# Szegedy Walk Unitaries for Quantum Maps

## Abstract

Szegedy developed a generic method for quantizing classical algorithms based on random walks (Szegedy, in: 45th annual IEEE Symposium on Foundations of Computer Science, pp 32–41, 2004. https://doi.org/10.1109/FOCS.2004.53). A major contribution of his work was the construction of a walk unitary for any reversible random walk. Such unitary posses two crucial properties: its eigenvector with eigenphase 0 is a quantum sample of the limiting distribution of the random walk and its eigenphase gap is quadratically larger than the spectral gap of the random walk. It was an open question if it is possible to generalize Szegedy’s quantization method for stochastic maps to quantum maps. We answer this in the affirmative by presenting an explicit construction of a Szegedy walk unitary for detailed balanced Lindbladians—generators of quantum Markov semigroups—and detailed balanced quantum channels. We prove that our Szegedy walk unitary has a purification of the fixed point of the Lindbladian as eigenvector with eigenphase 0 and that its eigenphase gap is quadratically larger than the spectral gap of the Lindbladian. To construct the walk unitary we leverage a canonical form for detailed balanced Lindbladians showing that they are structurally related to Davies generators. We also explain how the quantization method for Lindbladians can be applied to quantum channels. We give an efficient quantum algorithm for quantizing Davies generators that describe many important dynamics of open quantum systems, for instance, the relaxation of a quantum system coupled to a bath. Our algorithm extends known techniques for simulating dynamics of quantum systems on a quantum computer.