Learning and Generating Mixed States Prepared by Shallow Channel Circuits

2026-04-01

AI Summary

This work addresses the problem of efficiently learning and generating mixed quantum states residing in the trivial phase on finite-dimensional lattices using only measurement data. Focusing on states prepared by shallow local quantum channel circuits, the paper proposes a learning algorithm that does not require access to the original preparation circuit. By leveraging trace distance approximation and a mechanism preserving local invertibility, the method enables efficient reconstruction of such states. The study establishes, for the first time, that mixed states in the trivial phase can be learned and generated solely from measurement data with sample complexity and runtime scaling polynomially (or quasi-polynomially) in the number of qubits. Furthermore, it lays the theoretical foundation for shallow-channel quantum generative models, whose classical limit yields an efficient classical diffusion model with polynomial overhead for both training and generation.

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.

Problem

• quantum state learning
• mixed states
• shallow channel circuits
• trivial phase
• quantum generative models

Innovation

• shallow channel circuits
• mixed state learning
• trivial phase
• quantum generative models
• sample complexity