# Efficient Learning of Long-Range and Equivariant Quantum Systems

arXiv: https://arxiv.org/abs/2312.17019 · HTML: https://mlsanity.com/paper/2312.17019/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 30 references that are in the map.
- First posted on arXiv 28 Dec 2023.

## Builds on (most cited first, 7 of 7)

- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Machine Learning from a Continuous Viewpoint](https://mlsanity.com/paper/1912.12777/) (2019)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Improved machine learning algorithm for predicting ground state properties](https://mlsanity.com/paper/2301.13169/) (2023)

## Built on by (most cited first, 3 of 3)

- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [Accurate Learning of Equivariant Quantum Systems from a Single Ground State](https://mlsanity.com/paper/2405.12309/) (2024)
- [Quantum circuit complexity and unsupervised machine learning of topological order](https://mlsanity.com/paper/2508.04486/) (2025)

## Abstract (arXiv)

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 1D 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.