Quantum Algorithms, from VQE to SQD, Estimate Ground State Energy by Sampling Determinants

Estimating the ground state energy of molecules is a fundamental challenge in quantum chemistry, and researchers continually seek more efficient computational methods. Abdelmouheymen Rabah Khamadja and Mohamed Taha Rouabah, from University Frères Mentouri, Constantine 1, are pioneering a new approach to this problem, moving beyond the limitations of current variational algorithms. Their work investigates sampling-based methods, specifically Selected Configuration Interaction and Sample-Based Diagonalization, which sidestep complex optimisation procedures by directly sampling quantum states and performing calculations classically. Crucially, they have derived the first analytical formula to predict the number of measurements needed for these methods to succeed, a breakthrough that allows for accurate prediction of performance and optimisation of experimental design, and they validate this analysis through simulations and experiments on a large quantum processor. This achievement significantly advances the feasibility of solving complex molecular problems on near-term quantum computers.

First and foremost, the author expresses deepest gratitude to Dr Rouabah for the tremendous effort he invested over the past two years. His unwavering support and guidance proved invaluable throughout the research process. The author also extends sincere thanks to sister, extended family, and dearest friends L. S. and G. Z. These methods overcome limitations of the Variational Quantum Eigensolver by eliminating the need for variational optimization, instead relying on a quantum computer to sample relevant quantum states while performing Hamiltonian diagonalization classically. The study meticulously investigates the sampling process inherent in these algorithms, establishing a clear connection to the classical coupon-collector problem. This innovative mapping yields both an exact formula and a scalable lower-bound estimator for calculating the number of measurements required to identify all determinants contributing to the ground state.

Researchers began by preparing the ground state using standard quantum computational chemistry techniques, then implemented both QSCI and SQD. In QSCI, the quantum computer samples Slater determinants, which are then used in a classical diagonalization procedure to determine the ground state energy. SQD follows a similar principle, employing quantum sampling followed by classical post-processing of the results. To rigorously validate the theoretical analysis, the team conducted a comprehensive suite of simulations, starting with ideal state-vector simulations to establish baseline performance. The simulations were then extended to incorporate realistic noise models, mimicking the imperfections of actual quantum hardware.

Crucially, the study culminated in a real-device run on IBM’s 127-qubit Brisbane processor, demonstrating the practical operation of QSCI and SQD in a progressively realistic noise environment. This experimental execution involved careful calibration of the quantum hardware and meticulous data acquisition to assess the algorithms’ performance under real-world conditions. The results demonstrate the feasibility and potential of these sampling-based methods for near-term quantum computation, highlighting the critical role of sampling efficiency in achieving practical quantum advantage.

Ground State Energy via Quantum Sampling

This work presents a breakthrough in determining the ground state energy of molecules using quantum computation, focusing on algorithms that offer alternatives to the Variational Eigensolver. A central achievement is the derivation of an analytical formula for the sampling bottleneck, specifically the determinant-discovery step, mapping it to the classical coupon-collector problem. This provides both an exact formula and a scalable lower-bound estimator for calculating the number of measurements needed to accurately represent the ground state.

The team validated this analysis through simulations, noise-model studies, and crucially, execution on a 127-qubit IBM Brisbane quantum processor. Results demonstrate that sampling efficiency is the dominant factor determining the feasibility of QSCI and SQD for near-term quantum devices. For the H2O molecule, the study reports ground-state energies obtained using different estimators and varying the number of measurements. In ideal conditions, energies within chemical accuracy, defined as less than 1. 6 milli-Hartree, were achieved, and the team demonstrated the ability to accurately compute energies even with limited computational resources.

Further experiments, incorporating a simulated noise model of the IBM Brisbane processor, continued to yield results within chemical accuracy. Importantly, the team also successfully executed the algorithms on the actual IBM Brisbane quantum computer, achieving comparable accuracy with varying the number of measurements. Specifically, the study details ground-state energies obtained with different estimators and varying the number of measurements on the real quantum computer, confirming the practical viability of these sampling-based methods. The team established an analytical expression for the sampling bottleneck in these algorithms, specifically the determinant-discovery step, demonstrating its equivalence to the classical coupon-collector problem. This analytical solution provides both an exact formula and a scalable lower-bound estimator for determining the number of measurements needed to accurately represent the ground state. The findings demonstrate that sampling efficiency is the dominant factor influencing the feasibility of these algorithms on near-term quantum computers. Validating the analysis through simulations, noise-model studies, and execution on a 127-qubit processor, the researchers successfully computed ground-state energies for the water molecule with varying levels of accuracy depending on the number of measurements taken.

👉 More information
🗞 From VQE To SQD: Modern Quantum Algorithms For The Electronic Structure Problem
🧠 ArXiv: https://arxiv.org/abs/2509.21555

Rohail T.

Rohail T.

As a quantum scientist exploring the frontiers of physics and technology. My work focuses on uncovering how quantum mechanics, computing, and emerging technologies are transforming our understanding of reality. I share research-driven insights that make complex ideas in quantum science clear, engaging, and relevant to the modern world.

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