demonstrate that simply by changing the shift operator, we can generate widely different distributions. We also compare the behaviour of quantum walks with maximally entangled coins with that of quantum walks with non-entangled coins. Finally, we show that the use of different shift operators on two and three qubit coins leads to different position probability distributions in one- and two-dimensional graphs.
Quantum walks can be used either as tools for quantum algorithm development or as entanglement generators, potentially useful to test quantum hardware. We present a novel algorithm based on a discrete Hadamard quantum walk on a line with one coin and two walkers whose purpose is to generate entanglement between walkers. We provide several classical computer simulations of our quantum algorithm in which we show that, although the asymptotical amount of entanglement generated between walkers does not reach the highest degree of entanglement possible at each step for either coin measurement outcome, the entanglement ratio (entanglement generated/highest value of entanglement possible, for each step) tends to converge, and the actual convergence value depends on the coin initial state and on the coin measurement outcome. Furthermore, our numerical simulations show that, for the quantum walks used in our algorithm, the value towards which entanglement ratio converges also depends on the position probability distribution symmetry of a quantum walk computed with one single walker and the same coin initial state employed in the corresponding quantum walk with two walkers.
We offer theoretical explanations for some recent observations in numerical simulations of quantum random walks (QRWs). Specifically, in the case of a QRW on the line with one particle (walker) and two entangled coins, we explain the phenomenon, called ``localization,'' whereby the probability distribution of the walker's position is seen to exhibit a persistent major ``spike'' (or ``peak'') at the initial position and two other minor spikes which drift to infinity in either direction. Another interesting finding in connection with QRWs of this sort pertains to the limiting behavior of the position probability distribution. It is seen that the probability of finding the walker at any given location eventually becomes stationary and nonvanishing. We explain these observations in terms of the degeneration of some eigenvalue of the time evolution operator U(k)U(k)U(k). An explicit general formula is derived for the limiting probability, from which we deduce the limiting value of the height of the observed spike at the origin. We show that the limiting probability decreases quadratically for large values of the position xxx. We locate the two minor spikes and demonstrate that their positions are determined by the phases of nondegenerated eigenvalues of U(k)U(k)U(k). Finally, for fixed time ttt sufficiently large, we examine the dependence on ttt of the probability of finding a particle at a given location xxx.
We introduce a generalization of quantum walks with entangled coins consisting of a model of discrete quantum walks with coin pairs under various degrees of entanglement and walkers in quantum superposition as initial states. We introduce novel position probability distributions that may be used for algorithm development based on quantum-mechanical phenomena. Also, we numerically show that the skewness of position probability distribution produced by using coin initial state with various degrees of entanglement cannot be easily inferred.
Quantum walk (QW) is the quantum analog of the random walk. QW is an integral part of the development of numerous quantum algorithms. Hence, an in-depth understanding of QW helps us to grasp the quantum algorithms. We revisit the one-dimensional discrete-time QW and discuss basic steps in detail by incorporating the most general coin operator, constant in both space and time, and a localized initial state using numerical modeling. We investigate the impact of each parameter of the general coin operator on the probability distribution of the quantum walker. We show that by tuning the parameters of the general coin, one can regulate the probability distribution of the walker. We provide an algorithm for the one-dimensional quantum walk driven by the general coin operator. The study on general coin operators also includes the popular coins – Hadamard, Grover, and Fourier.
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 phenomena not described by the single-particle model, due to entanglement and interaction among the walkers. In this context, we present here a thorough numerical study of an one dimensional system of two quantum walkers exhibiting rich collective dynamics controlled by simple time-dependent unitary coins proposed in [Phys. Rev. A \textbf{80}, 042332(2009)] and [Phys. Rev. A \textbf{73},062304(2006)]. We study how the interplay of coin time-dependence, simple interaction schemes, entanglement and the relative phase between the coin states of the particles influences the evolution of the quantum walk. The results show that the system offers a rich variety of collective dynamical behavior while being controlled by time dependent coins. In particular, we find and characterize fascinating two-body localization phenomena with tunable quasiperiodic dynamics of correlations and entanglements which are quantities of quantum origin.