Quantum walks, the discrete wave equation and Chebyshev polynomials - Archive ouverte HAL Accéder directement au contenu
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.
Fichier principal
Vignette du fichier
2402.07809.pdf (575.92 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04455089 , version 1 (13-02-2024)

Identifiants

Citer

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

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More