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Quantum Walk on the Line

Nayak Ashwin, Ashvin Vishwanath

Published Dec 13, 2000• 122 citations
OpenAlex

Abstract

Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider {\em quantum\/} walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''''. In particular, we show that after~$t$ time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval~$[-t/\sqrt{2},\;t/\sqrt{2}]$. This implies that the same walk defined on the circle mixes in {\em linear\/} time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.

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Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider {\em quantum/} walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''''. In particular, we show that after~ttt time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval~[−t/2,  t/2][-t/\sqrt{2},\;t/\sqrt{2}][−t/2​,t/2​]. This implies that the same walk defined on the circle mixes in {\em linear/} time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.
Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider_quantum_ walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''. We show that after t time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval [-t/sqrt(2),t/sqrt(2)]. This implies that the same walk defined on the circle mixes in_linear_ time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.
Many classical randomized algorithms (e.g. approximation algorithms for #P-complete problems) utilize the following random walk algorithm for almost uniform sampling from a state space S of cardinality N : run a symmetric ergodic Markov chain P on S for long enough to obtain a random state from within total variation distance of the uniform distribution over S . The running time of this algorithm, the so-called mixing time of P , is O (δ −1 (log N +log −1 )), where δ is the spectral gap of P . We present a natural quantum version of this algorithm based on repeated measurements of the quantum walk U t = e −i Pt . We show that it samples almost uniformly from S with logarithmic dependence on −1 just as the classical walk P does; previously, no such quantum walk algorithm was known. We then outline a framework for analysing its running time and formulate two plausible conjectures which together would imply that it runs in time O (δ −1/2 log N log −1 ) when P is the standard transition matrix of a constant-degree graph. We prove each conjecture for a subclass of Cayley graphs.
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.
Quantum walks on graphs have been shown in certain cases to mix quadratically faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. Here, we construct a lifted Markov chain on a graph with n2D(G)n^2 D(G)n2D(G) vertices that mixes exactly to the average mixing distribution of a quantum walk on the graph GGG with nnn vertices, where D(G)D(G)D(G) is the diameter of GGG. Moreover, the mixing time of this chain is D(G)D(G)D(G) timesteps, and we prove that computing the transition probabilities for the lifted chain takes time polynomial in nnn. As an immediate consequence, for every quantum walk there is a lifted Markov chain with a faster mixing time that is polynomial-time computable, as the quantum mixing time is trivially lower bounded by the graph diameter. The result is based on a lifting presented by Apers, Ticozzi and Sarlette (arXiv:1705.08253).
We define and analyze quantum computational variants of \nrandom walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random \nwalk, which we call the Hadamard walk. Several striking \ndifferences between the quantum and classical cases are ob- served. For example, when unrestricted in either direction, \nthe Hadamard walk has position that is nearly uniformly \ndistributed in the range [-t/√2, t/√2] after t steps, which \nis in sharp contrast to the classical random walk, which has \ndistance O(√t) from the origin with high probability. With \nan absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is 2/π, and with an additional absorbing boundary at location n, the probability that the walk exits to the left actually increases, approaching 1/√2 in the limit. In the classical case both values are 1.
Recently, several groups have investigated quantum analogues of random walk algorithms, both on a line and on a circle. It has been found that the quantum versions have markedly different features to the classical versions. Namely, the variance on the line, and the mixing time on the circle increase quadratically faster in the quantum versions as compared to the classical versions. Here, we propose a scheme to implement the quantum random walk on a line and on a circle in an ion trap quantum computer. With current ion trap technology, the number of steps that could be experimentally implemented will be relatively small. However, we show how the enhanced features of these walks could be observed experimentally. In the limit of strong decoherence, the quantum random walk tends to the classical random walk. By measuring the degree to which the walk remains ``quantum,'' this algorithm could serve as an important benchmarking protocol for ion trap quantum computers.