HHL for Hermitian linear systems: indefinite extensions and hardware experiments
TL;DR
The results provide a reproducible baseline for evaluating resource--accuracy trade-offs in near-term HHL-type routines and support the view that such routines are most naturally used as subroutines for estimating task-relevant quantities from \(\ket{x}\), rather than as standalone tools for full vector reconstruction.
Abstract
Near-term demonstrations of HHL-type quantum linear-system solvers require implementations that make spectral assumptions, post-selection costs, and hardware-induced errors explicit. This is particularly important for small hardware experiments, where finite-shot sampling, phase-estimation resolution, and multi-controlled-gate overhead can dominate the observed performance. A hardware-executable HHL benchmark pipeline for Hermitian linear systems is presented and evaluated on small positive-definite and indefinite benchmark instances. By fixing the complete end-to-end workflow, including sign-aware handling of indefinite spectra and post-selected diagnostic readout, the pipeline provides a reproducible basis for comparing ideal simulation, backend-calibrated noisy simulation, and selected IBM Quantum hardware executions. The benchmark instances are organized into amplitude-resolvability classes, indexed by a threshold factor \(\tau\), enabling finite-shot sampling effects to be compared across increasingly well-resolved instance families. Experiments on \(8\times 8\) positive-definite and indefinite nonsingular systems show close agreement across ideal simulator backends. Within the constructed benchmark family, more restrictive resolvability classes yield lower reference-sign-assisted diagnostic vector errors. Indefinite instances are more noise-sensitive because sign-aware handling introduces additional controlled phase operations and reduces the effective phase resolution available within each sign sector. Selected IBM Quantum hardware runs show end-to-end executability and broad consistency with backend-calibrated noisy simulation in the reference-sign-assisted diagnostic vector error. However, for these selected runs, the noisy simulator systematically overestimates the number of shots that survive ancilla post-selection, revealing a limitation of calibrated noise models for predicting the usable sampling budget of deep HHL circuits. Overall, the results provide a reproducible baseline for evaluating resource--accuracy trade-offs in near-term HHL-type routines. They also support the view that such routines are most naturally used as subroutines for estimating task-relevant quantities from \(\ket{x}\), rather than as standalone tools for full vector reconstruction.