Noise correlations behind superdiffusive quantum walks

TL;DR

The paper addresses how discrete-time quantum walks respond to short-range correlated noise implemented as binary pair correlations in the gate sequence. The authors analyze both spatial and temporal regimes, showing that spatial and temporal correlations can drive superdiffusive transport, with robust spreading exponents across varied inhomogeneity levels and gate choices, while a fraction of the walker may remain localized due to resonant extended states. The study combines numerical simulations and analytical insights to reveal mechanisms behind the observed superdiffusion, including ballistic wavefronts and power-law tails in the wave-packet distributions. The findings have implications for understanding noise-induced transport in quantum systems and suggest potential experimental realizations using time-multiplexed quantum walk setups, along with the possibility of correlation-filtering gates that mitigate detrimental coherence loss.

Abstract

We study how discrete-time quantum walks behave under short-range correlated noise. By considering noise as a source of inhomogeneity of quantum gates, we introduce a primitive relaxation in the assumption of uncorrelated stochastic noise: binary pair correlations manifesting in the random distribution. Using different quantum gates, we examined the transport properties for both spatial and temporal noise regimes. For spatial inhomogeneities, we unveil noise correlations driving quantum walks from the well-known exponentially localized regime to superdiffusive spreading. This scenario displays an intriguing performance in which the superdiffusive exponent is almost invariant to the degree of inhomogeneity. The time-asymptotic regime and the finite-size scaling also unveil an emergent superdiffusive behavior for quantum walks undergoing temporal noise correlation, replacing the diffusive regime exhibited when noise is random and uncorrelated. However, some quantum gates are insensitive to correlations, contrasting with the spatial noise scenario. Numerical and analytical results provide valuable insights into the underlying mechanism of superdiffusive quantum walks, including those arising from deterministic aperiodic inhomogeneities.

Figures (9)

• Figure 1: Average probability distributions after 3000 time-steps for a quantum walker subjected to uncorrelated (brown circles) and spatially correlated noise (orange squares). In the absence of correlations, the quantum walker's profile exhibits a signature of Anderson localization, characterized by exponential decay and linear fitting in the semilog scaled plot (the dashed line is a guide for eyes). With binary pair correlations (BPC), the wave function spreads further, and the exponentially decaying tails give way to wave-packet fronts exhibiting sharp cutoff, which suggests a delocalized behavior.
• Figure 2: Average standard deviation of the quantum walker distribution vs. time for noiseless, random, and binary pair-correlated quantum walks. (a) $\theta_1=\pi/3$ and $\theta_2=\pi/4$ and (b) $\theta_1=4\pi/15$ and $\theta_2=\pi/4$. An asymptotic superdiffusive behavior emerges from the binary pair correlation, contrasting with the characteristic localized regime exhibited by quantum walks subjected to uncorrelated random noise.
• Figure 3: (a) The finite-size scaling computed for the long-time average of $\sigma(t)$ supports the previous findings, which unveils the superdiffusive behavior for lattices with binary pair correlation. However, the persistence of a nonvanishing return probability at long times (see b-c) reveals that a fraction of the walker remains localized around its initial position.
• Figure 4: Profile of the average probability distributions along the lattice sites ($\ell = n - n_0$) computed at different time steps $t$ for a quantum walker subjected to spatially correlated noise. Quantum gates are the same employed in Fig. \ref{['fig1']}. (a) Despite the concentration around the initial position $n_0$ (the magnified view shown in the inset confirms the nonvanishing return probability), the wavefront advances as time evolves. (b) Analysis of the distribution at distinct evolution times shows such wavefront advancing ballistically ($\sim t$) and a power-law tail $|\Psi_n|^2 \sim \ell^{-\varphi}$, with scaling exponent $\varphi = 1.46(5)$.
• Figure 5: Asymptotic exponent ($\alpha$) of standard deviation and long-time average of return probability for different quantum gates $\theta_1$, with (a-b) $\theta_2=\pi/4$ and (c-d) $\theta_2=\pi/3$. Binary pair correlation induces a notable transition from exponential localization to superdiffusive spreading. Despite correlated noise, a fraction of the walker remains localized around the initial position, a phenomenon not exclusive to Hadamard quantum gates appearing in pairs.