# Neural tensor contractions and the expressive power of deep neural quantum states

arXiv: https://arxiv.org/abs/2103.10293 · HTML: https://mlsanity.com/paper/2103.10293/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 59 references that are in the map.
- First posted on arXiv 18 Mar 2021.

## Builds on (most cited first, 8 of 8)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Deep autoregressive models for the efficient variational simulation of many-body quantum systems](https://mlsanity.com/paper/1902.04057/) (2019)
- [Quantum Entanglement in Deep Learning Architectures](https://mlsanity.com/paper/1803.09780/) (2018)
- [Neural networks and rational functions](https://mlsanity.com/paper/1706.03301/) (2017)
- [Neural-Network Quantum States, String-Bond States, and Chiral Topological States](https://mlsanity.com/paper/1710.04045/) (2017)
- [Equivalence of restricted Boltzmann machines and tensor network states](https://mlsanity.com/paper/1701.04831/) (2017)

## Built on by (most cited first, 21 of 21)

- [Learning to Robustly Aggregate Labeling Functions for Semi-supervised Data Programming](https://mlsanity.com/paper/2109.11410/) (2021)
- [Gauge Equivariant Neural Networks for 2+1D U(1) Gauge Theory Simulations in Hamiltonian Formulation](https://mlsanity.com/paper/2211.03198/) (2022)
- [Pairing-based graph neural network for simulating quantum materials](https://mlsanity.com/paper/2311.02143/) (2023)
- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [ANTN: Bridging Autoregressive Neural Networks and Tensor Networks for Quantum Many-Body Simulation](https://mlsanity.com/paper/2304.01996/) (2023)
- [NNQS-Transformer: an Efficient and Scalable Neural Network Quantum States Approach for Ab initio Quantum Chemistry](https://mlsanity.com/paper/2306.16705/) (2023)
- [Certifying almost all quantum states with few single-qubit measurements](https://mlsanity.com/paper/2404.07281/) (2024)
- [When can classical neural networks represent quantum states?](https://mlsanity.com/paper/2410.23152/) (2024)
- [Quantum computing and artificial intelligence: status and perspectives](https://mlsanity.com/paper/2505.23860/) (2025)
- [Approximately-symmetric neural networks for quantum spin liquids](https://mlsanity.com/paper/2405.17541/) (2024)
- [Infinite Neural Network Quantum States: Entanglement and Training Dynamics](https://mlsanity.com/paper/2112.00723/) (2021)
- [A Simple Structure For Building A Robust Model](https://mlsanity.com/paper/2204.11596/) (2022)
- [Towards Neural Variational Monte Carlo That Scales Linearly with System Size](https://mlsanity.com/paper/2212.11296/) (2022)
- [Artificial intelligence for artificial materials: moir'e atom](https://mlsanity.com/paper/2303.08162/) (2023)
- [An Empirical Study of Quantum Dynamics as a Ground State Problem with Neural Quantum States](https://mlsanity.com/paper/2206.09241/) (2022)
- [Neural-network quantum state study of the long-range antiferromagnetic Ising chain](https://mlsanity.com/paper/2308.09709/) (2023)
- [Simulating Non-Markovian Open Quantum Dynamics with Neural Quantum States](https://mlsanity.com/paper/2404.11093/) (2024)
- [A Theoretical Framework for an Efficient Normalizing Flow-Based Solution to the Electronic Schrodinger Equation](https://mlsanity.com/paper/2406.00047/) (2024)
- [Seeding neural network quantum states with tensor network states](https://mlsanity.com/paper/2506.23550/) (2025)
- [Neural-Quantum-States Impurity Solver for Quantum Embedding Problems](https://mlsanity.com/paper/2509.12431/) (2025)
- [Expressibility of neural quantum states: a Walsh-complexity perspective](https://mlsanity.com/paper/2604.03294/) (2026)

## Abstract (arXiv)

We establish a direct connection between general tensor networks and deep feed-forward artificial neural networks. The core of our results is the construction of neural-network layers that efficiently perform tensor contractions, and that use commonly adopted non-linear activation functions. The resulting deep networks feature a number of edges that closely matches the contraction complexity of the tensor networks to be approximated. In the context of many-body quantum states, this result establishes that neural-network states have strictly the same or higher expressive power than practically usable variational tensor networks. As an example, we show that all matrix product states can be efficiently written as neural-network states with a number of edges polynomial in the bond dimension and depth logarithmic in the system size. The opposite instead does not hold true, and our results imply that there exist quantum states that are not efficiently expressible in terms of matrix product states or PEPS, but that are instead efficiently expressible with neural network states.