[Paper Review] Variational Benchmarks for Quantum Many-Body Problems

Dian Wu, Riccardo Rossi+31 more|arXiv (Cornell University)|Feb 9, 2023

Cold Atom Physics and Bose-Einstein CondensatesPhysics and Astronomy131 references19 citations

TL;DR

The paper introduces the V-score, a variational accuracy metric based on mean energy and energy variance, and provides the largest curated dataset of variational and numerically exact results for lattice quantum models to benchmark classical and quantum variational methods.

ABSTRACT

The continued development of computational approaches to many-body ground-state problems in physics and chemistry calls for a consistent way to assess its overall progress. In this work, we introduce a metric of variational accuracy, the V-score, obtained from the variational energy and its variance. We provide an extensive curated dataset of variational calculations of many-body quantum systems, identifying cases where state-of-the-art numerical approaches show limited accuracy, and future algorithms or computational platforms, such as quantum computing, could provide improved accuracy. The V-score can be used as a metric to assess the progress of quantum variational methods toward a quantum advantage for ground-state problems, especially in regimes where classical verifiability is impossible.

Motivation & Objective

• Define a consistent accuracy metric for variational ground-state methods across classical and quantum approaches.
• Compile the largest curated dataset of variational and exact results for strongly correlated lattice models.
• Identify which Hamiltonians and regimes are hardest for state-of-the-art variational methods.
• Assess potential quantum advantage using the V-score as an absolute hardness and progress metric.

Proposed method

• Define V-score as V = N Var(E) / (E - E_infty)^2, where E is mean energy, Var(E) its variance, N the degrees of freedom, and E_infty a zero point energy.
• Compute mean energy and variance for a variety of variational techniques including tensor networks, VMC, neural networks, and parameterized quantum circuits.
• Assemble and analyze a large dataset of lattice Hamiltonians (spin and fermion models, impurities) with ED, QMC, TN, VMC, and PQC methods.
• Use ED or numerically exact QMC results as reference when available to validate the V-score as an estimator of energy relative error.
• Present visual rankings (V-scores) of Hamiltonians to delineate easy vs. hard problems for current variational methods.

Experimental results

Research questions

• RQ1 Can a single, dimensionless metric (the V-score) reliably indicate the proximity of variational results to ground-state energies across diverse lattice models and methods?
• RQ2 Which lattice Hamiltonians and geometries (e.g., frustrated vs unfrustrated, 1D vs higher dimensions) are hardest for current variational approaches?
• RQ3 How well does the V-score correlate with actual energy relative errors across classical and quantum variational methods?
• RQ4 Can the V-score guide targeting of quantum algorithms to regimes where classical methods struggle?

Key findings

• The V-score correlates linearly with the logarithm of energy relative error across a wide set of Hamiltonians and variational methods.
• Frustrated geometries (e.g., kagome, pyrochlore) and Hubbard-like fermionic models exhibit higher V-scores, indicating harder ground-state approximations.
• 1D and unfrustrated systems generally show small V-scores, implying easier variational approximation with existing methods.
• PQCs on classical hardware show V-scores that align with classical results, suggesting the metric's applicability to quantum variational approaches.
• The dataset highlights Hamiltonians and regimes where state-of-the-art methods underperform, flagging them as targets for quantum algorithms or improved classical techniques.
• The V-score provides an absolute hardness measure suitable for benchmarking progress toward quantum advantage in ground-state problems.