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Two-dimensional quantum central limit theorem by quantum walks

Keisuke Asahara1,*, Daiju Funakawa2,†, Motoki Seki3,‡, and Akito Suzuki4,§

  • 1Education and Research Center for Mathematical and Data Science, Hokkaido University, Kita 12, Nishi 7, Kita-ku, Sapporo, Hokkaido 060-0812, Japan
  • 2Department of Electronics and Information Engineering, Hokkai-Gakuen University, Sapporo, Hokkaido 062-8605, Japan
  • 3Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
  • 4Department of Production Systems Engineering and Sciences, Komatsu University, Awazu Campus, Nu 1-3 Shicho-machi, Komatsu, Ishikawa 923-8511, Japan

  • *Contact author: keisuke-asahara@mdsc.hokudai.ac.jp
  • †Contact author: funakawa@hgu.jp
  • ‡Contact author: marmot.motoki@gmail.com
  • §Contact author: akito.suzuki@komatsu-u.ac.jp
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Phys. Rev. A 113, 022439 – Published 23 February, 2026

DOI: https://doi.org/10.1103/fs8d-h2z8

Abstract

The weak limit theorem (WLT), the quantum analog of the central limit theorem, is foundational to quantum walk (QW) theory. Unlike the universal Gaussian limit of classical walks, deriving analytical forms of the limiting probability density function (PDF) in higher dimensions has remained a challenge since the one-dimensional (1D) Konno distribution was established. Previous explicit PDFs for two-dimensional (2D) models were limited to specific cases whose fundamental nature was unclear. This paper resolves this long-standing gap by introducing the notion of maximal speed vmax as a critical parameter. We demonstrate that all previous 2D solutions correspond to a degenerate regime where vmax=1. We then present the first exact analytical representation of the limiting PDF for the physically richer, unexplored regime vmax<1 of a general class of 2D two-state QWs. Our result reveals 2D Konno functions that govern these dynamics. We establish these as the proper 2D generalization of the 1D Konno distribution by demonstrating their convergence to the 1D form in the appropriate limit. Furthermore, our derivation, based on spectral analysis of the group-velocity map, analytically resolves the singular asymptotic structure: we explicitly determine the caustics loci where the PDF diverges and prove they define the boundaries of the distribution's support. By also providing a closed-form expression for the weight functions, this work offers a complete description of the 2D WLT.

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Phys. Rev. A 113, 022439– Published 23 February, 2026
  • Received 14 November 2025
  • Accepted 2 February 2026
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