Efficient Quantum Simulation of Variable-Coefficient Transport with Continuous Source Injection
Abstract
Quantum time-marching algorithms for transport PDEs often represent variable coefficients and forcing through register-expanding dilations, block-encoding oracles, or repeated postselection. We present an alternative algorithm for a forced variable-coefficient advection-diffusion equation in flow-inspired skew-symmetric form that incorporates spatially varying velocity, viscous dissipation, and persistent source injection with a peak logical requirement of nq+1 qubits. A centered skew-symmetric discretization makes the advection operator strictly skew-Hermitian for arbitrary velocity profiles, enabling an ancilla-free unitary realization using a Gray-code Trotter sequence of controlled-Ry rotations. Diffusion is applied in the Fourier basis through a uniformly controlled rotation on one postselected ancilla, which is measured, reset, and reused between the two diffusion half-steps, while the source is incorporated classically through second-order Strang splitting. Statevector simulations for N=16 and $32$ recover second-order temporal convergence against high-accuracy classical solutions, while Richardson extrapolation gives fourth-order accuracy and reduces kernel calls by factors of four to fourteen. Independent tests through N=256 confirm second-order spatial consistency. We further show that the per-step ancilla failure probability is proportional to the instantaneous viscous dissipation rate, making postselection cost self-regulating over a fifty-fold viscosity range. Stable evolution is demonstrated for 5×104 time steps without observable secular error growth, while Gray-code advection accounts for $71−−95\%$ of transpiled controlled-NOT gates. The fixed-width kernel provides a qubit-efficient building block for near-term hardware studies, although classical readout and state re-preparation remain the main obstacles to coherent multistep evolution.
Related
- Transport Polynomial Exactness for Direct ALE-DG Methods with Nonsmooth mesh motion: A Conservative Coefficient-Evolution Framework
- Efficient estimation methods for non-Gaussian regression models in continuous time
- Optimal Transport for Continuous-Time Mean-Variance Portfolio Selection
- Continuous data assimilation in steady Navier-Stokes equations with unknown viscosity: robust and efficient solvers and fast parameter recovery
- Stability, discretization, and optimality of continuous-time causal optimal transport
BibTeX
@misc{akash2026efficient,
title = {{Efficient Quantum Simulation of Variable-Coefficient Transport with Continuous Source Injection}},
author = {Mohammad Mehedi Hasan Akash and Turag Dev and Nhat-Quang Nguyen and Yanzhu Chen and Huixuan Wu and Kourosh Shoele},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27712},
eprint = {2608.27712},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27712},
}
RIS
TY - UNPB
TI - Efficient Quantum Simulation of Variable-Coefficient Transport with Continuous Source Injection
AU - Mohammad Mehedi Hasan Akash
AU - Turag Dev
AU - Nhat-Quang Nguyen
AU - Yanzhu Chen
AU - Huixuan Wu
AU - Kourosh Shoele
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27712
UR - https://arxiv.org/abs/2608.27712
AB - Quantum time-marching algorithms for transport PDEs often represent variable coefficients and forcing through register-expanding dilations, block-encoding oracles, or repeated postselection. We present an alternative algorithm for a forced variable-coefficient advection-diffusion equation in flow-inspired skew-symmetric form that incorporates spatially varying velocity, viscous dissipation, and persistent source injection with a peak logical requirement of n_q+1 qubits. A centered skew-symmetric discretization makes the advection operator strictly skew-Hermitian for arbitrary velocity profiles, enabling an ancilla-free unitary realization using a Gray-code Trotter sequence of controlled-R_y rotations. Diffusion is applied in the Fourier basis through a uniformly controlled rotation on one postselected ancilla, which is measured, reset, and reused between the two diffusion half-steps, while the source is incorporated classically through second-order Strang splitting. Statevector simulations for N=16 and $32$ recover second-order temporal convergence against high-accuracy classical solutions, while Richardson extrapolation gives fourth-order accuracy and reduces kernel calls by factors of four to fourteen. Independent tests through N=256 confirm second-order spatial consistency. We further show that the per-step ancilla failure probability is proportional to the instantaneous viscous dissipation rate, making postselection cost self-regulating over a fifty-fold viscosity range. Stable evolution is demonstrated for 5×10^4 time steps without observable secular error growth, while Gray-code advection accounts for $71−−95\%$ of transpiled controlled-NOT gates. The fixed-width kernel provides a qubit-efficient building block for near-term hardware studies, although classical readout and state re-preparation remain the main obstacles to coherent multistep evolution.
ER -