Memory-, Circuit-, and Ansatz-Efficient VQLS for CFD on Hybrid Quantum–HPC Systems

Research Context and Objective

Fluid dynamics simulations rely heavily on solving large linear systems, often as part of iterative nonlinear solvers or time-stepping loops. While Quantum Linear Systems Algorithms (QLSAs) offer potential asymptotic speedups, their practical implementation on current Noisy Intermediate-Scale Quantum (NISQ) hardware is hindered by high circuit depth and memory requirements. This paper investigates the Variational Quantum Linear Solver (VQLS) as a more viable near-term alternative, focusing on optimizing the end-to-end pipeline for Computational Fluid Dynamics (CFD) workloads on hybrid quantum-HPC systems.

Algorithmic Improvements

The authors address three primary bottlenecks in VQLS deployment: matrix encoding, ansatz selection, and hardware scalability. To manage the memory explosion associated with Linear Combination of Unitaries (LCU) encoding, they benchmark four strategies. They show that the Fast Walsh-Hadamard Transform (FWHT) approach reduces peak memory usage by up to 1298x compared to naive methods. Furthermore, they implement an SVD-based two-term LCU decomposition, which provides a 10,000x per-iteration speedup over standard Pauli-based decompositions at 8 qubits, effectively minimizing the number of quantum circuits required.

Ansatz Evaluation and HPC Deployment

A significant portion of the study evaluates 11 different ansatz families to determine if standard metrics—specifically expressibility and entanglement capability—predict solver convergence. The results reveal a weak correlation between these metrics and actual VQLS performance, suggesting that researchers should prioritize problem-aware ansatz design over generic circuit metrics. Finally, the authors validate their optimized workflow on the OLCF Frontier supercomputer, successfully simulating a 15-qubit tridiagonal Toeplitz system. This demonstration establishes a practical baseline for integrating VQLS into larger hybrid quantum-HPC CFD workflows.