Learning and Generating Mixed States Prepared by Shallow Channel Circuits

TL;DR

This paper demonstrates that mixed quantum states in the trivial phase can be efficiently learned and generated using shallow local channel circuits, with implications for quantum and classical generative models.

Contribution

It introduces an efficient learning algorithm for trivial phase mixed states using measurement data, relying solely on the existence of shallow preparation channels.

Findings

States in the trivial phase can be learned from measurement data with polynomial sample complexity.

The learned shallow local channel circuit approximates the unknown state in trace distance.

The framework extends to classical diffusion models with polynomial overhead.

Abstract

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.