Provably efficient machine learning for quantum many-body problems (2106.12627v4)

Abstract: Classical ML provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

• The paper establishes that classical ML algorithms can accurately predict quantum many-body ground states using classical shadows derived from randomized measurements.
• The paper introduces a novel ML framework that classifies quantum phases by learning discriminative features from few-body reduced density matrices.
• The paper offers rigorous theoretical guarantees with polynomial data and time complexity bounds that outperform traditional non-ML methods.

Provably Efficient Machine Learning for Quantum Many-body Problems: A Summary

The paper presents a rigorous exploration of classical ML techniques applied to quantum many-body problems, establishing both methodological advancements and theoretical guarantees. The authors prove that classical ML can efficiently predict ground state properties and classify quantum phases of matter for a family of gapped Hamiltonians in finite dimensions. This is achieved by training on data obtained through randomized measurements on states belonging to the same quantum phase. The work is significant as it lays the theoretical groundwork for leveraging classical ML to solve complex quantum problems in physics and chemistry, which are traditionally computationally intractable.

Key Contributions

1. Ground State Prediction: The paper establishes that classical ML algorithms can efficiently learn representations of quantum many-body ground states across a parameterized family of Hamiltonians with a constant spectral gap. Classical shadows, generated from randomized measurements, serve as compact classical descriptions enabling prediction. The authors present an algorithm that achieves a small average prediction error, supported by polynomial data and time complexity.
2. Phase Classification: The authors propose a classical ML framework for distinguishing between quantum phases of matter, including exotic topological phases. Utilizing shadow kernels, they demonstrate that ML models can classify phases accurately by learning from a few-body reduced density matrix representations of quantum states. This is contingent upon the existence of a nonlinear function based on reduced density matrices capable of discriminating phases.
3. Theoretical Guarantees: The classical ML algorithms' ability to outperform non-ML methods is rigorously justified. The paper proves a lower bound on the training data size needed for any classical algorithm to achieve similar predictive performance without leveraging data, highlighting where ML offers true computational advantages.

Numerical Experiments

The authors validate their theoretical claims through extensive numerical experiments. For instance, they demonstrate efficient prediction of ground state properties in quantum systems such as Rydberg atom chains and 2D Heisenberg models. They also show successful classification of phases in XXZ models, including symmetry-protected topological phases, using machine learning models trained on classical shadows.

Implications and Future Directions

The implications of this work are multifaceted. Practically, the methodology could be applied to quantum chemistry and materials science, potentially accelerating the discovery of novel compounds and materials. Theoretically, the paper provides a robust foundation for future studies exploring the application of advanced ML architectures in quantum many-body physics, as well as investigating the boundaries between quantum and classical computational capabilities. Future research directions might include extending the current framework to systems with long-range interactions, electronic Hamiltonians, and other quantum systems beyond those with a simple qubit structure.

Overall, this paper represents a substantial step towards integrating ML with traditional quantum mechanics, offering both new computational tools and foundational theory to support this interdisciplinary venture.