arXiv subjects

Yingzhou Li

Publications and source records attributed to Yingzhou Li.

At least 19 recordsLinked to original sources

arXiv2026

Virtual quantum neural networks

Quantum neural networks are a prominent model of quantum machine learning. Their training consists in the minimization of a given loss function over a parametrized family of quantum circuits, mathematically described by unitary operators, or, more generally, completely positive linear maps. In this work, we extend the notion of quantum neural network, using random sampling and classical data processing to enlarge the optimization space in a way that includes linear combinations of completely positive maps. Our extended model, called virtual quantum neural networks, leverages its enlarged optimization space to achieve increased expressivity and improved noise robustness. These benefits are illustrated in three representative tasks: quantum error mitigation, binary classification, and estimation of ground-state energies. Overall, virtual quantum neural networks offer a flexible learning paradigm that expands the space of achievable computations and strengthens the applications of near-term quantum hardware.

quant-ph↗

arXiv2026

Approximate Inversion of Discrete Fourier Integral Operators via Hierarchically Semiseparable Matrices

This paper introduces a novel method for approximating the inverse of discrete Fourier integral operators (FIOs). Given an $N \times N$ matrix representation $K$ of an FIO, the proposed algorithm consists of two stages. In the offline stage, we first construct a butterfly factorization (BF) $\tilde{K}$ of $K$, which enables fast forward matrix-vector multiplication. We then construct a hierarchically semiseparable (HSS) approximation $\tilde{G} \approx G$, where $G = K^{*} K$, using fast applications of $\tilde{K}$ and $\tilde{K}^{*}$ to random matrices. Finally, we apply the ULV factorization to the HSS matrix $\tilde{G}$ to obtain an approximation $\tilde{F} \approx G^{-1}$. Combining these approximations yields an approximate inverse $K^{-1} \approx \tilde{F} \tilde{K}^{*}$. The offline stage has complexity $O(N \log^{2} N)$ for 1D problems and $O(N^{1.5} \log N)$ for 2D problems. In the online stage, the proposed method approximates $K^{-1} u$ for a given input vector $u$ with complexity $O(N \log N)$ for both 1D and 2D problems. The proposed method can be used either as a direct solver or as a preconditioner for iterative methods. Numerical results for 1D and 2D FIOs demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2026

Nonadiabatic Molecular Dynamics on Real-time Excited-State Surfaces via Machine Learning Hamiltonians

Simulating the coupled, nonequilibrium dynamics of electrons and nuclei is a central challenge in chemistry, physics, and materials science, governing phenomena from photocatalysis to quantum information. The primary bottleneck has been the lack of a general, accurate, and efficient method for modeling the complete excited-state landscape: the potential energy surfaces, forces, and non-adiabatic couplings for multiple electronic states. While machine learning has revolutionized ground-state simulations and shown promise for excited states in molecules, a unified framework that solves the complete multi-state problem for general condensed matter systems has remained elusive. Here we introduce on-the-fly N${^2}$AMD (Neural network NAMD), a machine learning framework that makes on-the-fly NAMD in solids a reality. By employing an equivariant neural network to predict the system Hamiltonian, the framework delivers excited-state energies, forces, and non-adiabatic coupling vectors at a fraction of the cost of ab initio calculations. Crucially, it allows simulations with hybrid functional accuracy, a level of approach previously inaccessible for NAMD. We showcase its capabilities with three topical examples: correcting order-of-magnitude errors in carrier dynamics predicted by conventional procedure in a MoS$_2$/WS$_2$ heterostructure, simulating previously inaccessible photoinduced ferroelectric switching, and capturing real-time polaron formation in TiO$_2$ at the hybrid-functional level. On-the-fly N${^2}$AMD moves beyond the limitations of equilibrium theory, establishing a new paradigm for the predictive, first-principles design of materials operating far from equilibrium.

physics.comp-ph↗

arXiv2026

The Convergence and Error Analysis of Coordinate Descent Methods with Compression for Full Configuration Interaction

We study the effect of compression in Coordinate Descent Full Configuration Interaction (CDFCI) within an unconstrained optimization formulation of the full configuration interaction ground-state problem. Under suitable local assumptions, we prove that the compressed iteration converges linearly to the solution of an associated restricted problem. We also characterize the convergence point of the compressed algorithm. Under an additional exponential decay assumption on the target eigenvector, we show that the resulting eigenvalue error is of order $\tau^2$, where $\tau$ denotes the compression threshold. Numerical results support the analysis.

math.NA↗

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 na\"ive 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

A Superfast Direct Solver for 2D Type-II Inverse Nonuniform Discrete Fourier Transform Based on Hierarchically Semiseparable Matrix

