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

2026-08-10

AI Summary

This work addresses three key challenges hindering the application of Variational Quantum Linear Solvers (VQLS) to computational fluid dynamics (CFD) on hybrid quantum–HPC systems: memory explosion in matrix encoding, lack of convergence guidance in ansatz design, and absence of large-scale validation. The authors propose an efficient hybrid solver framework featuring a novel parallel Pauli decomposition via the Fast Walsh–Hadamard Transform (FWHT) and an SVD-based two-term Linear Combination of Unitaries (LCU) encoding, drastically reducing memory footprint and iteration time. Systematic evaluation of eleven ansatz families and multiple optimizers reveals that expressibility and entanglement have limited impact on convergence, advocating for problem-aware ansatz construction. The study reports the first end-to-end simulation of a 15-qubit Toeplitz system on a single node of the Frontier supercomputer, demonstrating a 1,298× peak memory reduction with FWHT on an 11×11 Hele-Shaw grid and over 10,000× speedup per iteration for 8-qubit SVD-VQLS.

Abstract

Fluid dynamics workloads are dominated by repeated solves of large, structured linear systems, motivating the search for quantum acceleration. The Variational Quantum Linear Solver (VQLS) is a leading near-term candidate, but practical deployment on hybrid quantum--high--performance computing (HPC) systems faces three persistent challenges: (i) the linear-combination-of-unitaries (LCU) encoding of the system matrix explodes in memory and runtime as the problem size grows, (ii) ansatz selection is largely empirical, with no clear link between standard circuit metrics and solver convergence, and (iii) end-to-end VQLS pipelines have rarely been exercised on production HPC hardware at non-trivial qubit counts. This work addresses these challenges through three contributions. First, we benchmark four matrix-encoding strategies---naive LCU, PennyLane-integrated, Fast Walsh--Hadamard Transform (FWHT)-based parallel Pauli decomposition, and an singular value decomposition (SVD)-based two-term LCU---and show that the FWHT approach reduces peak memory by up to $1298\times$ on an $11\times 11$ Hele--Shaw grid, while the SVD-based coherent VQLS delivers over $10{,}000\times$ per-iteration speedup over standard Pauli-based VQLS at 8 qubits. Second, we evaluate 11 ansatz families with gradient-free and gradient-based optimizers on canonical Hele--Shaw flow, and find that expressibility and entanglement metrics correlate only weakly with VQLS convergence, motivating problem-aware ansatz design. Third, we deploy the full workflow on the OLCF Frontier supercomputer and successfully simulate a 15-qubit tridiagonal Toeplitz system on a single node. Together, these results establish a practical baseline for VQLS in hybrid quantum--HPC computation fluid dynamic (CFD) workflows and identify the remaining bottlenecks for larger problems.

Problem

• VQLS
• quantum-HPC
• CFD
• ansatz design
• matrix encoding

Innovation

• VQLS
• matrix encoding
• ansatz design
• hybrid quantum-HPC
• computational fluid dynamics