# Exponentially improved efficient machine learning for quantum many-body states with provable guarantees

arXiv: https://arxiv.org/abs/2304.04353 · HTML: https://mlsanity.com/paper/2304.04353/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 135 references that are in the map.
- First posted on arXiv 10 Apr 2023.

## Builds on (most cited first, 30 of 31)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [A high-bias, low-variance introduction to Machine Learning for physicists](https://mlsanity.com/paper/1803.08823/) (2018)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Deep autoregressive models for the efficient variational simulation of many-body quantum systems](https://mlsanity.com/paper/1902.04057/) (2019)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders](https://mlsanity.com/paper/1703.02435/) (2017)
- [Quantum State Tomography with Conditional Generative Adversarial Networks](https://mlsanity.com/paper/2008.03240/) (2020)
- [Equivalence of restricted Boltzmann machines and tensor network states](https://mlsanity.com/paper/1701.04831/) (2017)
- [Improved machine learning algorithm for predicting ground state properties](https://mlsanity.com/paper/2301.13169/) (2023)
- [Experimental kernel-based quantum machine learning in finite feature space](https://mlsanity.com/paper/1906.04137/) (2019)
- [Efficient estimation of Pauli observables by derandomization](https://mlsanity.com/paper/2103.07510/) (2021)
- [Machine Learning Topological Invariants with Neural Networks](https://mlsanity.com/paper/1708.09401/) (2017)
- [Experimental neural network enhanced quantum tomography](https://mlsanity.com/paper/1904.05902/) (2019)
- [Classification and reconstruction of optical quantum states with deep neural networks](https://mlsanity.com/paper/2012.02185/) (2020)
- [Geometry of learning neural quantum states](https://mlsanity.com/paper/1910.11163/) (2019)
- [Machine learning assisted quantum state estimation](https://mlsanity.com/paper/2003.03441/) (2020)
- [A Unifying Framework in Vector-valued Reproducing Kernel Hilbert Spaces for Manifold Regularization and Co-Regularized Multi-view Learning](https://mlsanity.com/paper/1401.8066/) (2014)
- [Deep Learning Topological Invariants of Band Insulators](https://mlsanity.com/paper/1805.10503/) (2018)
- [Learning Temporal Quantum Tomography](https://mlsanity.com/paper/2103.13973/) (2021)
- [Inferring the quantum density matrix with machine learning](https://mlsanity.com/paper/1904.05903/) (2019)
- [Unveiling phase transitions with machine learning](https://mlsanity.com/paper/1904.01486/) (2019)
- [Reinforcement Learning for Many-Body Ground-State Preparation Inspired by Counterdiabatic Driving](https://mlsanity.com/paper/2010.03655/) (2020)
- [Bounding Multivariate Trigonometric Polynomials with Applications to Filter Bank Design](https://mlsanity.com/paper/1802.09588/) (2018)
- [Unsupervised machine learning of quantum phase transitions using diffusion maps](https://mlsanity.com/paper/2003.07399/) (2020)
- [Learning Disordered Topological Phases by Statistical Recovery of Symmetry](https://mlsanity.com/paper/1709.05790/) (2017)
- [Boltzmann Generators -- Sampling Equilibrium States of Many-Body Systems with Deep Learning](https://mlsanity.com/paper/1812.01729/) (2018)
- [Probing Criticality in Quantum Spin Chains with Neural Networks](https://mlsanity.com/paper/2005.02104/) (2020)

## Built on by (most cited first, 7 of 7)

- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithms](https://mlsanity.com/paper/2405.18489/) (2024)
- [Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits](https://mlsanity.com/paper/2408.12199/) (2024)
- [Concept learning of parameterized quantum models from limited measurements](https://mlsanity.com/paper/2408.05116/) (2024)
- [Sequence-Model-Guided Measurement Selection for Quantum State Learning](https://mlsanity.com/paper/2507.09891/) (2025)
- [Quantum circuit complexity and unsupervised machine learning of topological order](https://mlsanity.com/paper/2508.04486/) (2025)
- [Provably scalable learning from quantum many-body data via locality](https://mlsanity.com/paper/2509.13705/) (2025)

## Abstract (arXiv)

Solving the ground state and the ground-state properties of quantum many-body systems is generically a hard task for classical algorithms. For a family of Hamiltonians defined on an m-dimensional space of physical parameters, the ground state and its properties at an arbitrary parameter configuration can be predicted via a machine learning protocol up to a prescribed prediction error varepsilon, provided that a sample set (of size N) of the states can be efficiently prepared and measured. In a recent work [Huang et al., Science 377, eabk3333 (2022)], a rigorous guarantee for such a generalization was proved. Unfortunately, an exponential scaling for the provable sample complexity, N=m^calOleft(frac1varepsilonright), was found to be universal for generic gapped Hamiltonians. This result applies to the situation where the dimension of the parameter space is large while the scaling with the accuracy is not an urgent factor. In this work, we consider an alternative scenario where m is a finite, not necessarily large constant while the scaling with the prediction error becomes the central concern. By jointly preserving the fundamental properties of density matrices in the learning protocol and utilizing the continuity of quantum states in the parameter range of interest, we rigorously obtain a polynomial sample complexity for predicting quantum many-body states and their properties, with respect to the uniform prediction error varepsilon and the number of qubits n. Moreover, if restricted to learning local quantum-state properties, the number of samples with respect to n can be further reduced exponentially. Our results provide theoretical guarantees for efficient learning of quantum many-body states and their properties, with model-independent applications not restricted to ground states of gapped Hamiltonians.