Unusual scaling in a discrete quantum walk with random long range steps

Abstract

A discrete time quantum walker is considered in one dimension, where at each step, the translation can be more than one unit length chosen randomly. In the simplest case, the probability that the distance travelled is ℓ is taken as P (ℓ) = αδ(ℓ − 1) + (1 − α)δ(ℓ − 2 n ) with n ≥ 1. Even the n = 1 case shows a drastic change in the scaling behaviour for any α = 0, 1. Specifically, x 2 ∝ t 3/2 for 0 < α < 1, implying the walk is slower compared to the usual quantum walk. This scaling behaviour, which is neither conventional quantum nor classical, can be justified using a simple form for the probability density. The decoherence effect is characterized by two parameters which vanish in a power law manner close to α = 0 and 1 with an exponent ≈ 0.5. It is also shown that randomness is the essential ingredient for the decoherence effect.

Persistent quantum walks: Dynamic phases and diverging timescales

“…The walk clearly has periodicity 2, and there is no randomness in the choice of steps. This particular walk has already been studied in [9] and was found to have the same scaling behavior as the ordinary quantum walk. The probability distribution resembles an overlap of distributions obtained for the ordinary walks with l = 1 and 2 [9].…”

“…This particular walk has already been studied in [9] and was found to have the same scaling behavior as the ordinary quantum walk. The probability distribution resembles an overlap of distributions obtained for the ordinary walks with l = 1 and 2 [9]. On the other hand, when p = 1, the walk becomes deterministic with a unique step length and thus identical to the usual quantum walk.…”

“…For larger values of p, the walker behaves as that with random step lengths and persistence is apparently merely the tool that provides the stochasticity. Since it was found in [9] that the slightest randomness alters the exponent to 3/2, it is not surprising to see that for large p = 1, one gets the same exponent.…”

Scaling and crossover behaviour in a truncated long range quantum walk

“…Clearly, the asymptotic variation is x ∝ t 1/2 and x 2 ∝ t 3/2 . Following [6], we attempt to fit the moments using the equations…”

“…With a binary choice of step lengths, the main result was the observation that there is a sub-ballistic but super-diffusive scaling for the second moment; x 2 ∝ t 1.5 asymptotically [6]. In [7], Poissonian and other exponentially decaying distributions for the step lengths were used and a similar scaling was found (the exponent was reportedly ∼ 1.4 obtained from a short time range).…”