This paper proposes a direct inversion method for the 2D type-II nonuniform discrete Fourier transform~(NUDFT). The NUDFT matrix $A$ is factored as $A = G F$, where $G$ can be expressed as a kernel matrix and $F$ is the 2D DFT matrix. We show that $G$ can be approximated by a hierarchically semiseparable~(HSS) matrix and give an estimate of the HSS rank. Then, using the least-squares solver for HSS matrix and the two-dimensional inverse fast Fourier transform, the inverse NUDFT problem can be solved efficiently. Our algorithm has an offline complexity of $O\bigl(M+ N^{3 / 2} \log^{3} N\bigr)$ where $M$ and $N$ are the size of rows and columns of the NUDFT matrix, respectively. Once the direct solver is built, it can be applied to a vector with an online complexity of $O\bigl(M+ N \log^{3} N\bigr)$. The proposed method can be used as a preconditioner for iterative methods, especially when the sample points are distributed on a grid such that $A$ is ill-conditioned. Numerical results are provided to show the scaling performance of the inversion method and demonstrate the efficiency and robustness of it as a preconditioner.

math.NA↗

arXiv2026

Random Grover Search

Grover's algorithm achieves a quadratic speedup for unstructured search given a global oracle for the target set. In many applications, however, the target set is specified as the intersection of multiple constraint sets. Constructing a global oracle for the intersection can be costly, whereas the individual constraint oracles are often much simpler to implement. We study a randomized Grover search algorithm that directly uses these constraint oracles. At each iteration, one of the corresponding Grover operators is selected at random. For the two-operator case with uniform sampling, we prove that the success probability approaches one after \[ \Theta \left(\frac\pi4\sqrt{\frac{N}{r}}\right) \] iterations, where $r$ is the size of the intersection. Thus, the algorithm achieves the same asymptotic query complexity as standard Grover search but without requiring a global oracle. We then generalize the analysis to arbitrary sampling distributions and an arbitrary number of Grover operators through an auxiliary operator that approximates the expected Grover evolution, while retaining the same asymptotic complexity. We further show that highly biased sampling distributions can still achieve near-unit success probability, enabling cheaper Grover operators to be used more frequently. Finally, we prove asymptotic optimality and support the theoretical results with numerical simulations.

quant-ph↗

arXiv2026

A stochastic gradient algorithm for non-separable optimization with convergence guarantee

We study non-separable objectives in which the loss depend on dataset-level quantities. We introduce an SGD-style framework that employs two batch-gradient constructs: the ideal per-batch gradient `$G$' and a cached surrogate `$H$' for cases where full-data terms are expensive. Notably, in the sample-wise separable case, our method reduces to standard mini-batch SGD. Our main contribution is a unified local convergence theory: under mild smoothness and Jacobian-boundedness assumptions, we prove local linear convergence under local strong convexity and local $O(1/k)$ sublinear convergence under local convexity for both `$G$'-driven and `$H$'-driven updates. Crucially, these guarantees hold for fixed step sizes within explicitly characterized ranges; we provide explicit bounds showing how cache staleness, surrogate approximation error, batch size, and step size influence the convergence constants and allowable step-size ranges.

math.OC↗

arXiv2026

On the representation for stochastic graph delay propagation

In this work, we utilize the Hermite expansion to approximate the distributions of the sum and maximum of independent random variables. We model distributions with a three-segment representation, where the left and right tails are respectively modeled as combinations of Hermite functions, and the intermediate segment is approximated by piecewise polynomials. This approximation admits rigorous $L^2$- and pointwise convergence properties supported by classical results. We develop an algorithmic framework for applying our model to the graph delay propagation problem, where sum and max operations are performed on the proposed model structure. Numerical experiments demonstrate that our model can capture the quantile values with high accuracy compared to Monte Carlo simulation results, significantly outperforming classical Gaussian-based models.

math.NA↗

arXiv2026

CDFCI: High-Performance Parallel Software for Many-Body Large-Scale Eigenvalue Problems

CDFCI is a shared-memory parallel numerical program for computing low-lying eigenpairs of large-scale, non-relativistic fermionic Hamiltonians. The software is designed to handle a broad class of many-body quantum models, including both ab initio electronic structure Hamiltonians and lattice-based Hamiltonians arising in condensed matter physics. CDFCI combines an efficient coordinate-descent-based selected configuration interaction algorithm with dedicated parallelization strategies, achieving high performance on modern multi-core architectures. Benchmark results on representative quantum chemistry and condensed matter test cases demonstrate that CDFCI attains state-of-the-art accuracy with competitive performance compared to established selected configuration interaction (such as CIPSI or SHCI) and DMRG implementations. The software is open-source, extensively documented, and provides a Python interface for seamless integration with PySCF and other many-body simulation workflows.

