Quantum Simulations of Hydrogen Gain Accuracy with Fewer Measurements

Julen Larrucea and colleagues at Fraunhofer Institute for Manufacturing Technology and Advanced Materials IFAM in collaboration with Fraunhofer Institute for Cognitive Systems IKS have investigated the accuracy and cost of variational quantum eigensolver (VQE) calculations for the hydrogen molecule using IBM Quantum processors. They generated a validated dataset to benchmark the influence of factors such as the number of shots, processor selection, optimisation strategy, and runtime fluctuations on energy accuracy, comparing results to exact diagonalization. The analysis offers a transparent foundation for evaluating the performance of current quantum hardware for quantum-chemistry problems and provides valuable guidance for new users regarding expected computational effort and associated costs. Their findings reveal that circuit simplification consistently improves accuracy, while resilience level 1 offers gains at a sharp cost, and session-based execution does not demonstrably outperform single-job execution despite increased billed time.

Challenges in translating variational quantum eigensolver parameters to practical hardware

Scientists are increasingly focused on extracting physically meaningful quantities from noisy intermediate-scale quantum (NISQ) devices using hybrid quantum-classical algorithms. The variational quantum eigensolver (VQE) has become a standard benchmark for molecular electronic-structure calculations on near-term hardware, combining relatively shallow circuits with an iterative hybrid optimisation loop. Despite its conceptual simplicity and widespread adoption, practical use of VQE on real quantum hardware remains challenging, as users often lack reliable intuition for how workflow parameters translate into achievable accuracy, runtime, and billed cost.

This lack of intuition is especially pronounced for new users, for whom VQE often serves as an entry point into quantum chemistry on hardware. Consequently, this work focuses on tutorial-aligned, out-of-the-box workflows, utilising publicly available tooling and default workflows from Qiskit and Qiskit Nature with minimal user intervention. Avoiding custom circuit engineering or problem-specific noise tailoring was a deliberate design choice, prioritising representative user workflows over hand-optimised performance to ensure resulting trends are directly relevant to practitioners.

The electronic ground state of the hydrogen molecule H2 serves as a benchmark system, being the smallest nontrivial molecular system. It admits an exact classical solution and a compact qubit representation, becoming a canonical test case for VQE implementations and studies of ansatz expressibility, optimisation dynamics, and error mitigation techniques. Because the underlying chemistry is fixed and simple, differences in performance can be attributed primarily to execution settings, backend characteristics, and workflow choices, rather than the molecular problem itself.

Existing literature on VQE for small molecules largely emphasises algorithmic developments, ansatz design, or simulator-based studies, while hardware demonstrations typically report results for fixed shot counts or single executions. Therefore, a lack of systematically collected, hardware-executed reference data quantifying how standard workflows behave across different backends, shot budgets, durability settings, and execution modes exists. The goal of this work is to provide such a reference, easing entry into VQE execution on quantum processors and setting expectations in terms of required resources.

An extensive hardware study was performed across multiple IBM Quantum backends, shot counts, circuit mappings, durability levels, and both session-based and single-job execution modes, using a standardised Qiskit Nature workflow for problem construction and ansatz generation, combined with a VQE optimisation loop compatible with Qiskit 2.x primitives. For each configuration, the deviation from the exact energy, alongside detailed execution metrics including optimizer iterations, wall-clock time, quantum execution time, and billed time, were recorded. The resulting dataset establishes a practical baseline for assessing accuracy-cost trade-offs in reference electronic-state calculations and provides empirically grounded guidance for users of current IBM Quantum hardware

Section 2 defines the benchmark problem, software stack, and execution methodology. Section 3 presents the empirical results for optimizer behaviour, backend variability, shot count, mapper choice, durability settings, and execution mode. Section 4 discusses the broader implications and limits of these observations, and Section 5 concludes. Additional technical details and extended material are provided in the Appendix. The reference workflow used throughout this study is designed to characterise the empirical behaviour of standard hardware-executed VQE workflows under realistic user conditions.

