The Simplest 2D Quantum Walk Detects Chaoticity

César Alonso-Lobo, Gabriel G. Carlo and Florentino Borondo ()
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César Alonso-Lobo: Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingenería Agronómica, Agroambiental y de Biosistemas, Universidad Politécnica de Madrid, Avenida Puerta de Hierro 2-4, 28040 Madrid, Spain
Gabriel G. Carlo: CONICET, Comisión Nacional de Energía Atómica, Avenida del Libertador 8250, Buenos Aires 1429, Argentina
Florentino Borondo: Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

Mathematics, 2025, vol. 13, issue 19, 1-17

Abstract: Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium—a chaotic billiard—level statistics described by a Brody distribution with parameter δ ≃ 0.1 . This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio ( PR ) ≃ 1150 compared to the rectangular billiard (regular) case, where the average PR ≃ 1500 . Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.

Keywords: quantum walks; quantum random walks; discrete quantum processes; quantum computation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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