Noise-Resilient Spatial Search with Lackadaisical Quantum Walks

TL;DR

This paper investigates the robustness of spatial search using lackadaisical quantum walks (LQWs) on a 2D torus under dynamic percolation decoherence. The authors model decoherence by randomly breaking edges each step and show that incorporating self-loops (parameterized by $\ell$) mitigates the degradation caused by noise, preserving a detectable signal at the marked vertex. The study finds that the optimal self-loop weight remains near $\ell \approx 4/N$ across noise levels, and that the marked vertex maintains a probability above the uniform baseline even in noisy environments. These results extend the known noiseless advantages of LQWs to more realistic, noisy scenarios, highlighting self-loops as a practical resource for robust quantum search algorithms.

Abstract

Quantum walks are a powerful framework for the development of quantum algorithms, with lackadaisical quantum walks (LQWs) standing out as an efficient model for spatial search. In this work, we investigate how broken-link decoherence affects the performance of LQW-based search on a two-dimensional toroidal grid. We show through numerical simulations that, while decoherence drives the loopless walk toward a uniform distribution and eliminates its search capability, the inclusion of self-loops significantly mitigates this effect. In particular, even under noise, the marked vertex remains identifiable with probability well above uniform, demonstrating that self-loops enhance the robustness of LQWs in realistic scenarios. These findings extend the known advantages of LQWs from the noiseless setting to noisy environments, consolidating self-loops as a valuable resource for designing resilient quantum search algorithms.

Figures (7)

• Figure 1: A two-dimensional grid with $4\times 4$ vertices, self-loops of weight $\ell$ on each vertex, and periodic boundary. The marked vertex is indicated by a white circle.
• Figure 2: Success probability as a function of time for the search with Grover coin on the two-dimensional grid of $N=32\times 32$ vertices and self-loop of weight $\ell = 4/N$ at each vertex, assuming absence of noise. (Color online.)
• Figure 3: The elimination of an edge makes the amplitude flux be redirected. (a) No edge has been removed. (b) Edge $(v,w)$ has been removed.
• Figure 4: Success probability over time under broken-link decoherence, for a $32\times 32$ toroidal grid. Each curve represents the average of 50 independent noisy simulations. The shaded region indicates the standard deviation across runs. The dotted horizontal line marks the uniform distribution for reference. (Color online.)
• Figure 5: Success probability over time under broken-link probability ($p=0.01$) and different self-loop weights, for a $32\times 32$ toroidal grid. Each curve represents the average of 50 independent noisy simulations. The shaded region indicates the standard deviation across runs. The dotted horizontal line marks the uniform distribution for reference. (Color online.)