A Unified Poisson Summation Framework for Generalized Quantum Matrix Transformations

• URL: http://arxiv.org/abs/2604.02874v1
• Date: Fri, 03 Apr 2026 08:42:08 GMT
• Title: A Unified Poisson Summation Framework for Generalized Quantum Matrix Transformations
• Abstract summary: We present a unified algorithmic framework for quantum simulation of non-unitary dynamics and matrix functions.<n>By reinterpreting discretization errors as spectral folding in dual domains, we synthesize two distinct algorithmic paths.<n>We demonstrate the versatility of this framework by efficiently simulating diverse phenomena.
• Score: 2.1496959255800188
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

Related papers

• Quantum algorithms for the fractional Poisson equation via rational approximation [20.497126318288906]
  We present a quantum algorithm for solving the fractional equation ((-)s u = f) with (s in (0,1)) on bounded domains.<n>The proposed approach combines rational approximation techniques with quantum linear system solvers to achieve exponential quantum advantage.
  arXiv  Detail & Related papers  (2026-04-01T08:08:33Z)
• Amplitude-Phase Separation toward Optimal and Fast-Forwardable Simulation of Non-Unitary Dynamics [39.740772144144366]
  Amplitude-Phase Separation (APS) methods formulates any non-unitary evolution into separate simulation of a unitary operator and a Hermitian operator.<n>APS provides an effective and generic pathway for developing efficient quantum algorithms for general non-unitary dynamics.
  arXiv  Detail & Related papers  (2026-02-10T09:23:55Z)
• Variational Entropic Optimal Transport [67.76725267984578]
  We propose Variational Entropic Optimal Transport (VarEOT) for domain translation problems.<n>VarEOT is based on an exact variational reformulation of the log-partition $log mathbbE[exp(cdot)$ as a tractable generalization over an auxiliary positive normalizer.<n> Experiments on synthetic data and unpaired image-to-image translation demonstrate competitive or improved translation quality.
  arXiv  Detail & Related papers  (2026-02-02T15:48:44Z)
• Preconditioning Benefits of Spectral Orthogonalization in Muon [50.62925024212989]
  We study the effectiveness of a simplified variant of Muon in two case studies: matrix factorization and in-context learning of linear transformers.<n>Our analysis reveals that the Muon dynamics decouple into a collection of independent scalar sequences in the spectral domain, each exhibiting similar convergence behavior.
  arXiv  Detail & Related papers  (2026-01-20T00:08:31Z)
• Contour-integral based quantum eigenvalue transformation: analysis and applications [5.838361906016925]
  Eigenvalue transformations are beyond the reach of the quantum singular value transformation framework.<n>We propose a quantum algorithm for estimating observables of eigenvalue transformations using only $3$ qubits.<n>We show that the contour integral algorithm can outperform all the existing quantum algorithms in the case of solving ordinary differential equations.
  arXiv  Detail & Related papers  (2026-01-17T08:29:36Z)
• Quantization through Piecewise-Affine Regularization: Optimization and Statistical Guarantees [13.571671030124604]
  Piecewise regularization (PAR) provides a flexible modelingization based on the statistical perspectives.<n>We show how to use the PAR method to solve problems using gradient, and Alternating Direction Multipliers.
  arXiv  Detail & Related papers  (2025-08-14T23:35:21Z)
• Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation [55.88862563823878]
  In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics.<n>We demonstrate performance of the algorithm on two PDEs: a linear equation which governs slightly compressible fluid flow in porous media and the wave equation.<n>Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest.
  arXiv  Detail & Related papers  (2025-07-24T11:02:13Z)
• A Unified Theory of Stochastic Proximal Point Methods without Smoothness [52.30944052987393]
  Proximal point methods have attracted considerable interest owing to their numerical stability and robustness against imperfect tuning. This paper presents a comprehensive analysis of a broad range of variations of the proximal point method (SPPM)
  arXiv  Detail & Related papers  (2024-05-24T21:09:19Z)
• Stochastic parameter optimization analysis of dynamical quantum critical phenomena in long-range transverse-field Ising chain [0.0]
  We explore the quantum phase transition of the one-dimensional long-range transverse-field Ising model. In our simulations, the simulator automatically determines the parameters to sample from, even without prior knowledge of the critical point and universality class. We successfully obtained numerical evidence supporting $sigma = 7/4$ as the universality boundary between the latter two.
  arXiv  Detail & Related papers  (2023-05-23T14:46:16Z)
• Linearization Algorithms for Fully Composite Optimization [61.20539085730636]
  This paper studies first-order algorithms for solving fully composite optimization problems convex compact sets. We leverage the structure of the objective by handling differentiable and non-differentiable separately, linearizing only the smooth parts.
  arXiv  Detail & Related papers  (2023-02-24T18:41:48Z)
• Joint Network Topology Inference via Structured Fusion Regularization [70.30364652829164]
  Joint network topology inference represents a canonical problem of learning multiple graph Laplacian matrices from heterogeneous graph signals. We propose a general graph estimator based on a novel structured fusion regularization. We show that the proposed graph estimator enjoys both high computational efficiency and rigorous theoretical guarantee.
  arXiv  Detail & Related papers  (2021-03-05T04:42:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.