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Pré-Publication, Document De Travail Année : 2024

Quantum walks, the discrete wave equation and Chebyshev polynomials

Résumé

A quantum walk is the quantum analogue of a random walk. While it is relatively well understood how quantum walks can speed up random walk hitting times, it is a long-standing open question to what extent quantum walks can speed up the spreading or mixing rate of random walks on graphs. In this expository paper, inspired by a blog post by Terence Tao, we describe a particular perspective on this question that derives quantum walks from the discrete wave equation on graphs. This yields a description of the quantum walk dynamics as simply applying a Chebyshev polynomial to the random walk transition matrix. This perspective decouples the problem from its quantum origin, and highlights connections to earlier (non-quantum) work and the use of Chebyshev polynomials in random walk theory as in the Varopoulos-Carne bound. We illustrate the approach by proving a weak limit of the quantum walk dynamics on the lattice. This gives a different proof of the quadratically improved spreading behavior of quantum walks on lattices.
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hal-04455089 , version 1 (13-02-2024)

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Simon Apers, Laurent Miclo. Quantum walks, the discrete wave equation and Chebyshev polynomials. 2024. ⟨hal-04455089⟩
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