A Unified Poisson Summation Framework for Generalized Quantum Matrix Transformations

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 $e^{-tH^α}$, 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.

Theorems & Definitions (8)

• Lemma 1
• Theorem 1: Complexity for $\alpha \in \mathbb{Z}^+$
• Corollary 2
• Theorem 3: Complexity for $\alpha \notin \mathbb{Z}$ and $\alpha \geq 0.5$
• Theorem 4: Logarithmic Scaling of $L^1$-Norm
• proof
• proof
• proof