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.
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.
We experimentally study error mitigation and finite-shot sampling on superconducting quantum hardware under a constrained execution budget. The study combines calibration-aware qubit selection, circuit-depth scaling, zero-noise extrapolation, dynamical decoupling, readout-error mitigation, and repeated-shot estimation on an IBM Quantum processor. Experiments are organized across ideal simulation, noise-model simulation, and physical-device execution to separate sampling uncertainty from device-induced error. We investigate how mitigation performance changes with circuit depth, effective noise scale, qubit connectivity, and measurement budget, and quantify accuracy using expectation-value error, mean-squared error, statistical uncertainty, and mitigation gain. A fixed hardware-execution budget is used to evaluate shot allocation strategies and repeated measurements without relying on unlimited sampling. The resulting analysis provides a hardware-aware characterization of when mitigation improves expectation-value estimation and when finite-shot fluctuations offset the benefit of additional mitigation overhead. The implementation uses contemporary Qiskit and IBM Quantum Runtime workflows and is designed to provide reproducible experimental evidence for error-mitigation studies on current quantum processors.
Solving systems of linear equations, Ax=b, is a ubiquitous task in computational science. The Harrow-Hassidim-Lloyd algorithm requires phase estimation, controlled rotations, and post-selection, impractical on noisy intermediate-scale quantum hardware. We extend sample-based quantum diagonalization (SQD), developed for quantum chemistry, to linear systems. Following the sample-based Krylov variant, the Hermitian matrix A is decomposed in the Pauli basis, the Trotterized time evolution of ∣b⟩ is sampled at several times to select computational-basis configurations, and the system is solved classically by a Galerkin projection onto the selected subspace. We prove that the subspace solution is exact when the sampled configurations contain the Krylov support of b and optimal in the energy norm for symmetric positive-definite matrices, so that noise-induced configurations cannot degrade it as long as they do not displace the ideal ones. On IBM Quantum hardware, the sampled distribution of a 4×4 example deviates from the exact one by eleven times the shot-noise level, yet the solution is recovered exactly. For a localized problem of dimension N=1024, SQD selects 33 configurations, reaches a relative error of 10−5, and its truncated subspaces coincide with the optimal choice; readout errors enlarge the subspace without degrading the accuracy. The tests also show the limitations: the selected subspace coincides with that of inexpensive classical selection, banded matrices require exponentially many Pauli terms in the binary encoding, and Trotter errors populate spurious configurations unless the circuits are deep. An MIT-licensed Python package and notebooks that reproduce the numerical results are provided.
Quantum PDE solvers are difficult to evaluate in practice because published studies use different discretizations, output models, reconstruction rules, and hardware assumptions. We present a reproducible, application-driven benchmark for the 1-D Dirichlet heat equation that compares eleven kernels under the same problem instances and readout contract. The benchmark covers coherent linear solvers (HHL, QSVT, and QLS-Fourier), VQLS, imaginary-time methods (QITE, var-QITE, and AVQDS), real-time Hamiltonian simulation and unitary dilations (Hamiltonian simulation, Schade-Hamiltonian, and Schr"odingerisation), and the spectral quantum simulation method (QSM). We use three initial conditions, four grid sizes from n=4 to 7 qubits (N=16 to 128), a CFL-like ratio r≈0.4, and final time T=1. Statevector, ideal-shot (105 shots per step), and noisy Aer backends separate algorithmic, sampling, and device-noise errors. On statevector, QSM and Schade-Hamiltonian reproduce the semi-discrete reference to floating-point precision, Schr"odingerisation reaches approximately 10−4 error, and QITE is the strongest non-transform method for smooth data. Under the fixed-shot setting, HHL degrades to approximately 0.79 relative ℓ2 error, while several low-depth or postselected methods become readout-limited. A norm-mismatch ablation attributes 23--29% of the n=7 smooth-initial-condition error of Hamiltonian simulation, AVQDS, and QLS-Fourier to reconstruction normalization. Compact observables, including total thermal energy and individual Fourier-mode weights, require 1--3 orders of magnitude fewer shots than full-field reconstruction. The resulting public benchmark provides a practical guide for selecting quantum PDE solvers.
