# Learning quantum states prepared by shallow circuits in polynomial time

arXiv: https://arxiv.org/abs/2410.23618 · HTML: https://mlsanity.com/paper/2410.23618/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 23 references that are in the map.
- First posted on arXiv 31 Oct 2024.

## Builds on (most cited first, 3 of 3)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Learning shallow quantum circuits](https://mlsanity.com/paper/2401.10095/) (2024)

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

- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Generative quantum advantage for classical and quantum problems](https://mlsanity.com/paper/2509.09033/) (2025)
- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [Local Diffusion Models and Phases of Data Distributions](https://mlsanity.com/paper/2508.06614/) (2025)
- [Learning and Generating Mixed States Prepared by Shallow Channel Circuits](https://mlsanity.com/paper/2604.01197/) (2026)
- [Learning to erase quantum states: thermodynamic implications of quantum learning theory](https://mlsanity.com/paper/2504.07341/) (2025)

## Abstract (arXiv)

We give a polynomial time algorithm that, given copies of an unknown quantum state vertpsirangle=Uvert 0nrangle that is prepared by an unknown constant depth circuit U on a finite-dimensional lattice, learns a constant depth quantum circuit that prepares vertpsirangle. The algorithm extends to the case when the depth of U is polylog(n), with a quasi-polynomial run-time. The key new idea is a simple and general procedure that efficiently reconstructs the global state vertpsirangle from its local reduced density matrices. As an application, we give an efficient algorithm to test whether an unknown quantum state on a lattice has low or high quantum circuit complexity.