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An Algebraic Model for Simulation of Percolation Decoherence on Quantum-Walk-Based Search Algorithms

Gabriel Mauricio Oswald Vieira, Rui Aldé-Lopes, Nelson Maculan, Franklin de Lima Marquezino

Published Jul 19, 2024• 0 citations
Publisher

Abstract

In this paper, we propose a linear algebra based model for percolation noise simulation in quantum-walk-based searches to analyze the impact on the success probability of the search. The model was built to allow an open and simultaneous simulation of any type of noise that represents an edge-break, which we show to be a well-behaved family of permutation matrices that maintain a particular set theory relation when acting as some system evolution operators. The simulation evolves using only linear algebra operations, opening avenues for easier analyses of unexpected observed convergence behaviour. As a use case, we investigate the simulation of a lackadaisical quantum walk with percolation noise.

DOI
10.5753/etc.2024.2031

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In this paper, we propose a linear algebra based model for percolation noise simulation in quantum-walk-based searches to analyze the impact on the success probability of the search. The model was built to allow an open and simultaneous simulation of any type of noise that represents an edge-break, which we show to be a well-behaved family of permutation matrices that maintain a particular set theory relation when acting as some system evolution operators. The simulation evolves using only linear algebra operations, opening avenues for easier analyses of unexpected observed convergence behaviour. As a use case, we investigate the simulation of a lackadaisical quantum walk with percolation noise.
The decoherence phenomenon has been widely studied in different types of quantum walks. In this work we show how to model decoherence inspired by percolation on staggered quantum walks. Two models of unitary noise are described: breaking polygons and breaking vertices. The evolution operators subject to these noises are obtained, and the equivalence to the coined quantum walk model is presented. Further, we numerically analyze the effect of these decoherence models on the two-dimensional grid of 4-cliques. We examine how these perturbations affect the quantum walk based search algorithm in this graph and how expanding the tessellation intersection can make it more robust against decoherence.
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.
This paper investigates the effects of decoherence generated by broken-link-type noise in the hypercube on an optimized quantum random-walk search algorithm. When the hypercube occurs with random broken links, the optimized quantum random-walk search algorithm with decoherence is depicted through defining the shift operator which includes the possibility of broken links. For a given database size, we obtain the maximum success rate of the algorithm and the required number of iterations through numerical simulations and analysis when the algorithm is in the presence of decoherence. Then the computational complexity of the algorithm with decoherence is obtained. The results show that the ultimate effect of broken-link-type decoherence on the optimized quantum random-walk search algorithm is negative.
Quantum walks obey unitary dynamics: they form closed quantum systems. The system becomes open if the walk suffers from imperfections represented as missing links on the underlying basic graph structure, described by dynamical percolation. Openness of the system's dynamics creates decoherence, leading to strong mixing. We present a method to analytically solve the asymptotic dynamics of coined, percolated quantum walks for a general graph structure. For the case of a circle and a linear graph we derive the explicit form of the asymptotic states. We find that a rich variety of asymptotic evolutions occur: not only the fully mixed state, but other stationary states; stable periodic and quasiperiodic oscillations can emerge, depending on the coin operator, the initial state, and the topology of the underlying graph.
We study the time evolution of continuous-time quantum walks on randomly changing graphs. At certain moments, edges of the graph appear or disappear with a given probability, as in percolation. We treat this problem in a strong noise limit. We focus on the case when the time interval between the subsequent changes of the graph tends to zero. We derive explicit formulae for the general evolution in this limit. We find that the percolation in this limit causes an effective time rescaling. Independently of the graph and the initial state of the walk, the time is rescaled by the probability of keeping an edge. Both the individual trajectories for a single system and average properties with a superoperator formalism are discussed. We give an analytical proof for our theorem and we also present results from the numerical simulations of the phenomena for different graphs. We also analyze the effect of finite step-size on the evolution.
Quantum walks on graphs can model physical processes and serve as efficient tools in quantum information theory. Once we admit random variations in the connectivity of the underlying graph, we arrive at the problem of percolation, where the long-time behaviour appears untreatable with direct numerical methods. We develop novel analytic methods based on the theory of random unitary operations which help us to determine explicitly the asymptotic dynamics of quantum walks on two-dimensional finite integer lattices with percolation. Based on this theory, we find new unexpected features of percolated walks like asymptotic position inhomogeneity or special directional symmetry breaking.