# Learning and Generating Mixed States Prepared by Shallow Channel Circuits

arXiv: https://arxiv.org/abs/2604.01197 · HTML: https://mlsanity.com/paper/2604.01197/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 65 references that are in the map.
- First posted on arXiv 1 Apr 2026.

## Builds on (most cited first, 13 of 13)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Learning shallow quantum circuits](https://mlsanity.com/paper/2401.10095/) (2024)
- [Generative quantum advantage for classical and quantum problems](https://mlsanity.com/paper/2509.09033/) (2025)
- [On the average-case complexity of learning output distributions of quantum circuits](https://mlsanity.com/paper/2305.05765/) (2023)
- [Local Diffusion Models and Phases of Data Distributions](https://mlsanity.com/paper/2508.06614/) (2025)
- [Learning quantum states prepared by shallow circuits in polynomial time](https://mlsanity.com/paper/2410.23618/) (2024)
- [Information-Computation Gaps in Quantum Learning via Low-Degree Likelihood](https://mlsanity.com/paper/2505.22743/) (2025)
- [Quantum Gibbs states are locally Markovian](https://mlsanity.com/paper/2504.02208/) (2025)
- [Unlearnable phases of matter](https://mlsanity.com/paper/2602.11262/) (2026)
- [The Hardness of Learning Quantum Circuits and its Cryptographic Applications](https://mlsanity.com/paper/2504.15343/) (2025)

## Built on by (most cited first, 2 of 2)

- [Concurrence of Symmetry Breaking and Nonlocality Phase Transitions in Diffusion Models](https://mlsanity.com/paper/2605.04830/) (2026)
- [Renormalization Group Flow Matching for Scalable Local Generative Modeling](https://mlsanity.com/paper/2608.23696/) (2026)

## Abstract (arXiv)

Learning quantum states from measurement data is a central problem in quantum information and computational complexity. In this work, we study the problem of learning to generate mixed states on a finite-dimensional lattice. Motivated by recent developments in mixed state phases of matter, we focus on arbitrary states in the trivial phase. A state belongs to the trivial phase if there exists a shallow preparation channel circuit under which local reversibility is preserved throughout the preparation. We prove that any mixed state in this class can be efficiently learned from measurement access alone. Specifically, given copies of an unknown trivial phase mixed state, our algorithm outputs a shallow local channel circuit that approximately generates this state in trace distance. The sample complexity and runtime are polynomial (or quasi-polynomial) in the number of qubits, assuming constant (or polylogarithmic) circuit depth and gate locality. Importantly, the learner is not given the original preparation circuit and relies only on its existence. Our results provide a structural foundation for quantum generative models based on shallow channel circuits. In the classical limit, our framework also inspires an efficient algorithm for classical diffusion models using only a polynomial overhead of training and generation.