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arXiv:2511.11494 · Computer Methods in Applied Mechanics and Engineering

A Quantum Spectral Method for Non-Periodic Boundary Value Problems

Eky Febrianto, Yiren Wang, Burigede Liu, Michael Ortiz, Fehmi Cirak

math.NAquant-phSubmitted Nov 14, 2025• Announced Nov 17, 2025• 2 citations• Revised Apr 21, 2026 (v3)
v1 · Nov 14, 2025v2 · Mar 27, 2026v3 · Apr 21, 2026
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Abstract

Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as $\mathscr{O}((\log N)^c)$, where $N$ is the problem size and $c$ a constant. We propose a quantum spectral method with polylogarithmic complexity for solving non-periodic boundary value problems with arbitrary Dirichlet boundary conditions. Our method extends the recently proposed approach by Liu et al. (2025), in which periodic problems are discretised using truncated Fourier series. In such spectral methods, the discretisation of boundary value problems with constant coefficients leads to a set of algebraic equations in the Fourier space. We implement the respective diagonal solution operator by first approximating it with a polynomial and then quantum encoding the polynomial. The mapping between the physical and Fourier spaces is accomplished using the quantum Fourier transform (QFT). To impose zero Dirichlet boundary conditions, we double the domain size and reflect all physical fields antisymmetrically. The respective reflection matrix defines the quantum sine transform (QST) by pre- and post-multiplying with the QFT. For non-zero Dirichlet boundary conditions, the solution is decomposed into a boundary-conforming and a homogeneous part. The homogenous part is determined by solving a problem with a suitably modified forcing vector. We illustrate the basic approach with a Dirichlet-Poisson problem and demonstrate its generality by applying it to a fractional stochastic PDE for modelling spatial random fields. We discuss the circuit implementation of the proposed approach and provide numerical evidence confirming its polylogarithmic complexity.

DOI
10.1016/j.cma.2026.118934

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Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as O((log⁡N)c)\mathscr{O}((\log N)^c)O((logN)c), where NNN is the problem size and ccc a constant. We propose a quantum spectral method with polylogarithmic complexity for solving non-periodic boundary value problems with arbitrary Dirichlet boundary conditions. Our method extends the recently proposed approach by Liu et al. (2025), in which periodic problems are discretised using truncated Fourier series. In such spectral methods, the discretisation of boundary value problems with constant coefficients leads to a set of algebraic equations in the Fourier space. We implement the respective diagonal solution operator by first approximating it with a polynomial and then quantum encoding the polynomial. The mapping between the physical and Fourier spaces is accomplished using the quantum Fourier transform (QFT). To impose zero Dirichlet boundary conditions, we double the domain size and reflect all physical fields antisymmetrically. The respective reflection matrix defines the quantum sine transform (QST) by pre- and post-multiplying with the QFT. For non-zero Dirichlet boundary conditions, the solution is decomposed into a boundary-conforming and a homogeneous part. The homogenous part is determined by solving a problem with a suitably modified forcing vector. We illustrate the basic approach with a Dirichlet-Poisson problem and demonstrate its generality by applying it to a fractional stochastic PDE for modelling spatial random fields. We discuss the circuit implementation of the proposed approach and provide numerical evidence confirming its polylogarithmic complexity.
We present a hybrid numerical-quantum method for solving the Poisson equation under homogeneous Dirichlet boundary conditions, leveraging the Quantum Fourier Transform (QFT) to enhance computational efficiency and reduce time and space complexity. This approach bypasses the integration-heavy calculations of classical methods, which have to deal with high computational costs for large number of points. The proposed method estimates the coefficients of the series expansion of the solution directly within the quantum framework. Numerical experiments validate its effectiveness and reveal significant improvements in terms of time and space complexity and solution accuracy, demonstrating the capability of quantum-assisted techniques to contribute in solving partial differential equations (PDEs). Despite the inherent challenges of quantum implementation, the present work serves as a starting point for future researches aimed at refining and expanding quantum numerical methods.
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Quantum computing offers the potential to solve computational mechanics problems with a more favourable complexity scaling than classically possible. To realise this potential, a quantum-compatible formulation of the finite element method is essential. We present such a formulation for Poisson problems with spatially varying coefficients on general domains discretised by an immersed uniform Cartesian grid. The resulting banded sparse system matrices are encoded as the sum of diagonal matrices with polynomial entries and their products with shift matrices. The sum of the matrices is formed using the linear combination of unitaries (LCU) technique. The resulting block-encoded system matrix can be implemented using a polylogarithmic number of elementary gates. For immersed boundaries, the block-encoded system matrix is further processed using a diagonal indicator matrix with zeros and ones. The indicator matrix is defined in terms of an integer-valued level-set function composed of simple geometric primitives via Boolean set operations. The required integer operations are computed on the fly using quantum arithmetic. The resulting block-encoded linear system of equations is solved using quantum singular value transformation (QSVT). The overall gate count, including the QSVT, is O(Ntot2/dpolylog⁡(Ntot))O(N_{\text{tot}}^{2/d} \operatorname{polylog}(N_{\text{tot}}))O(Ntot2/d​polylog(Ntot​)), where ddd is the problem dimension and NtotN_{\text{tot}}Ntot​ the number of grid points. In numerical examples that confirm the theoretical complexity estimates, the solutions converge approximately linearly for immersed problems and quadratically for simply connected domains.
We present a quantum spectral solver for the steady incompressible Stokes equations on a two-dimensional periodic domain. The method uses the Quantum Fourier Transform as a coherent change of basis and exploits the resulting spectral structure of the Stokes operator: the Laplacian becomes diagonal, while incompressibility is enforced mode by mode through a Helmholtz projection. In two dimensions, this projection is realized by a mode-dependent rotation from Cartesian velocity components to longitudinal–transverse coordinates, followed by component-conditioned inverse-Laplacian scaling. The velocity and pressure fields are encoded as quantum states over Fourier modes and physical components, and the corresponding spectral factors are implemented through polynomially encoded amplitude blocks. The construction extends recent quantum spectral methods in computational mechanics to an incompressible flow operator with explicit pressure–velocity splitting and divergence-free projection. The approach is also compatible with multiscale finite-element architectures in which quantum parallelism can simultaneously update all representative volume element (RVE) states. Numerical verification includes a steady vortex, a regularized periodic force-dipole benchmark, and an RVE-inspired Kolmogorov-like fluctuation benchmark. The latter illustrates how the circuit can recover a homogenized kinetic-energy observable without reconstructing the full velocity field, consistent with the role of averaged quantities in multiscale flow calculations. Subject to application-dependent assumptions about efficient state preparation and observable estimation, the circuit has polylogarithmic dependence on the grid resolution, with the polynomial degree and tile count appearing as explicit approximation and implementation parameters.
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