arXiv subjects

Xiajie Huang

Publications and source records attributed to Xiajie Huang.

3 recordsLinked to original sources

arXiv2026

A Discontinuous Galerkin Method for H(curl)-Elliptic Hemivariational Inequalities

In this paper, we develop a Discontinuous Galerkin (DG) method for solving H(curl)-elliptic hemivariational inequalities. By selecting an appropriate numerical flux, we construct an Interior Penalty Discontinuous Galerkin (IPDG) scheme. A comprehensive numerical analysis of the IPDG method is conducted, addressing key aspects such as consistency, boundedness, stability, and the existence, uniqueness, uniform boundedness of the numerical solutions. Building on these properties, we establish a priori error estimates, demonstrating the optimal convergence order of the numerical solutions under suitable solution regularity assumptions. Finally, a numerical example is presented to illustrate the theoretically predicted convergence order and to show the effectiveness of the proposed method.

math.NA↗

arXiv2025

Quantum Framework for Simulating Linear PDEs with Robin Boundary Conditions

We propose an explicit, oracle-free quantum framework for numerically simulating general linear partial differential equations (PDEs), extending previous work to incorporate (a) Robin boundary conditions - which include Neumann and Dirichlet conditions as special cases - (b) inhomogeneous terms, and (c) variable coefficients in space and time. Our approach begins with a general finite-difference discretization and applies the Schrodingerisation technique to transform the resulting system into one that admits unitary quantum evolution, enabling quantum simulation. For the Schrodinger equation corresponding to the discretized PDE, we construct an efficient block-encoding of the Hamiltonian $H$ that scales polylogarithmically with the number of grid points $N$. This encoding is compatible with quantum signal processing and allows for the implementation of the evolution operator $e^{-iHt}$. The oracle-free nature of our method permits complexity to be measured in fundamental gate units-namely, CNOT gates and single-qubit rotations-bypassing the inefficiencies of oracle queries. Consequently, the overall algorithm scales polynomially with $N$ and linearly with the spatial dimension $d$, achieving a polynomial speedup in $N$ and an exponential advantage in $d$, thereby mitigating the classical curse of dimensionality. The validity and efficiency of the proposed approach are further substantiated by numerical simulations. By explicitly defining the quantum operations and quantifying their resource requirements, our approach offers a practical alternative for numerically solving PDEs, distinct from others that rely on oracle queries and purely asymptotic scaling methods.

quant-ph↗

arXiv2025

Efficient explicit gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations

One of the most promising applications of quantum computers is solving partial differential equations (PDEs). By using the Schrodingerisation technique - which converts non-conservative PDEs into Schrodinger equations - the problem can be reduced to Hamiltonian simulations. The particular class of Hamiltonians we consider is shown to be sufficient for simulating almost any linear PDE. In particular, these Hamiltonians consist of discretizations of polynomial products and sums of position and momentum operators. This paper addresses an important gap by efficiently loading these Hamiltonians into the quantum computer through block-encoding. The construction is explicit and efficient in terms of one- and two-qubit operations, forming a fundamental building block for constructing the unitary evolution operator for that class of Hamiltonians. The proposed algorithm demonstrates a squared logarithmic scaling with respect to the spatial partitioning size, offering a polynomial speedup over classical finite-difference methods in the context of spatial partitioning for PDE solving. Furthermore, the algorithm is extended to the multi-dimensional case, achieving an exponential acceleration with respect to the number of dimensions, alleviating the curse of dimensionality problem. This work provides an essential foundation for developing explicit and efficient quantum circuits for PDEs, Hamiltonian simulations, and ground state and thermal state preparation.

quant-ph↗