# When can classical neural networks represent quantum states?

arXiv: https://arxiv.org/abs/2410.23152 · HTML: https://mlsanity.com/paper/2410.23152/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 77 references that are in the map.
- First posted on arXiv 30 Oct 2024.

## Builds on (most cited first, 16 of 16)

- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [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)
- [Representation Benefits of Deep Feedforward Networks](https://mlsanity.com/paper/1509.08101/) (2015)
- [A quantum-inspired classical algorithm for recommendation systems](https://mlsanity.com/paper/1807.04271/) (2018)
- [Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning](https://mlsanity.com/paper/1910.06151/) (2019)
- [NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems](https://mlsanity.com/paper/2112.10526/) (2021)
- [Neural tensor contractions and the expressive power of deep neural quantum states](https://mlsanity.com/paper/2103.10293/) (2021)
- [Neural network wave functions and the sign problem](https://mlsanity.com/paper/2002.04613/) (2020)
- [From Tensor Network Quantum States to Tensorial Recurrent Neural Networks](https://mlsanity.com/paper/2206.12363/) (2022)
- [(Pseudo) Random Quantum States with Binary Phase](https://mlsanity.com/paper/1906.10611/) (2019)
- [Certifying almost all quantum states with few single-qubit measurements](https://mlsanity.com/paper/2404.07281/) (2024)
- [Investigating Topological Order using Recurrent Neural Networks](https://mlsanity.com/paper/2303.11207/) (2023)
- [Empirical Sample Complexity of Neural Network Mixed State Reconstruction](https://mlsanity.com/paper/2307.01840/) (2023)
- [Positive bias makes tensor-network contraction tractable](https://mlsanity.com/paper/2410.05414/) (2024)

## Built on by (most cited first, 6 of 6)

- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [Probing Quantum Spin Systems with Kolmogorov-Arnold Neural Network Quantum States](https://mlsanity.com/paper/2506.01891/) (2025)
- [Adaptive Neural Quantum States: A Recurrent Neural Network Perspective](https://mlsanity.com/paper/2507.18700/) (2025)
- [Geometry-Induced Long-Range Correlations in Recurrent Neural Network Quantum States](https://mlsanity.com/paper/2604.08661/) (2026)
- [Mechanistic Interpretability and Causal Feature Steering of Neural Quantum States via Sparse Autoencoders](https://mlsanity.com/paper/2607.01336/) (2026)
- [Exploring the Effect of Basis Rotation on NQS Performance](https://mlsanity.com/paper/2512.17893/) (2025)

## Abstract (arXiv)

A naive classical representation of an n-qubit state requires specifying exponentially many amplitudes in the computational basis. Past works have demonstrated that classical neural networks can succinctly express these amplitudes for many physically relevant states, leading to computationally powerful representations known as neural quantum states. What underpins the efficacy of such representations? We show that conditional correlations present in the measurement distribution of quantum states control the performance of their neural representations. Such conditional correlations are basis dependent, arise due to measurement-induced entanglement, and reveal features not accessible through conventional few-body correlations often examined in studies of phases of matter. By combining theoretical and numerical analysis, we demonstrate how the state's entanglement and sign structure, along with the choice of measurement basis, give rise to distinct patterns of short- or long-range conditional correlations. Our findings provide a rigorous framework for exploring the expressive power of neural quantum states.