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arXiv:2604.01197

Learning and Generating Mixed States Prepared by Shallow Channel Circuits

Fangjun Hu, Christian Kokail, Milan Kornjača, Pedro L. S. Lopes, Weiyuan Gong, Sheng-Tao Wang, Xun Gao, Stefan Ostermann

quant-phcond-mat.stat-mechcs.CCcs.LGSubmitted Apr 1, 2026• Announced Apr 2, 2026• 0 citations• Revised Apr 16, 2026 (v2)

v1 · Apr 1, 2026 v2 · Apr 16, 2026 v3 · May 13, 2026 v4 · Jun 15, 2026

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.

Comments: 44 pages, 14 figures, 1 table. Updated references to include additional relevant works, applications, and discussions

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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.

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Abstract. Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown [Formula: see text]-qubit shallow quantum circuit [Formula: see text] (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of [Formula: see text]. We also provide a polynomial-time classical algorithm for learning the description of any unknown [Formula: see text]-qubit state [Formula: see text] prepared by a shallow quantum circuit [Formula: see text] (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of [Formula: see text]. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.

We give a polynomial time algorithm that, given copies of an unknown quantum state

\vertψ\rangle=U\vert 0^n\rangle

that is prepared by an unknown constant depth circuit

U

on a finite-dimensional lattice, learns a constant depth quantum circuit that prepares

\vertψ\rangle

. The algorithm extends to the case when the depth of

U

is

\mathrm{polylog}(n)

, with a quasi-polynomial run-time. The key new idea is a simple and general procedure that efficiently reconstructs the global state

\vertψ\rangle

from its local reduced density matrices. As an application, we give an efficient algorithm to test whether an unknown quantum state on a lattice has low or high quantum circuit complexity.

We present the first computationally-efficient algorithm for average-case learning of shallow quantum circuits with many-qubit gates. Specifically, we provide a quasi-polynomial time and sample complexity algorithm for learning unknown QAC

^0

circuits -- constant-depth circuits with arbitrary single-qubit gates and polynomially many

CZ

gates of unbounded width -- with at most logarithmic ancilla, up to inverse-polynomially small error. Furthermore, we show that the learned unitary can be efficiently synthesized in poly-logarithmic depth. This work expands the family of efficiently learnable quantum circuits, notably since in finite-dimensional circuit geometries, QAC

^0

circuits require polynomial depth to implement.

We construct a family of 2D-local constant-depth quantum circuits that output states whose entanglement entropy across a specified cut cannot be estimated in quantum polynomial time. As constant-depth quantum circuits can be learned from polynomially many quantum samples, our resulting pseudoentangled states are implicitly public-key and not pseudorandom. This separates pseudoentanglement from pseudorandomness in the shallow-circuit regime: the former is possible, while the latter is not. The construction is based on the quantum intractability of the Dense-Sparse Learning Parity with Noise problem introduced in [DJ25] and uses a bounded-fan-in, bounded-fan-out classical randomized encoding for linear maps

\mathbf{x} \mapsto \mathbf{Mx},

which could be of independent interest. As applications, we obtain quantum hardness for the problem of learning the entanglement structure (across a fixed cut) of the ground-state of 1D and 2D local Hamiltonians. The 1D Hamiltonian has an inverse polynomial gap, whereas the 2D one has a constant gap. This complements the result of [BZZ24] that showed only factoring-based hardness for the 1D case, though achieving a volume versus area entanglement difference.

Preparing complex quantum states is of limited value without efficient methods for extracting useful classical information from them. This thesis studies when such information can be extracted efficiently, given copies of an unknown quantum state or query access to an unknown unitary transformation. We begin with pure state tomography. We present an algorithm that uses only nonadaptive single-qubit Pauli measurements yet achieves optimal copy and time complexities up to logarithmic factors. This is the first full state tomography algorithm with near-optimal running time. We then show how additional structure can make tomography time-efficient, giving algorithms for two structured classes of quantum states: non-interacting fermions and near-stabilizer states. We also introduce a new learning task, agnostic tomography, where the goal is to find the best approximation to an arbitrary quantum state from a structured class of states, and give efficient algorithms in this model. Turning to property testing, we give an efficient tester for stabilizer dimension, which implies that Ω(n) T-gates are necessary for any Clifford+T circuit to prepare n-qubit computationally pseudorandom states. We also prove copy-complexity lower bounds for testing low-degree phase states, showing that testing is nearly as hard as learning these states. More strongly, for every fixed degree d, ensembles of n-qubit degree-d phase states form exponentially accurate approximate state t-designs for t=Ω(n^{⌊(d-1)/2⌋}). Finally, we extend these questions from states to unitary transformations. We develop a general framework for learning unitary transformations by exploiting low-complexity Pauli spectra. This framework unifies and extends prior learning algorithms for Clifford and near-Clifford circuits, shallow circuits, quantum juntas, and fermionic matchgate circuits.