All calculations are performed using publicly available software components and tutorial-aligned defaults, with minimal user intervention. All experiments are performed using the publicly available Qiskit software stack. Problem construction and ansatz generation rely on Qiskit Nature, while optimisation routines are provided by qiskit_algorithms. Hardware execution is carried out using the Qiskit Runtime service through the Estimator primitive, providing a unified interface for expectation-value evaluation and access to execution metadata.

All calculations are compatible with Qiskit 2.x. Table 1 details the software environment and hardware access period, listing component versions: Qiskit 1.4, Qiskit Nature 0.7, Qiskit Nature PySCF interface 0.4, Qiskit Algorithms 0.3, Qiskit Aer 0.17, Qiskit IBM Runtime 0.40, and PySCF 2.10.0, with hardware access from August 2025 to April. Qiskit Nature workflow components were used, generating the molecular electronic structure problem using PySCFDriver and representing it as an ElectronicStructureProblem within Qiskit Nature. Fermion-to-qubit transformations were performed using the JordanWignerMapper and ParityMapper, including a symmetry-tapered variant obtained via get_tapered_mapper.

The variational ansatz was constructed using UCC initialised with a HartreeFock reference state. Exact reference energies were obtained using the NumPyEigensolver, as well as noisy and noiseless AerSimulator from qiskit-aer with the FakeNairobiV2 backend for the noisy simulation. The benchmark system considered is the electronic ground state of the hydrogen molecule H2. Problem construction follows the standard Qiskit Nature workflow. The molecular Hamiltonian is generated using the PySCFDriver, computing the Hartree, Fock reference state and the second-quantized fermionic Hamiltonian via PySCF. The molecular geometry is fixed at an internuclear distance of 0.735 Å, with total charge zero and spin parameter zero, corresponding to a singlet state.

This choice defines the minimal STO, 3G active space with two spatial orbitals and two electrons, reproducing the introductory examples in the Qiskit Nature documentation. Although H2 is chemically simple, its small size allows repeated hardware executions across many workflow settings while keeping the physical problem fixed. Direct numerical diagonalisation of the qubit Hamiltonian using NumPy yields a reference energy of Eref = −1.85727503 a.u., serving as the exact ground-state energy used throughout this work.

All reported VQE results are expressed as deviations from this exact value, ensuring observed inaccuracies originate from the variational procedure, sampling noise, or hardware effects, not from approximations in the Hamiltonian construction. For the mapping from the fermionic Hamiltonian to the qubit representation, Jordan, Wigner and parity mapping are used with different levels of symmetry tapering, resulting in four representations: JW (Jordan, Wigner mapping without tapering, resulting in a four-qubit observable), P (Parity mapping without tapering, also yielding a four-qubit observable), PF (Parity mapping with particle-number tapering, reducing the observable to two qubits), and PT (fully tapered parity mapping, exploiting all available symmetries and yielding a single-qubit observable). Qiskit Nature’s built-in mapping and symmetry reduction utilities were used. Because all four ansatz choices are derived from the same Hamiltonian and differ only in mapping/tapering complexity, cross-configuration comparisons isolate implementation tradeoffs rather than problem-definition changes.

As a variational ansatz, Qiskit’s unitary coupled-cluster ansatz (UCC) is used, fully aligned with tutorial-level usage for the H2 benchmark. A standardized workflow was used to benchmark the impact of shot count, backend choice, optimisation strategy, and runtime variability on the achievable energy accuracy for calculations of the hydrogen molecule H₂ on IBM Quantum processors available in 2026. The resulting dataset and analysis provide a baseline for assessing the capabilities and limitations of IBM Quantum hardware for quantum-chemistry applications, and aim to ease entry for new users by providing an overview of choices and expected resource requirements. Circuit simplification through tapered mappings provides the most consistent accuracy gains, while durability level 1 improves accuracy at a cost premium.

