Learning and Generating Mixed States Prepared by Shallow Channel Circuits
2026-04-01 • Computational Complexity
Computational ComplexityMachine Learning
AI summaryⓘ
The authors study how to learn and recreate certain complex quantum states called mixed states that can be made by simple, shallow circuits. They focus on a special group called trivial phase states, which can be prepared without complicated changes in small parts. Their main result is an efficient method that learns how to make these states just from measurements, without knowing the original way they were made. This method runs fairly quickly and needs a manageable amount of data when the circuits are shallow and local. They also show that their approach can inspire efficient algorithms for classical systems that use similar ideas.
quantum statesmixed statestrivial phaseshallow circuitslocal reversibilitytrace distancesample complexityquantum generative modelsquantum channelsdiffusion models
Authors
Fangjun Hu, Christian Kokail, Milan Kornjača, Pedro L. S. Lopes, Weiyuan Gong, Sheng-Tao Wang, Xun Gao, Stefan Ostermann
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.