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INHOMOGENEOUS QUANTUM WALKS

Angelica Deibel, Kevin Schwenkler, Laura Veith

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

We examined three different types of one-dimensional quantum walks, all exhibit some degree of inhomogeneity. The first is inhomogeneous only insofar as the coin operator differs at even and odd positions along the line. The second has a coin operator that is inhomogeneous and oscillatory in time. The third has a coin operator that is inhomogeneous and oscillatory in space, differing at every position along the line. Much progress is made towards attaining asymptotics for the first walk, although our analytical solution does not yet agree with the numerics. An asymptotic approach is attempted for the second walk, although it seems that this approach may not be well- suited to the walk. The third walk, it seems, requires more machinery, although progress is made using symmetry and a pentadiagonal transition operator is obtained. This general pentadiagonal operator gives hope for much progress in the near future.

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We examined three different types of one-dimensional quantum walks, all exhibit some degree of inhomogeneity. The first is inhomogeneous only insofar as the coin operator differs at even and odd positions along the line. The second has a coin operator that is inhomogeneous and oscillatory in time. The third has a coin operator that is inhomogeneous and oscillatory in space, differing at every position along the line. Much progress is made towards attaining asymptotics for the first walk, although our analytical solution does not yet agree with the numerics. An asymptotic approach is attempted for the second walk, although it seems that this approach may not be well- suited to the walk. The third walk, it seems, requires more machinery, although progress is made using symmetry and a pentadiagonal transition operator is obtained. This general pentadiagonal operator gives hope for much progress in the near future.
In this paper, a study on discrete-time coined quantum walks on the line is presented. Clear mathematical foundations are still lacking for this quantum walk model. As a step toward this objective, the following question is being addressed: Given a graph, what is the probability that a quantum walk arrives at a given vertex after some number of steps? This is a very natural question, and for random walks it can be answered by several different combinatorial arguments. For quantum walks this is a highly non-trivial task. Furthermore, this was only achieved before for one specific coin operator (Hadamard operator) for walks on the line. Even considering only walks on lines, generalizing these computations to a general SU(2) coin operator is a complex task. The main contribution is a closed-form formula for the amplitudes of the state of the walk (which includes the question above) for a general symmetric SU(2) operator for walks on the line. To this end, a coin operator with parameters that alters the phase of the state of the walk is defined. Then, closed-form solutions are computed by means of Fourier analysis and asymptotic approximation methods. We also present some basic properties of the walk which can be deducted using weak convergence theorems for quantum walks. In particular, the support of the induced probability distribution of the walk is calculated. Then, it is shown how changing the parameters in the coin operator affects the resulting probability distribution.
Discrete-time quantum walk evolve by a unitary operator which involves two operators a conditional shift in position space and a coin operator. This operator entangles the coin and position degrees of freedom of the walker. In this paper, we investigate the asymptotic behavior of the coin position entanglement (CPE) for an inhomogeneous quantum walk which determined by two orthogonal matrices in one-dimensional lattice. Free parameters of coin operator together provide many conditions under which a measurement perform on the coin state yield the value of entanglement on the resulting position quantum state. We study the problem analytically for all values that two free parameters of coin operator can take and the conditions under which entanglement becomes maximal are sought.
Quantum walks are a powerful tool for developing efficient algorithms in quantum computing. This research explores two discrete-time one-dimensional quantum walks where the coin operator varies along even and odd positions on the line. We find closed-form expressions for the coefficients of the wave function for both walks and also arrive at a formula for the probability distribution for one of the walks. A significant discovery is a way to model the well-known Hadamard walk using two alternating coins.
Long-time limit distributions are key quantities for understanding the asymptotic dynamics of quantum walks, and they are known for most forms of one-dimensional quantum walks using two-state coin systems. For two-dimensional quantum walks using a four-state coin system, however, the only known limit distribution is for a walk using a parameterized Grover coin operation and analytical complexities have been a major obstacle for obtaining long-time limit distributions for other coins. In this work however, we present two new types of long-time limit distributions for walks using different forms of coin-flip operations in a four-state coin system. This opens the road towards understanding the dynamics and asymptotic behaviour for higher state coin system from a mathematical view point.
Quantum random walks have been much studied recently, largely due to their highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum random walk on the line: the use of multiple quantum ``coins'' (or more generally, coins of higher dimension) in order to diminish the effects of interference between paths. We find solutions to this system in terms of the single-coin random walk, and compare the asymptotic limit of these solutions to numerical simulations. We find exact analytical expressions for the time dependence of the first two moments, and show that in the long-time limit the ``quantum-mechanical'' behavior of the one-coin walk persists, even if each coin is flipped only twice. We further show that this is generic for a very broad class of possible walks, and that this behavior disappears only in the limit of a new coin for every step of the walk.
In this Chapter, we present some interesting properties of quantum walks on the line.We concentrate our attention in the emergence of invariance and provide some insights into the ultimate origin of the observed behavior.In the first part of the Chapter, we review the building blocks of the quantum-mechanical version of the standard random walk in one dimension.The most distinctive difference between random and quantum walks is the replacement of the random coin in the former by the action of a unitary operator upon some internal property of the later.We provide explicit expressions for the solution to the problem when the most general form for the homogeneous unitary operator is considered, and we analyze several key features of the system as the presence of symmetries or stationary limits.After that, we analyze the consequences of letting the properties of the coin operator change from site to site, and from time step to time step.In spite of this lack of homogeneity, the probabilistic properties of the motion of the walker can remain unaltered if the coin variability is chosen adequately.Finally, we show how this invariance can be connected to the gauge freedom of electromagnetism.