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Quantum walks, the discrete wave equation and Chebyshev polynomials

Simon Apers, Laurent Miclo

TL;DR

This work reframes quantum walks as discrete-wave dynamics on graphs, deriving a unitary evolution from Chebyshev polynomials applied to the random-walk operator. By embedding the discrete wave equation into a unitary dilation and linking it to block encodings, the authors connect quantum walks to established polynomial tools and to the Varopoulos–Carne bound, enabling a weak-limit analysis on lattices. The main technical achievement is a rigorous proof that the $n$-step quantum walk on the lattice spreads ballistically, via a detailed moment method that identifies a limiting measure on $[-1,1]^2$ and a corresponding pushforward measure on $[0,1]^2$. This approach offers a versatile framework for studying spreading/mixing beyond highly symmetric graphs and points toward extensions to higher-dimensional lattices and other Cayley graphs, with potential links to the wave-geometry viewpoint of Friedman–Tillich.

Abstract

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.

Quantum walks, the discrete wave equation and Chebyshev polynomials

TL;DR

This work reframes quantum walks as discrete-wave dynamics on graphs, deriving a unitary evolution from Chebyshev polynomials applied to the random-walk operator. By embedding the discrete wave equation into a unitary dilation and linking it to block encodings, the authors connect quantum walks to established polynomial tools and to the Varopoulos–Carne bound, enabling a weak-limit analysis on lattices. The main technical achievement is a rigorous proof that the -step quantum walk on the lattice spreads ballistically, via a detailed moment method that identifies a limiting measure on and a corresponding pushforward measure on . This approach offers a versatile framework for studying spreading/mixing beyond highly symmetric graphs and points toward extensions to higher-dimensional lattices and other Cayley graphs, with potential links to the wave-geometry viewpoint of Friedman–Tillich.

Abstract

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.
Paper Structure (18 sections, 10 theorems, 114 equations, 2 figures)

This paper contains 18 sections, 10 theorems, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Random walks from the diffusion equation
  3. Quantum walks from the wave equation
  4. Discrete approximation.
  5. Quantum walks and block encodings
  6. Varopoulos-Carne bound and quantum fast-forwarding
  7. Quantum walks on lattices
  8. Quantum walk on the line
  9. Quantum walk on the lattice
  10. Proof of weak convergence on the lattice
  11. Second moment as an integral
  12. Negligible boundary region
  13. Simplifying the integral
  14. First simplification
  15. Second simplification
  16. ...and 3 more sections

Key Result

Theorem 1

There exists a continuous measure $\mu$ on $[-1,1]^2$ such that $\mu_n \rightarrow \mu$.

Figures (2)

  • Figure 1: (left) Probability measure induced by $[P_L^n]_{0,\cdot}$ for $n = 50$, corresponding to an $n$-step lazy random walk on the line. (right) Probability measure induced by $[T_n(P_L)]_{0,\cdot}^2$ for $n = 50$, associated to an $n$-step quantum walk on the line. (left and right) Lines drawn continuously for clarity.
  • Figure 2: (left) Probability measure induced by $[P^n]_{0,\cdot}$ for $n = 50$, corresponding to an $n$-step random walk on the lattice. (right) Probability measure induced by $[T_n(P)]_{0,\cdot}^2$ for $n = 50$, associated to an $n$-step quantum walk on the lattice.

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Corollary 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • ...and 1 more