arXiv subjects

Hengzhun Chen

Publications and source records attributed to Hengzhun Chen.

5 recordsLinked to original sources

arXiv2026

SinCoTrap: A High-Order Locally Corrected Trapezoidal Rule for Periodic Singular Integrals in Arbitrary Dimensions

We present SinCoTrap (Singularity-Corrected Trapezoidal Rule), a high-order locally corrected trapezoidal method for periodic singular integrals in arbitrary dimension $d$ with kernel $|\boldsymbol{x}|^{-s}$, $0<s<d$. The scheme preserves the uniform tensor grid and modifies only a fixed, small stencil of weights near the singularity. For a correction order $p$, the resulting quadrature attains the error rate $O(h^{2p+2+d-s})$. We derive explicit, mesh-independent limiting correction weights via analytic continuation of a special generalization of the Riemann zeta function, yielding rapidly computable formulas that can be pretabulated for each $(d,s,p)$. This makes SinCoTrap both efficient in application and robust for high-order accuracy across a broad class of periodic singular integrals.

math.NA↗

arXiv2026

Structure-Aware Compilation for Scalable Neutral-Atom Quantum Computing

We study the compilation of structured quantum gate families on two-dimensional neutral-atom arrays, aiming to reduce addressing and transport overhead under realistic hardware constraints. For single-qubit gates, we exploit the algebraic structures of gate families at the matrix level, enabling efficient rank-one decompositions over appropriate algebraic structures and thereby reducing the number of addressing layers. For controlled-Z (C-Z) gates, we formulate the transport scheduling problem using graph-theoretic models, leading to efficient compilation algorithms under realistic transport constraints. We provide provable performance guarantees for the proposed methods and validate them through extensive numerical experiments. Across representative single-qubit gate families, our methods reduce the number of addressing layers by up to a factor of two compared with naïve row- or column-wise implementations. For C-Z gates, our scheduling strategy reduces the required number of atom transport operations by approximately 50\%. When applied to QAOA circuits for MaxCut, the proposed framework reduces transport cost by more than 30\% on average. These results show that the physical constraints of neutral-atom hardware can be converted into algebraic and graph-theoretic structure, turning a hardware-level scheduling bottleneck into tractable decomposition and coloring problems.

quant-ph↗

arXiv2026

On the Continuity of Schur-Horn Mapping

The Schur-Horn theorem is a well-known result that characterizes the relationship between the diagonal elements and eigenvalues of a symmetric (Hermitian) matrix. In this paper, we extend this theorem by exploring the eigenvalue perturbation of a symmetric (Hermitian) matrix with fixed diagonals, which is referred to as the continuity of the Schur-Horn mapping. We introduce a concept called strong Schur-Horn continuity, characterized by minimal constraints on the perturbation. We demonstrate that several categories of matrices exhibit strong Schur-Horn continuity. Leveraging this notion, along with a majorization constraint on the perturbation, we prove the Schur-Horn continuity for general symmetric (Hermitian) matrices. The Schur-Horn continuity finds applications in oblique manifold optimization related to quantum computing.

math.NA↗

arXiv2025

Landscape Analysis of Excited States Calculation over Quantum Computers

The variational quantum eigensolver (VQE) is one of the most promising algorithms for low-lying eigenstates calculation on Noisy Intermediate-Scale Quantum (NISQ) computers. Specifically, VQE has achieved great success for ground state calculations of a Hamiltonian. However, excited state calculations arising in quantum chemistry and condensed matter often requires solving more challenging problems than the ground state as these states are generally further away from a mean-field description, and involve less straightforward optimization to avoid the variational collapse to the ground state. Maintaining orthogonality between low-lying eigenstates is a key algorithmic hurdle. In this work, we analyze three VQE models that embed orthogonality constraints through specially designed cost functions, avoiding the need for external enforcement of orthogonality between states. Notably, these formulations possess the desirable property that any local minimum is also a global minimum, helping address optimization difficulties. We conduct rigorous landscape analyses of the models' stationary points and local minimizers, theoretically guaranteeing their favorable properties and providing analytical tools applicable to broader VQE methods. A comprehensive comparison between the three models is also provided, considering their quantum resource requirements and classical optimization complexity.

math.NA↗

arXiv2025

Frozen Gaussian approximation for the fractional Schrödinger equation

We develop a refined Frozen Gaussian approximation (FGA) for the fractional Schrödinger equation in the semi-classical regime, where the solution exhibits rapid oscillations as the scaled Planck constant $\varepsilon$ becomes small. Our approach utilizes an integral representation based on asymptotic analysis, offering a highly efficient computational framework for high-frequency wave function evolution. Crucially, we introduce the momentum space representation of the FGA and a regularization parameter $δ$ to address singularities in the higher-order derivatives of the Hamiltonian flow coefficients, which are typically assumed to be second-order differentiable or smooth in conventional analysis. We rigorously prove convergence of the method to the true solution and provide numerical experiments that demonstrate its precision and robust convergence behavior.

math.NA↗