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.

• The paper introduces a unified dual-path Poisson Summation framework that enables efficient simulation of non-unitary quantum dynamics and generalized matrix functions.
• It rigorously develops Fourier-PSF and Contour-PSF methods that achieve exponential convergence for holomorphic functions and polynomial scaling for fractional dynamics.
• The framework resolves convergence challenges and provides sharp complexity bounds along with circuit-level improvements for a wide range of quantum matrix transformations.

A Unified Poisson Summation Framework for Generalized Quantum Matrix Transformations

Introduction and Motivation

This work introduces a unified, dual-path algorithmic framework for quantum simulation of non-unitary dynamics and generalized matrix functions, fundamentally grounded in an operator-theoretic reinterpretation of discretization as spectral aliasing via the Poisson Summation Formula (PSF). The approach provides systematic, rigorous tools for simulating operator functions $f(A)$, effectively bridging and extending previous methods in quantum simulation, with applicability to both fractional and holomorphic matrix transformations. The framework resolves structural and convergence challenges in existing schemes and yields provably improved complexity scaling in several regimes. Notably, the formalism enables efficient handling of cases such as fractional anomalous diffusion, high-order dissipation, and general matrix polynomials.

Mathematical Foundation: Dual-Path Poisson Summation Principle

The PSF serves as the analytical cornerstone, connecting two domains:

• Fourier-PSF (Path A): Discretizations in the time or real spectral domain correspond, via PSF, to a sum of operator-valued time evolutions weighted by Fourier coefficients. The spectral aliasing incurred by finite truncation is rigorously analyzed in terms of operator-valued spectral copies.
• Contour-PSF (Path B): In the complex domain, a novel finite Poisson summation is formulated using discrete contour integration over the cyclic group of roots of unity. The discretization of Cauchy integrals expressing $f(A)$ is reinterpreted as geometric aliasing in the dual cyclic group, making explicit the group-theoretic structure underlying contour-sampled LCU decompositions.

This duality provides an analytical toolset to optimize between smoothness and sparsity: the smoother (analytic) the target function in the dual domain, the sparser the LCU decomposition of the non-unitary operator in the primary domain.

Algorithmic Synthesis

Path A: Fourier-PSF for Singular, Fractional, and Dissipative Dynamics

In this regime, the target is simulating non-unitary dynamics generated by expressions like $e^{-tH^\alpha}$, including fractional powers and associated with dissipative PDEs (heat, biharmonic, and fractional diffusion). The framework generalizes classical transmutation techniques—including the Kannai transform—beyond even integers $\alpha$ to fractional and non-integer exponents.

The construction utilizes LCU decompositions of the target evolution in terms of cosine (or more general exponentiated Hermitian) operator polynomials, with weights derived from the inverse Fourier transform of the spectral function. The tradeoff between aliasing and truncation is exactly quantified, revealing that in the analytic regime (e.g., $\alpha \in 2\mathbb{Z}^+$), exponential convergence is achieved and the complexity is near-optimal.

For fractional $\alpha$ (branch-point singularities), convergence is polynomial in the required precision. The inherent singularity leads to a heavy-tailed algebraic decay of the weight coefficients, imposing a fundamental bottleneck, in full accord with Tauberian theorems. Block-encoding oracles for local, sparse matrices suffice, as the nonlocality is absorbed into the classical coefficient computation.

Path B: Discrete Contour-PSF for Holomorphic Matrix Functions

This path addresses the case where $f(A)$ is a holomorphic function over a neighborhood containing $\sigma(A)$. The method discretizes the Cauchy integral formula for $f(A)$ using $m$ roots of unity sampled uniformly on a circular contour, leading to an LCU representation where terms are resolved into shifted resolvent operations.

The discretization error is decomposed rigorously into geometric aliasing (corresponding to powers $f(A)$0, $f(A)$1) and an analytic truncation integral. In contrast to standard quadrature analysis, this yields an exponential convergence rate in $f(A)$2: the required number of nodes to achieve error $f(A)$3 is $f(A)$4. The circuit-level overhead, owing to LCU control register preparation and amplitude amplification, is thus exponentially improved relative to legacy approaches.

Complexity is linked to the analytic structure of $f(A)$5 as well as spectral geometry; a key implication is that, for well-behaved $f(A)$6, quantum simulation of functions of large sparse matrices enjoys polylogarithmic scaling with error, independent of polynomial degree.

Complexity Results and Formal Claims

• For integer $f(A)$7: The quantum query complexity for simulating $f(A)$8 to $f(A)$9-precision is

$e^{-tH^\alpha}$0

where $e^{-tH^\alpha}$1, matching and unifying tight bounds from recent quantum PDE simulation analyses (2604.02874).

• For non-integer fractional $e^{-tH^\alpha}$2: The complexity is polynomial in $e^{-tH^\alpha}$3 with exponent determined by $e^{-tH^\alpha}$4, specifically:

$e^{-tH^\alpha}$5

This scaling is strictly necessary due to the nonanalyticity of the kernel at the origin.

• For holomorphic $e^{-tH^\alpha}$6 (discrete contour): The required number of resolvent LCU nodes $e^{-tH^\alpha}$7, with query complexity depending on the spectral gap and magnitude bounds for $e^{-tH^\alpha}$8 on the contour. This result is markedly sharper than previous polynomial-in-$e^{-tH^\alpha}$9 bounds for similar tasks, as established rigorously via the group-theoretic PSF formalism.

Applications and Numerical Regimes

The paper includes prototypical applications illustrating the breadth and generality of the framework:

• High-order and fractional diffusion in quantum simulation: E.g., simulating the biharmonic equation or anomalous (Lévy) diffusion using only sparse, local block-encodings; the nonlocality is handled in the spectral domain, not at the hardware layer.
• Matrix polynomial and rational transforms: Enables high-precision quantum evaluation of polynomials or rational transforms of general normal or diagonalizable matrices, with complexity sharply determined by contour optimization.

In all instances, the unified PSF approach either recovers or outperforms prior optimal bounds using cost metrics strictly dominated by analytic properties of the spectral function and not by ad hoc structural assumptions.

Implications and Future Directions

The theoretical implications extend beyond quantum algorithms for linear ODEs and PDEs:

• Simulations of non-unitary and non-Hermitian dynamics: The framework captures a broad array of physical phenomena (e.g., open quantum systems, diffusion, fractional dynamics) and provides clear prescriptions for selecting the optimal simulation path.
• Design of optimal polynomial bases: The results point toward the existence of new classes of polynomials or series expansions, adapted to dissipative or non-unitary transformations, potentially setting new lower bounds for quantum simulation complexity.
• Circuit-level advantages: By decoupling hardware-level operator structure from problem-level analytic complexity, the methodology enhances scalability and practicality on near-term quantum devices.

A promising direction for future research is the application of conformal mapping and Faber polynomial expansions to general spectral geometries, further tightening the convergence and complexity constants and enabling the treatment of operators with highly non-circular spectra. Refinements may also be developed for multi-matrix and multi-variable function settings.

Conclusion

The unified Poisson summation framework presented establishes spectral aliasing as the primary mechanism underlying discretization error in quantum simulation of matrix functions. By synthesizing continuous (Fourier) and discrete (contour) manifestations of the PSF within a rigorous operator-theoretic language, the work synthesizes, clarifies, and extends existing techniques in quantum PDE/PDE simulation, LCU expansion, and quantum matrix arithmetic. It delivers sharp, regime-adaptive complexity results, and paves the way for both theoretical advances in harmonic analysis-inspired quantum algorithms and practical improvements in circuit-level simulation of complex non-unitary dynamics and general matrix functions (2604.02874).