Variational Benchmarks for Quantum Many-Body Problems

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.

• The paper defines the V-score, a novel metric that quantifies the accuracy of variational methods without requiring exact energy comparisons.
• It curates a comprehensive benchmark dataset spanning models like the Hubbard and Heisenberg systems using tensor networks, neural quantum states, and more.
• Empirical validation shows a linear relation between the V-score and energy error, identifying challenges for classical methods and opportunities for quantum computing.

Variational Benchmarks for Quantum Many-Body Problems: An Overview

The paper "Variational Benchmarks for Quantum Many-Body Problems" introduces an empirical metric called the V-score to assess the accuracy of variational methods used for solving quantum many-body ground-state problems. This work is driven by the need to establish a consistent benchmark that can effectively evaluate the progress in classical and quantum computational techniques targeting these problems.

Key Contributions

1. Definition of the V-score: The paper articulates a dimensionless metric, the V-score, which combines the mean energy and its variance obtained from variational states. The formulation of this V-score is based on ensuring it is an intensive metric, invariant under energy shifts, and scales independently with system size. The V-score is designed to give an absolute measure of the accuracy of a variational approach without requiring comparison to known exact energies.
2. Benchmark Dataset: A substantial dataset of variational calculations is curated, spanning various quantum many-body models and computational methods like tensor networks, neural quantum states, parameterized quantum circuits, and traditional variational Monte Carlo methods. This dataset covers a range of lattice models, including the Hubbard and Heisenberg models, and aids in identifying models challenging to current computational methods.
3. Empirical Validation: The paper validates the V-score by demonstrating its consistency with known numerical results, showing linear scaling between the V-score and the relative energy error across various Hamiltonians and variational strategies.
4. Application to Quantum Computation: The V-score allows for the assessment of quantum advantage in solving many-body problems. This metric identifies which problems pose significant challenges to classical methods, thus highlighting potential areas where quantum computing might have a practical impact.

Implications and Future Directions

• Benchmarking Quantum Algorithms: The V-score offers a pathway to provision effective benchmarks for gauging the success of quantum variational algorithms, particularly when classical verification isn't feasible due to computational limits.
• Identifying Difficult Problems: Lattice models with high V-scores indicate robust targets for quantum computational approaches that may outperform classical counterparts, especially in higher-dimensional frustrated spin systems and impurity models in multi-band scenarios.
• Progress in Computational Methods: The availability of the referenced dataset serves as a baseline to evaluate and improve computational techniques over time. Future advancements could refine the definition and application of the V-score, further aiding in distinguishing between quantum and classical computational boundaries.
• Improving Variational Techniques: The dataset and the V-score feedback loop suggest routes to refine existing algorithms and develop novel methodologies for energy estimation in many-body systems.

This work provides a solid framework for advancing quantum many-body problem-solving strategies, applying both classical and evolving quantum resources, and drives the community toward a standardized approach for benchmarking computational progress. The impact of this research is wide-ranging, serving as a catalyst for enhanced algorithm design, deeper understanding of many-body physics, and effective deployment of quantum computations.