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Lecture notes in computer science

Evaluating Quantum Algorithms for Linear Solver Workflows

Sophia Kolak, Hamed Mohammadbagherpoor, Konstantis Daloukas, Kostas Kafousas, François-Henry Rouet, Yorgos Koutsoyannopoulos, Nathan Earnest, Robert F. Lucas

Published Jan 1, 2023• 1 citation
Publisher
DOI
10.1007/978-3-031-40843-4_47

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Electromagnetic analysis based on the finite element method is widely used in antenna design. At higher frequencies, however, the required mesh refinement makes solving large sparse linear systems the main contributor to computational cost. Quantum linear system solvers offer a potential means of accelerating this step, but their output is a superposition state: reconstructing all components of the classical solution vector would forfeit the quantum speedup. Prior studies have proposed methods for obtaining physical quantities such as scattering cross sections and scattering parameters from the solution state. To support antenna design using fault-tolerant quantum computers, we formulate the total radiated power, the radiated power within a prescribed solid angle, and the dielectric and conductive losses as observables for efficient evaluation from a quantum solution state. These observables avoid the measurement cost of reconstructing all components of the output quantum state. We develop an end-to-end hybrid quantum--classical workflow that combines finite-element electromagnetic analysis, matrix preconditioning, a quantum linear system solver, and the evaluation of antenna design metrics. To validate the workflow, we performed numerical experiments evaluating these observables for an axisymmetric coaxially fed monopole antenna. Under idealized conditions, the relative errors in the total radiated power and the dissipated power with respect to the classical finite-element solution were on the order of 10−310^{-3}10−3, confirming the consistency of the workflow. The proposed method is expected to provide a basis for using quantum linear system solvers to evaluate antenna performance and, in the future, to design high-frequency wireless devices.
Quantum linear solvers are well studied as standalone quantum algorithms; however, in hybrid classical-quantum routines, their practical value must be evaluated at the level of the full non-linear application. A central issue is whether the approximation error of the quantum linear solver remains controlled once embedded in a full iterative workflow. We study this question in the context of a hybrid computational fluid dynamics (CFD) scheme. Through numerical simulations, we analyze how an approximate quantum linear solver affects the convergence of the overall CFD iteration. We show that convergence can be preserved for a non-exact quantum solver with only a moderate overhead in iteration count, provided that the high-frequency components of the linear system are resolved with sufficient accuracy. In addition, we develop an approximate qubitization-based solver (Cheb-LCU) that can reduce quantum resource requirements relative to a Quantum Singular Value Transformation (QSVT)-based solver while inducing only a small loss in convergence performance. This claim is demonstrated through explicit implementation and compilation of the quantum algorithms and by examining their impact on the convergence of the full CFD scheme. We find that our approximate approach reduces the required number of single-qubit rotations by over an order of magnitude relative to the QSVT-based solver, while requiring only a modest increase in CFD iteration count.
We study how to evaluate hybrid quantum programs as end-to-end workflows rather than as isolated devices or algorithms. Building on the Hybrid Quantum Program Evaluation Framework (HQPEF), we formalize a workflow-aware Quantum Readiness Level (QRL) score; define a normalized speedup under quality constraints for the Utility of Quantumness (UQ); and provide a timing-and-drift audit for hybrid pipelines. We complement these definitions with concise Python reference implementations that illustrate how to instantiate the metrics and audit procedures with state-of-the-art classical and quantum solvers (e.g., via Qiskit or PennyLane), while preserving matched-budget discipline and reproducibility.