physics.comp-ph↗

arXiv2026

Windowed Fourier Propagator: A Frequency-Local Neural Operator for Wave Equations in Inhomogeneous Media

Wave equations are fundamental to describing a vast array of physical phenomena, yet their simulation in inhomogeneous media poses a computational challenge due to the highly oscillatory nature of the solutions. To overcome the high costs of traditional solvers, we propose the Windowed Fourier Propagator (WFP), a novel neural operator that efficiently learns the solution operator. The WFP's design is rooted in the physical principle of frequency locality, where wave energy scatters primarily to adjacent frequencies. By learning a set of compact, localized propagators, each mapping an input frequency to a small window of outputs, our method avoids the complexity of dense interaction models and achieves computational efficiency. Another key feature is the explicit preservation of superposition, which enables remarkable generalization from simple training data (e.g., plane waves) to arbitrary, complex wave states. We demonstrate that the WFP provides an explainable, efficient and accurate framework for data-driven wave modeling in complex media.

cs.LG↗

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

A Superfast Direct Solver for Type-III Inverse Nonuniform Discrete Fourier Transform

The nonuniform discrete Fourier transform (NUDFT) and its inverse are widely used in various fields of scientific computing. In this article, we propose a novel superfast direct inversion method for type-III NUDFT. The proposed method approximates the type-III NUDFT matrix as a product of a type-II NUDFT matrix and an HSS matrix, where the type-II NUDFT matrix is further decomposed into the product of an HSS matrix and an uniform discrete Fourier transform (DFT) matrix as in [Wilber, Epperly, and Barnett, SIAM Journal on Scientific Computing, 47(3):A1702-A1732, 2025]. This decomposition enables both the forward application and the backward inversion to be accomplished with quasi-linear complexity. The fast inversion can serve as a high-accuracy direct solver or as an efficient preconditioner. Additionally, we provide an error bound for the approximation under specific sample distributions. Numerical results are presented to verify the relevant theoretical properties and demonstrate the efficiency of the proposed methods.

math.NA↗

arXiv2025

State-Specific Orbital Optimization for Enhanced Excited-States Calculation on Quantum Computers

We propose a state-specific orbital optimization scheme for improving the accuracy of excited states of the electronic structure Hamiltonian for the use on near-term quantum computers, which can be combined with any overlap-based excited-state quantum eigensolver. We derived the gradient of the overlap term between different states generated by different orbitals with respect to the orbital rotation matrix and use the gradient-based optimization methods to optimize the orbitals. This scheme allows for more flexibility in the choice of orbitals. We implement the state-specific orbital optimization scheme with the variational quantum deflation (VQD) algorithm, and show that it achieves higher accuracy than the state-averaged orbital optimization scheme on various molecules including H4 and LiH.

quant-ph↗

arXiv2025

Products between block-encodings

Block-encoding is a standard framework for embedding matrices into unitary operators in quantum algorithms. Efficient implementation of products between block-encoded matrices is crucial for applications such as Hamiltonian simulation and quantum linear algebra. We present resource-efficient methods for matrix-matrix, Kronecker, and Hadamard products between block-encodings that apply to rectangular matrices of arbitrary dimensions. Our constructions significantly reduce the number of ancilla qubits, achieving exponential qubit savings for sequences of matrix-matrix multiplications, with a moderate increase in gate complexity. These product operations also enable more complex block-encodings, including a compression gadget for time-dependent Hamiltonian simulation and matrices represented as sums of Kronecker products, each with improved resource requirements.

quant-ph↗

arXiv2025

Randomized coordinate gradient descent almost surely escapes strict saddle points

We analyze the behavior of randomized coordinate gradient descent for nonconvex optimization, proving that under standard assumptions, the iterates almost surely escape strict saddle points. By formulating the method as a nonlinear random dynamical system and characterizing neighborhoods of critical points, we establish this result through the center-stable manifold theorem.

math.OC↗

arXiv2025

Hierarchical Tucker Low-Rank Matrices: Construction and Matrix-Vector Multiplication

In this paper, a hierarchical Tucker low-rank (HTLR) matrix is proposed to approximate non-oscillatory kernel functions in linear complexity. The HTLR matrix is based on the hierarchical matrix, with the low-rank blocks replaced by Tucker low-rank blocks. Using high-dimensional interpolation as well as tensor contractions, algorithms for the construction and matrix-vector multiplication of HTLR matrices are proposed admitting linear and quasi-linear complexities respectively. Numerical experiments demonstrate that the HTLR matrix performs well in both memory and runtime. Furthermore, the HTLR matrix can also be applied on quasi-uniform grids in addition to uniform grids, enhancing its versatility.

math.NA↗