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.

翻译：从测量数据中学习量子态是量子信息与计算复杂性的核心问题。本文研究有限维度晶格上混合态的学习生成问题。受近期混合态物质相研究进展的启发，我们聚焦于平凡相中的任意态。若存在一个浅层制备通道电路，使得整个制备过程中保持局域可逆性，则该态属于平凡相。我们证明此类混合态可通过测量访问高效学习。具体而言，给定未知平凡相混合态的副本，我们的算法输出一个近似生成该态的浅层局域通道电路（以迹距离衡量）。在假设恒定（或对数多项式）电路深度与门局域性的条件下，样本复杂度与运行时间与量子比特数呈多项式（或拟多项式）关系。关键的是，学习过程无需原始制备电路，仅依赖其存在性。该结果为基于浅层通道电路的量子生成模型提供了结构基础。在经典极限下，本框架亦启发了一种仅需多项式训练与生成开销的经典扩散模型高效算法。