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A Unified Poisson Summation Framework for Generalized Quantum Matrix Transformations

Chao Wang, Xi-Ning Zhuang, Menghan Dou, Zhao-Yun Chen, Guo-Ping Guo

Apr 3, 2026arXiv:2604.02874v1
quant-ph
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#232 of 3296 · Quantum Physics
Tournament Score
1519±31
10501750
66%
Win Rate
31
Wins
16
Losses
47
Matches
Rating
5.8/ 10
Significance6
Rigor6.5
Novelty5.5
Clarity7

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−tHαe^{-tH^α}, here H⪰0H\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.

AI Impact Assessments

(3 models)

Scientific Impact Assessment

1. Core Contribution

This paper proposes a unified algorithmic framework for quantum simulation of non-unitary dynamics and general matrix functions, built around the conceptual lens of *spectral aliasing* derived from the Poisson Summation Formula (PSF). The framework synthesizes two algorithmic paths:

  • Path A (Fourier-PSF): Generalizes transmutation methods (e.g., Kannai transform) to arbitrary fractional decay orders α, handling e^{-tH^α} for H ≥ 0. The key insight is that discretization errors in time-domain LCU expansions correspond to aliasing in the spectral domain, governed by the classical PSF.
  • Path B (Contour-PSF): Introduces a discrete contour transform for holomorphic matrix functions f(A), reinterpreting the discretized Cauchy integral as a finite Poisson summation over the cyclic group Z_m. This yields an explicit decomposition of error into geometric aliasing (matrix powers A^{km}) and analytic truncation, proving exponential convergence for analytic functions with m ~ O(log(1/ε)).

    • The central conceptual contribution is the smoothness-sparsity trade-off: the regularity of the target function in the dual domain dictates the sparsity (and hence efficiency) of the LCU decomposition in the primary domain. Smooth (analytic) functions yield exponentially sparse representations, while branch-point singularities (fractional α) impose polynomial scaling.

    2. Methodological Rigor

    The mathematical framework is carefully constructed. The operator-valued PSF identity (Eq. 7-8) and the finite group PSF identity (Eq. 12) provide clean, rigorous decompositions of simulation error. The proofs in the appendices for Theorems 1 and 3 follow standard techniques (saddle-point asymptotics, Tauberian theorems, ML inequality) and appear technically sound.

    Strengths in rigor:

  • The error decomposition into aliasing and truncation components is explicit and clean, making the trade-offs transparent.
  • The analysis of the QSVT norm-bounding issue (the domain-shifting technique from H̃ to 2H̃ - I) is practically important and carefully handled.
  • The distinction between integer α (exponential convergence) and non-integer α (polynomial scaling O(ε^{-1/2α})) is well-motivated by the branch-point singularity argument.

    • Potential concerns:

  • The paper claims to "outperform existing methods in their respective optimal regimes," but direct numerical comparisons are absent. The applications section (Sec. VI) provides complexity expressions but no concrete benchmarks against competing algorithms.
  • For Path B, the bound in Eq. (42) for matrix polynomials involves the condition number κ_S of the eigenvector matrix, which can be exponentially large for non-normal matrices. The paper does not deeply discuss how this affects practical applicability.
  • Theorem 4 (Appendix A) on the L¹-norm scaling as O(log α) relies on heuristic decomposition arguments (Sinc regime vs. decay regime) rather than fully rigorous bounds, though the authors acknowledge this.

    • 3. Potential Impact

    The framework has meaningful potential impact in several directions:

  • Fractional diffusion simulation: The ability to simulate e^{-tH^α} for non-integer α using only sparse local operators (avoiding dense fractional Laplacian encodings) is a genuine practical advantage. Lévy flights and anomalous diffusion are important in physics, finance, and biology.
  • Stiff differential equations: The contour-PSF path with exponential convergence for holomorphic functions could benefit quantum algorithms for stiff systems where high precision is needed.
  • Algorithmic design guidance: The smoothness-sparsity trade-off provides a principled criterion for choosing between Fourier-based and contour-based approaches, which could guide future algorithm development.

    • However, the impact is somewhat incremental rather than transformative. Path A generalizes known transmutation methods (particularly the recent work of Jin, Ma, and Zuazua [42]) to fractional orders, and Path B provides a cleaner analytical framework for contour-based methods that were already known to achieve similar query complexities [38]. The paper's primary value is in the unification and conceptual clarity rather than in achieving fundamentally new complexity bounds.

    4. Timeliness & Relevance

    The paper is well-timed. Quantum simulation of non-unitary dynamics is an active research frontier, with several concurrent works (Schrödinerization [27-30], LCHS [31-33], CBMD [34], contour-based eigenvalue transformation [38]) published in 2024-2026. The paper engages substantively with this recent literature and positions itself as a unifying lens. The fractional dynamics application addresses a genuine gap—most existing quantum algorithms focus on integer-order PDEs, and the extension to fractional orders is timely given growing interest in anomalous transport phenomena.

    5. Strengths & Limitations

    Key Strengths:

  • Elegant unifying principle: the PSF perspective provides genuine conceptual economy, connecting disparate algorithmic approaches.
  • Clear exposition of the smoothness-sparsity trade-off as a fundamental design principle.
  • Practical hardware-friendliness: lifting dynamical complexity into classical spectral weights while keeping quantum circuits structurally simple.
  • The domain-shifting technique for QSVT compliance is a useful practical contribution.

    • Notable Limitations:

  • No numerical experiments or concrete circuit implementations; all results are asymptotic complexity statements.
  • The comparison with prior art is largely qualitative. Claims of "outperforming existing methods" lack quantitative substantiation.
  • The contour-PSF analysis assumes diagonalizability and involves κ_S, limiting applicability to highly non-normal matrices.
  • The discussion of conformal mapping extensions (Sec. VII) is speculative and undeveloped.
  • The paper does not address lower bounds—it remains unclear whether the polynomial scaling O(ε^{-1/2α}) for fractional dynamics is fundamental or improvable.

    • 6. Additional Observations

    The paper builds substantially on very recent concurrent work [38, 42], raising questions about the degree of independent novelty. The relationship to [34] (by some of the same authors) as an extension is noted but could be more precisely delineated. The paper would benefit significantly from even simple numerical demonstrations validating the predicted convergence rates.

    Rating:5.8/ 10
    Significance 6Rigor 6.5Novelty 5.5Clarity 7

    Generated Apr 6, 2026

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