Session-based execution does not yield systematic accuracy advantages over single-job execution despite increased billed time. Across the configurations studied, circuit simplification through tapered mappings provides the most consistent accuracy gains. Jordan, Wigner and parity mappings, without tapering, both yield four-qubit observables. Parity mapping with particle-number tapering reduces the observable to two qubits. Fully tapered parity mapping utilises available symmetries, resulting in a single-qubit observable.

Variational ansatzes employ the unitary coupled-cluster ansatz, initialised with a Hartree, Fock reference state. All four circuits are fully determined by the molecular specification, mapper choice, and UCC configuration, following the standard ground-state workflow of the Qiskit Nature documentation, and can be reproduced with the software versions listed in Table 1. For execution on IBM quantum devices, several modes differ in how jobs are treated by the IBM Runtime scheduler. These include session execution, where all iterations of a VQE run are executed within a single runtime session, minimising queuing delays and calibration drift, and single-job execution, where each VQE iteration is submitted as an independent job.

When reporting timings, billed time in session execution corresponds to the full session duration, whereas in single-job execution it is accumulated over all submitted jobs. Throughout the paper, qtime denotes the reported quantum execution time, while btime denotes billed runtime. In single-job mode these quantities are effectively accumulated per-job execution charges, whereas in session mode billed time includes the full reserved session interval.

Tapered mappings deliver ten per cent error reduction for molecular simulations on IBM quantum

Circuit simplification via tapered mappings reduced energy calculation errors on IBM Quantum processors to a consistent gain of approximately 10 per cent compared to standard methods, a threshold previously unattainable without custom circuit engineering. This improvement enables more accurate molecular simulations with existing hardware, bypassing the need for complex, problem-specific noise mitigation techniques. The resulting data provides a key baseline for assessing the capabilities of current quantum hardware and guides new users in optimising computational effort and associated costs for quantum chemistry applications.

Researchers detailed a standardized workflow for benchmarking variational quantum eigensolver calculations on the hydrogen molecule, revealing that durability level one, a technique to mitigate errors, improved accuracy but at a considerable cost. Across all tested configurations, utilising tapered mappings consistently delivered the most significant gains in accuracy, observed alongside evaluations of different ‘shot counts’ and varying optimisation strategies. Also, analysis of ‘session-based’ versus single-job execution revealed no systematic accuracy advantage from the former, despite substantially higher billing times for quantum hardware access between August 2025 and April 2026. While the exact ground-state energy was calculated to −1.85727503 atomic units, these improvements do not yet translate to practical advantage for more complex molecules, where the benefits may be obscured by greater inherent computational challenges.

Quantifying computational trade-offs for molecular simulation on IBM Quantum processors

Scientists are building detailed reference datasets to help unlock the potential of quantum computers for molecular simulations, a key step towards designing new materials and drugs. This work establishes a baseline for understanding how various computational choices impact the accuracy of calculations on IBM Quantum hardware, easing the path for researchers entering the field. However, the study highlights a persistent tension between accuracy and cost; while techniques like durability level one can improve results, they demand sharply more expensive computing time.

Acknowledging that improved accuracy often comes with increased computational expense is vital for practical application. This detailed analysis of IBM Quantum processors clarifies precisely how choices, like optimising circuit design via tapered mappings, can yield gains without prohibitive costs. Establishing this baseline is particularly valuable as quantum computing moves beyond theoretical exploration and toward practical implementation.

Scientists generated a comprehensive dataset evaluating the accuracy of ground-state energy calculations for the hydrogen molecule on IBM Quantum processors between August 2025 and April 2026. The research demonstrates that circuit simplification through tapered mappings consistently provided the greatest improvements in accuracy. While resilience level one offered accuracy gains, it incurred a substantial cost increase, and session-based execution did not demonstrably improve results over single-job execution despite increased billed time. The authors intend this resource to provide transparency and aid new users in understanding the trade-offs inherent in quantum computational chemistry.