# Learning topological states from randomized measurements using variational tensor network tomography

arXiv: https://arxiv.org/abs/2406.00193 · HTML: https://mlsanity.com/paper/2406.00193/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 100 references that are in the map.
- First posted on arXiv 31 May 2024.

## Builds on (most cited first, 10 of 10)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning](https://mlsanity.com/paper/1907.03741/) (2019)
- [A Hierarchy for Replica Quantum Advantage](https://mlsanity.com/paper/2111.05874/) (2021)
- [Scalably learning quantum many-body Hamiltonians from dynamical data](https://mlsanity.com/paper/2209.14328/) (2022)
- [Certifying almost all quantum states with few single-qubit measurements](https://mlsanity.com/paper/2404.07281/) (2024)
- [Unifying (Quantum) Statistical and Parametrized (Quantum) Algorithms](https://mlsanity.com/paper/2310.17716/) (2023)
- [Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows](https://mlsanity.com/paper/2402.17911/) (2024)
- [Hardware-efficient learning of quantum many-body states](https://mlsanity.com/paper/2212.06084/) (2022)
- [Approximately-symmetric neural networks for quantum spin liquids](https://mlsanity.com/paper/2405.17541/) (2024)

## Built on by (most cited first, 4 of 4)

- [No-Free-Lunch Theories for Tensor-Network Machine Learning Models](https://mlsanity.com/paper/2412.05674/) (2024)
- [Network Tomography with Path-Centric Graph Neural Network](https://mlsanity.com/paper/2502.16430/) (2025)
- [Sketch Tomography: Hybridizing Classical Shadow and Matrix Product State](https://mlsanity.com/paper/2512.03333/) (2025)
- [Gaussian Universality and Its Breakdown in Tensor-Network Machine Learning](https://mlsanity.com/paper/2610.06080/) (2026)

## Abstract (arXiv)

Learning faithful representations of quantum states is crucial to fully characterizing the variety of many-body states created on quantum processors. While various tomographic methods such as classical shadow and MPS tomography have shown promise in characterizing a wide class of quantum states, they face unique limitations in detecting topologically ordered two-dimensional states. To address this problem, we implement and study a heuristic tomographic method that combines variational optimization on tensor networks with randomized measurement techniques. Using this approach, we demonstrate its ability to learn the ground state of the surface code Hamiltonian as well as an experimentally realizable quantum spin liquid state. In particular, we perform numerical experiments using MPS ans"atze and systematically investigate the sample complexity required to achieve high fidelities for systems of sizes up to 48 qubits. In addition, we provide theoretical insights into the scaling of our learning algorithm by analyzing the statistical properties of maximum likelihood estimation. Notably, our method is sample-efficient and experimentally friendly, only requiring snapshots of the quantum state measured randomly in the X or Z bases. Using this subset of measurements, our approach can effectively learn any real pure states represented by tensor networks, and we rigorously prove that random-XZ measurements are tomographically complete for such states.