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。本文研究有限维格点上混合态的生成学习问题。受混合态物相研究进展启发，我们聚焦平凡相中的任意态：若存在浅层制备通道电路使得整个制备过程保持局域可逆性，则该态属于平凡相。我们证明该类任意混合态均可仅通过测量访问高效学习。具体而言，给定未知平凡相混合态副本，本算法输出一个在迹距离上近似生成该态的浅层局域通道电路。在电路深度与门局域性分别保持常数（或多对数级）的前提下，样本复杂度和运行时间随量子比特数呈多项式（或拟多项式）增长。值得注意的是，学习器无需知晓原始制备电路，仅需其存在性即可。本研究为基于浅层通道电路的量子生成模型提供了结构性基础。在经典极限下，本框架还启发出仅需训练与生成多项式开销的经典扩散模型高效算法。