A Unified Poisson Summation Framework for Generalized Quantum Matrix Transformations

TL;DR

This paper introduces a unified quantum simulation framework based on the Poisson Summation Formula, enabling efficient and accurate simulation of non-unitary dynamics and matrix functions across various regimes.

Contribution

It develops a dual algorithmic framework combining Fourier-PSF and contour-PSF paths, improving simulation of singular, fractional, and holomorphic matrix functions.

Findings

Efficient simulation of fractional anomalous diffusion.

High-precision solutions for stiff differential equations.

Outperforms existing methods in optimal regimes.

Abstract

We present a unified algorithmic framework for quantum simulation of non-unitary dynamics and matrix functions, governed by the principle of spectral aliasing derived from the Poisson Summation Formula (PSF). By reinterpreting discretization errors as spectral folding in dual domains, we synthesize two distinct algorithmic paths: (i) the Fourier-PSF path, generalizing transmutation methods for time-domain filtering, which is optimal for singular and fractional dynamics $e^{-tH^\alpha}$, here $H\succeq 0$; and (ii) the contour-PSF path, a novel discrete contour transform based on the resolvent formalism, which achieves exponential convergence for holomorphic matrix functions via radius optimization. This dual framework resolves the smoothness-sparsity trade-off: it utilizes the Fourier basis to handle branch-point singularities where analyticity fails, and the Resolvent basis to exploit complex-plane regularity where it exists. We demonstrate the versatility of this framework by efficiently simulating diverse phenomena, from fractional anomalous diffusion to high-precision solutions of stiff differential equations, outperforming existing methods in their respective optimal regimes.