INTRODUCTION Solving quantum many-body problems, such as finding ground states of quantum systems, has far-reaching consequences for physics, materials science, and chemistry. Classical computers have facilitated many profound advances in science and technology, but they often struggle to solve such problems. Scalable, fault-tolerant quantum computers will be able to solve a broad array of quantum problems but are unlikely to be available for years to come. Meanwhile, how can we best exploit our powerful classical computers to advance our understanding of complex quantum systems? Recently, classical machine learning (ML) techniques have been adapted to investigate problems in quantum many-body physics. So far, these approaches are mostly heuristic, reflecting the general paucity of rigorous theory in ML. Although they have been shown to be effective in some intermediate-size experiments, these methods are generally not backed by convincing theoretical arguments to ensure good performance. RATIONALE A central question is whether classical ML algorithms can provably outperform non-ML algorithms in challenging quantum many-body problems. We provide a concrete answer by devising and analyzing classical ML algorithms for predicting the properties of ground states of quantum systems. We prove that these ML algorithms can efficiently and accurately predict ground-state properties of gapped local Hamiltonians, after learning from data obtained by measuring other ground states in the same quantum phase of matter. Furthermore, under a widely accepted complexity-theoretic conjecture, we prove that no efficient classical algorithm that does not learn from data can achieve the same prediction guarantee. By generalizing from experimental data, ML algorithms can solve quantum many-body problems that could not be solved efficiently without access to experimental data. RESULTS We consider a family of gapped local quantum Hamiltonians, where the Hamiltonian H ( x ) depends smoothly on m parameters (denoted by x ). The ML algorithm learns from a set of training data consisting of sampled values of x , each accompanied by a classical representation of the ground state of H ( x ). These training data could be obtained from either classical simulations or quantum experiments. During the prediction phase, the ML algorithm predicts a classical representation of ground states for Hamiltonians different from those in the training data; ground-state properties can then be estimated using the predicted classical representation. Specifically, our classical ML algorithm predicts expectation values of products of local observables in the ground state, with a small error when averaged over the value of x . The run time of the algorithm and the amount of training data required both scale polynomially in m and linearly in the size of the quantum system. Our proof of this result builds on recent developments in quantum information theory, computational learning theory, and condensed matter theory. Furthermore, under the widely accepted conjecture that nondeterministic polynomial-time (NP)–complete problems cannot be solved in randomized polynomial time, we prove that no polynomial-time classical algorithm that does not learn from data can match the prediction performance achieved by the ML algorithm. In a related contribution using similar proof techniques, we show that classical ML algorithms can efficiently learn how to classify quantum phases of matter. In this scenario, the training data consist of classical representations of quantum states, where each state carries a label indicating whether it belongs to phase A or phase B . The ML algorithm then predicts the phase label for quantum states that were not encountered during training. The classical ML algorithm not only classifies phases accurately, but also constructs an explicit classifying function. Numerical experiments verify that our proposed ML algorithms work well in a variety of scenarios, including Rydberg atom systems, two-dimensional random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases. CONCLUSION We have rigorously established that classical ML algorithms, informed by data collected in physical experiments, can effectively address some quantum many-body problems. These rigorous results boost our hopes that classical ML trained on experimental data can solve practical problems in chemistry and materials science that would be too hard to solve using classical processing alone. Our arguments build on the concept of a succinct classical representation of quantum states derived from randomized Pauli measurements. Although some quantum devices lack the local control needed to perform such measurements, we expect that other classical representations could be exploited by classical ML with similarly powerful results. How can we make use of accessible measurement data to predict properties reliably? Answering such questions will expand the reach of near-term quantum platforms. Classical algorithms for quantum many-body problems. Classical ML algorithms learn from training data, obtained from either classical simulations or quantum experiments. Then, the ML algorithm produces a classical representation for the ground state of a physical system that was not encountered during training. Classical algorithms that do not learn from data may require substantially longer computation time to achieve the same task.
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This content will become publicly available on May 31, 2024
Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems
Physics-informed machine learning (PIML) is a set of methods and tools that systematically integrate machine learning (ML) algorithms with physical constraints and abstract mathematical models developed in scientific and engineering domains. As opposed to purely data-driven methods, PIML models can be trained from additional information obtained by enforcing physical laws such as energy and mass conservation. More broadly, PIML models can include abstract properties and conditions such as stability, convexity, or invariance. The basic premise of PIML is that the integration of ML and physics can yield more effective, physically consistent, and data-efficient models. This paper aims to provide a tutorial-like overview of the recent advances in PIML for dynamical system modeling and control. Specifically, the paper covers an overview of the theory, fundamental concepts and methods, tools, and applications on topics of: 1) physics-informed learning for system identification; 2) physics-informed learning for control; 3) analysis and verification of PIML models; and 4) physics-informed digital twins. The paper is concluded with a perspective on open challenges and future research opportunities.
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- NSF-PAR ID:
- 10491565
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- Proceedings of the American Control Conference
- ISSN:
- 2378-5861
- Page Range / eLocation ID:
- 3735 to 3750
- Format(s):
- Medium: X
- Location:
- San Diego, CA, USA
- Sponsoring Org:
- National Science Foundation
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