We propose a hybrid quantum-classical implementation of the Harrow-Hassidim-Lloyd (HHL) algorithm for the matrix inversion (linear-systems) problem, a key subroutine in quantum computing. Rather than claiming a new complexity result, we provide an assumption-explicit proof sketch, aligned with established literature, that contextualizes matrix inversion as Bounded-Error Quantum Polynomial Time (BQP)-complete and offers reduction-based intuition via Grover-type amplitude amplification. We implement HHL for 2 × 2, 4 × 4, and 8 × 8 Hermitian matrices using Qiskit and evaluate performance using simulator-based experiments with calibrated noise models and executions on IBM Quantum and IonQ backends. To enhance robustness on Noisy Intermediate-Scale Quantum (NISQ) hardware, we integrate measurement-error mitigation, zero-noise extrapolation, and post-selection, and we quantify performance using fidelity and mean squared error. Results from noiseless simulation, calibrated-noise simulation, and real-device execution show that mitigation improves reconstruction fidelity for small Hermitian systems, particularly for deeper circuits. These findings support mitigation-aware feasibility on current NISQ devices; however, they do not establish present-day runtime advantage over optimized classical solvers. The proposed workflow is relevant to quantum optimization, quantum machine learning, and quantum chemistry, where linear-system primitives arise frequently. Future work will focus on reducing phase-estimation depth (e.g., iterative or semiclassical variants), improving compilation and benchmarking transparency, and re-evaluating larger instances on higher-fidelity hardware.
Accuracy-Cost Trade-offs for Reference VQE Calculations of H2 on IBM Quantum Hardware
We present a hardware-validated reference dataset for variational ground-state energy calculations of the hydrogen molecule H2 on several IBM Quantum processors available in 2026. Using a standardized workflow, we benchmark the impact of shot count, backend choice, optimization strategy, and runtime variability on the achievable energy accuracy relative to exact diagonalization. The resulting dataset and analysis provide a transparent baseline for assessing the current capabilities and limitations of IBM Quantum hardware for quantum-chemistry applications, and are meant to ease the entry for new users by providing a comprehensive overview of choices and their effects as well as runtime efforts and costs that can be expected.
Across the configurations studied here, circuit simplification through tapered mappings provides the most consistent accuracy gains, resilience level 1 improves accuracy at a substantial cost premium, and session-based execution yields no systematic accuracy advantage over single-job execution despite markedly higher billed time.
Hardware noise and finite sampling perturb the fidelity estimates forming a quantum-kernel Gram matrix. We measured how far three hardware-reconstructed Gram matrices depart from an exact statevector reference for one frozen four-qubit ZZ feature map on N=24 indoor air-quality windows, executed on ibm_fez at 1024 shots per circuit in three single, non-interleaved jobs: baseline, dynamical decoupling alone, and gate twirling alone. All were complete, finite, and positive-semidefinite. Off-diagonal root-mean-squared error (RMSE) against the reference was 0.0878, 0.0864, and 0.0427; full-matrix centered kernel alignment (CKA) ranged 0.933-0.989 and the post hoc diagonal-excluded (U-centered) CKA 0.816-0.986. The gate-twirled job deviated least on every reported geometry axis; its baseline contrasts are deletion-stable for the Spearman, mean-absolute-error, RMSE, and full-matrix CKA diagnostics, while the Pearson and diagonal-excluded contrasts fall just below that convention. Dynamical decoupling was not separated from the baseline. The observed error exceeded both finite-shot reference scales, so, under those sampling-only models, sampling does not explain it. Centered kernel-target alignment did not track reconstruction fidelity and stayed at or below each label-permutation reference: implementation fidelity and task relevance are distinct diagnostic axes. All configuration-level statements describe three realized jobs on one backend; no mitigation-efficacy, classifier-superiority, forecasting, or quantum-advantage claim is made.