A Quantum Spectral Solver for Periodic Incompressible Stokes Flow
Abstract
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 standard 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.
Versions
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BibTeX
@misc{gimenez2026quantum,
title = {{A Quantum Spectral Solver for Periodic Incompressible Stokes Flow}},
author = {Juan M. Gimenez and Fehmi Cirak and Michael Ortiz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.30447},
eprint = {2606.30447},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30447},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Quantum Spectral Solver for Periodic Incompressible Stokes Flow
AU - Juan M. Gimenez
AU - Fehmi Cirak
AU - Michael Ortiz
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2606.30447
UR - https://arxiv.org/abs/2606.30447
AB - 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 standard 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.
ER -