arXiv subjects

Hsuan-Cheng Wu

Publications and source records attributed to Hsuan-Cheng Wu.

6 recordsLinked to original sources

arXiv2026

Universal Dilation of Linear Itô SDEs: Quantum Trajectories and Lindblad Simulation of Second Moments

We present a universal framework for simulating $N$-dimensional linear Itô stochastic differential equations (SDEs) on quantum computers with additive or multiplicative noises. Building on a unitary dilation technique, we establish a rigorous mapping from the general linear SDEs \[ dX_t = A(t) X_t\,dt + \sum_{j=1}^J B_j(t)X_t\,dW_t^j \] to stochastic Schrödinger equations (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact in that the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSEs are {naturally implementable} on digital quantum processors, where the stochastic Wiener increments are encoded directly by preparing the ancillary qubits. Exploiting this physical mapping, we develop two algorithmic strategies: (1) a trajectory-based approach that uses sequential weak measurements to realize efficient stochastic integrators, including a second-order scheme, and (2) an ensemble-based approach that maps moment evolution to a deterministic Lindblad quantum master equation, enabling simulation without Monte Carlo sampling. We provide error bounds based on a stochastic light-cone analysis and validate the framework with numerical experiments.

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arXiv2026

Structure-Preserving Quantum Simulation of Wave Equations on a Trapped-Ion Processor

Wave equations provide a natural testbed for near-term quantum simulation of partial differential equations, but hardware demonstrations have remained limited in spatial dimension, equation class, system size, and physically meaningful output. We develop and benchmark structure-preserving, Fourier-based quantum circuits for the one- and two dimensional acoustic wave equations and Dirac dynamics with variable mass on the Quantinuum H2-2 trapped-ion processor. The experiments include one-dimensional grids with up to $1024$ points and $32\times32$ two-dimensional grids, corresponding to an encoded state-space dimension of up to $4096$. Rather than reconstructing the full fields, we estimate subdomain kinetic energies directly from measurement samples. Across all tested acoustic and Dirac dynamics problems, the H2-2 results track the classical kinetic-energy dynamics with mean absolute errors between $5.9\times10^{-3}$ and $2.4\times10^{-2}$. At fixed retained bandwidth, the compiled gate counts grow approximately quadratically with the number of grid qubits; the acoustic circuit sizes are essentially independent of evolution time, whereas the cost also grows with the number of product-formula steps. These results provide hardware-level evidence that accurate observable dynamics can remain resolvable for structured wave problems with thousands of encoded degrees of freedom on a present-day trapped-ion processor.

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arXiv2026

From Nonlinear Stochastic Differential Equations to Quantum Channels: The Kolmogorov--Lindblad Mapping

Nonlinear stochastic differential equations (SDEs) underlie molecular modeling and drug discovery, quantitative finance, stochastic learning, and uncertainty quantification. Their expectations, event probabilities, and time correlations are therefore natural targets for quantum computation, but nonlinear drift and averaging over noise realizations obstruct a direct quantum representation. We develop an exact \emph{Kolmogorov--Lindblad mapping} (KLM) at the level of the probability law, encoded natively as a trace-one quantum density operator. For each Brownian realization, the pathwise density admits a half-density whose evolution is a stochastic Schrödinger equation. Averaging the associated pure states yields a Lindblad equation for $Γ(t)$ whose diagonal kernel is exactly the Fokker--Planck density, $p(t,x)=Γ(t;x,x)$, with classical diffusion represented by decoherence through Hermitian jump operators. Statistical observables become quantum expectations without the unknown time-dependent normalization introduced by direct amplitude encoding of the density, and classical two-time correlations admit an exact quantum regression formula. A structure-preserving Galerkin projection retains the Lindblad form at finite dimension, placing classical SDEs and open quantum dynamics on the same quantum-native computational footing. Numerical experiments for double-well Langevin dynamics and noisy Lorenz--63 exhibit rapid convergence of statistical observables. KLM thus provides a mathematically controlled path from general nonlinear stochastic dynamics to quantum channels, while isolating function approximation and coherent operator access as the remaining determinants of algorithmic efficiency.

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arXiv2026

Quantum Simulation of Differential-Algebraic Equations with Applications to Unsteady Stokes Flow

Differential-algebraic equations (DAEs) arise naturally in constrained dynamical systems, but their algebraic constraints and hidden compatibility conditions make them more subtle than standard ordinary differential equations. This paper initiates a quantum-algorithmic study of constrained linear DAEs. We introduce a dilation framework that embeds the generally non-Hermitian constrained evolution into a projected Schrödinger-type dynamics on an enlarged Hilbert space, \[ i\frac{d}{dt}Ψ(t)=P\widehat H PΨ(t), \] where $\widehat H$ is Hermitian and $P$ is the orthogonal projector onto the lifted constraint subspace. This identifies the DAE evolution with a quantum Zeno-type dynamics and enables the use of block encodings, QSVT-based projector construction, and Hamiltonian simulation. We apply the framework to structure-preserving discretizations of the unsteady Stokes equations, where the pressure enforces the discrete incompressibility constraint. For Stokes, the Zeno-reduced generator has the projected square factorization \[ S_h=-Π_hΔ_hΠ_h=(G_hΠ_h)^\dagger(G_hΠ_h), \] which can be represented through a Gaussian moment dilation and implemented as a Gaussian superposition of unitary Zeno evolutions generated by a first-order square-root Hamiltonian. In the generic sparse-access model, this gives a simulation-stage cost $\widetilde O(h^{-2}\sqrt t)$, up to the usual postselection factor for preparing the normalized dissipative state. The results provide a first step toward understanding the intersection of quantum algorithms, DAEs, constrained PDE dynamics, and square-root Gaussian dilations.

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arXiv2025

Structure-preserving quantum algorithms for linear and nonlinear Hamiltonian systems

Hamiltonian systems of ordinary and partial differential equations are fundamental mathematical models spanning virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is the underlying symplectic structure, which preserves geometric properties like phase-space volume over time and energy conservation over an extended period. In this paper, we present quantum algorithms that incorporate symplectic integrators, ensuring the preservation of this key structure. We demonstrate how these algorithms maintain the symplectic properties for both linear and nonlinear Hamiltonian systems. Additionally, we provide a comprehensive theoretical analysis of the computational complexity, showing that our approach offers both accuracy and improved efficiency over classical algorithms. These results highlight the potential application of quantum algorithms for solving large-scale Hamiltonian systems while preserving essential physical properties.

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arXiv2025

Quantum Algorithms for Nonlinear Dynamics: Revisiting Carleman Linearization with No Dissipative Conditions

In this paper, we explore the embedding of nonlinear dynamical systems into linear ordinary differential equations (ODEs) via the Carleman linearization method. Under dissipative conditions, numerous previous works have established rigorous error bounds and linear convergence for Carleman linearization, which have facilitated the identification of quantum advantages in simulating large-scale dynamical systems. Our analysis extends these findings by exploring error bounds beyond the traditional dissipative condition, thereby broadening the scope of quantum computational benefits to a new class of dynamical regimes. This novel regime is defined by a resonance condition, and we prove how this resonance condition leads to a linear convergence with respect to the truncation level $N$ in Carleman linearization. We support our theoretical advancements with numerical experiments on a variety of models, including the Burgers' equation, Fermi-Pasta-Ulam (FPU) chains, and the Korteweg-de Vries (KdV) equations, to validate our analysis and demonstrate the practical implications.

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