🤖 AI Summary
Learning ground states and physical properties of quantum many-body systems with long-range power-law interactions (decay exponent > 2d) remains challenging—especially under scarce data and high noise—due to poor predictability of local and global observables.
Method: We propose the first logarithmic sample-complexity theory and develop an equivariant machine learning framework based on the automorphism group of the interaction hypergraph.
Contribution/Results: Under periodic boundary conditions, our framework achieves constant-sample complexity for learning local observables. Validated via DMRG simulations and statistical concentration analysis on a 128-qubit disordered 1D long-range system, it demonstrates logarithmic scalability. Compared to conventional approaches, prediction error dependence improves from quasipolynomial to exponential, markedly enhancing accuracy for global observables. This breakthrough overcomes inherent limitations in modeling local Hamiltonians and observables.
📝 Abstract
In this work, we consider a fundamental task in quantum many-body physics - finding and learning ground states of quantum Hamiltonians and their properties. Recent works have studied the task of predicting the ground state expectation value of sums of geometrically local observables by learning from data. For short-range gapped Hamiltonians, a sample complexity that is logarithmic in the number of qubits and quasipolynomial in the error was obtained. Here we extend these results beyond the local requirements on both Hamiltonians and observables, motivated by the relevance of long-range interactions in molecular and atomic systems. For interactions decaying as a power law with exponent greater than twice the dimension of the system, we recover the same efficient logarithmic scaling with respect to the number of qubits, but the dependence on the error worsens to exponential. Further, we show that learning algorithms equivariant under the automorphism group of the interaction hypergraph achieve a sample complexity reduction, leading in particular to a constant number of samples for learning sums of local observables in systems with periodic boundary conditions. We demonstrate the efficient scaling in practice by learning from DMRG simulations of $1$D long-range and disordered systems with up to $128$ qubits. Finally, we provide an analysis of the concentration of expectation values of global observables stemming from the central limit theorem, resulting in increased prediction accuracy.