[Paper Review] Walk/Zeta Correspondence

Takashi Komatsu, Norio Konno+1 more|arXiv (Cornell University)|Apr 21, 2021

Quantum Computing Algorithms and Architecture15 references4 citations

TL;DR

This paper establishes a Walk/Zeta Correspondence for a broad class of walks—including random walks, correlated random walks, quantum walks, and open quantum random walks—on the d-dimensional torus using Fourier analysis. It derives explicit formulas for walk-type zeta functions, generalizing previous Grover/Zeta Correspondence results and showing convergence to integral expressions involving cosine terms in the infinite-size limit.

ABSTRACT

Our previous work presented explicit formulas for the generalized zeta function and the generalized Ihara zeta function corresponding to the Grover walk and the positive-support version of the Grover walk on the regular graph via the Konno-Sato theorem, respectively. This paper extends these walks to a class of walks including random walks, correlated random walks, quantum walks, and open quantum random walks on the torus by the Fourier analysis.

Motivation & Objective

• To extend the Grover/Zeta Correspondence to a broader class of walks, including quantum and classical stochastic processes, on the d-dimensional torus.
• To develop a unified framework using Fourier analysis for walks with F- and M-type dynamics, beyond the Konno-Sato theorem.
• To derive explicit expressions for walk-type zeta functions and their asymptotic behavior as the torus size N → ∞.
• To include open quantum random walks as a special case within the generalized walk model.

Proposed method

• Formalizes a 2d-state discrete-time walk on the d-dimensional torus using shift operators and a general 2d×2d coin matrix A.
• Applies Fourier transform techniques to analyze the walk dynamics, leveraging the torus’s translational symmetry.
• Decomposes the walk evolution operator into components involving projections P_j and coin matrix A acting on shifted states.
• Derives the walk-type zeta function via path enumeration and spectral analysis, leading to integral representations in the limit N → ∞.
• Uses uniform measure on [0, 2π)^d to express the asymptotic zeta function as a multidimensional integral of logarithmic terms involving cosine functions.
• Establishes equivalence between the zeta function and spectral measures of the transition operator, generalizing earlier results on Grover walks.

Experimental results

Research questions

• RQ1 How can the Grover/Zeta Correspondence be generalized beyond the Grover walk to include random walks, correlated random walks, and open quantum random walks on the torus?
• RQ2 What role does Fourier analysis play in deriving the zeta function for a class of walks with F- and M-type dynamics on the d-dimensional torus?
• RQ3 What is the asymptotic form of the walk-type zeta function as the torus size N → ∞ for such walks?
• RQ4 How do the resulting zeta functions relate to spectral measures of the transition operator or Laplacian on the torus?
• RQ5 Can the generalized zeta function be expressed as a multidimensional integral involving cosine terms in the infinite-size limit?

Key findings

• The walk-type zeta function for the d-dimensional torus converges in the limit N → ∞ to an expression involving a d-fold integral over [0, 2π)^d with a logarithmic kernel depending on cosine terms.
• For the generalized zeta function, the limit is given by (1−u²)^{d−1} times the exponential of the integral of log{(1+u²) − (2u/d)∑_{j=1}^d cosθ_j} with respect to the uniform measure.
• For the generalized Ihara zeta function, the limit is (1−u²)^{d−1} times the exponential of the integral of log{(1+(2d−1)u²) − 2u∑_{j=1}^d cosθ_j} over the same domain.
• In the 2D case, the asymptotic zeta function becomes (1−u²) times the exponential of the double integral of log{(1+u²)−u(cosθ₁+cosθ₂)} over [0,2π)² with uniform measure.
• The results recover and generalize previous Grover/Zeta Correspondence results for the Grover walk and its positive-support version on the torus.
• The framework successfully includes open quantum random walks as a special case, extending the scope of zeta function analysis to non-unitary, dissipative quantum processes.