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arXiv:2107.08866 · Journal of Physics A Mathematical and Theoretical

Quantum walk on a comb with infinite teeth

François David, Thórdur Jónsson

Published Jan 6, 2022• 1 citation
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Abstract

Abstract We study continuous time quantum walk on a comb with infinite teeth and show that the return probability to the starting point decays with time t as t −1 . We analyse the diffusion along the spine and into the teeth and show that the walk can escape into the teeth with a finite probability and goes to infinity along the spine with a finite probability. The walk along the spine and into the teeth behaves qualitatively as a quantum walk on a line. This behaviour is quite different from that of classical random walk on the comb.

DOI
10.1088/1751-8121/ac4897

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Abstract We study continuous time quantum walk on a comb with infinite teeth and show that the return probability to the starting point decays with time t as t −1 . We analyse the diffusion along the spine and into the teeth and show that the walk can escape into the teeth with a finite probability and goes to infinity along the spine with a finite probability. The walk along the spine and into the teeth behaves qualitatively as a quantum walk on a line. This behaviour is quite different from that of classical random walk on the comb.
We study continuous time quantum walk on a random comb graph with infinite teeth. Due to localization effects along the spine, the walk cannot go to infinity in the spine direction, while it can escape to infinity along the teeth of the comb. Starting from an initial vertex, the walk has a nonzero probability to stay trapped in a finite region. These results are obtained by studying the spectrum and eigenstates of the random Hamiltonian for the graphand analysing its properties. We use both analytic and numerical methods, many of which come from the theory of Anderson localization in one dimension.
Continuous-time quantum walks with long-range steps ${R}^{\ensuremath{-}\ensuremath{\gamma}}$ (RRR being the distance between sites) on a discrete line behave in similar ways for all $\ensuremath{\gamma}\ensuremath{\ge}2$. This is in contrast to classical random walks, which for $\ensuremath{\gamma}>3$ belong to a different universality class than for $\ensuremath{\gamma}\ensuremath{\le}3$. We show that the average probabilities to be at the initial site after time ttt as well as the mean square displacements are of the same functional form for quantum walks with $\ensuremath{\gamma}=2$, 444, and with nearest neighbor steps. We interpolate this result to arbitrary $\ensuremath{\gamma}\ensuremath{\ge}2$.
The position density of a "particle" performing a continuous-time quantum walk on the integer lattice, viewed on length scales inversely proportional to the time , converges (as tends to infinity) to a probability distribution that depends on the initial state of the particle. This convergence behavior has recently been demonstrated for the simplest continuous-time random walk [N. Konno, Phys. Rev. E 72, 026113 (2005)]. In this Brief Report, we use a different technique to establish the same convergence for a very large class of continuous-time quantum walks, and we identify the limit distribution in the general case.
Interplay between quantum interference and classical randomness can enhance performance of various quantum information tasks. In the present paper we analyze recurrence phenomena in the discrete-time quantum stochastic walk on a line, which is a quantum stochastic process that interpolates between quantum and classical walk dynamics. Surprisingly, we find that introducing classical randomness can reduce the recurrence probability – despite the fact that the classical random walk returns with certainty – and we identify the conditions under which this intriguing phenomenon occurs. Numerical evaluation of the first-return generating function allows us to investigate the asymptotics of the return probability as the step number approaches infinity. This provides strong evidence that the suppression of recurrence probability is not a transient effect but a robust feature of the underlying quantum-classical interplay in the asymptotic limit. Our results show that for certain tasks discrete-time quantum stochastic walks outperform both classical random walks and unitary quantum walks.
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.