Efficient Computation of Decoherent Quantum Walks through Eigenvalue Perturbation

TL;DR

This paper introduces a perturbation theory method to efficiently approximate the eigendecomposition of Lindblad super-operators in decoherent quantum walks, significantly reducing computational complexity for large graphs.

Contribution

It presents a novel approach using eigenvalue perturbation to compute decoherent quantum walks more efficiently than existing methods.

Findings

Reduces computational complexity of quantum walk simulations

Enables analysis of large graphs with decoherence effects

Provides approximation techniques for Lindblad super-operator eigendecomposition

Abstract

A number of recent studies have investigated the introduction of decoherence in quantum walks and the resulting transition to classical random walks. Interestingly, it has been shown that algorithmic properties of quantum walks with decoherence such as the spreading rate are sometimes better than their purely quantum counterparts. Not only quantum walks with decoherence provide a generalization of quantum walks that naturally encompasses both the quantum and classical case, but they also give rise to new and different probability distribution. The application of quantum walks with decoherence to large graphs is limited by the necessity of evolving a state vector whose size is quadratic in the number of nodes of the graph, as opposed to the linear state vector of the purely quantum (or classical) case. In this technical report, we show how to use perturbation theory to reduce the computational complexity of evolving a continuous-time quantum walk subject to decoherence. More specifically, given a graph over n nodes, we show how to approximate the eigendecomposition of the n^2 x n^2 Lindblad super-operator from the eigendecomposition of the n x n graph Hamiltonian.