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SIAM Journal on Computing

Learning Shallow Quantum Circuits

Hsin-Yuan Huang, Yunchao Liu, Michael Broughton, Isaac Kim, Anurag Anshu, Zeph A. Landau, Jarrod R. McClean

Published Oct 16, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1137/24m1689946

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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.
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.
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 n-qubit shallow quantum circuit U (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of U. We also provide a polynomial-time classical algorithm for learning the description of any unknown n-qubit state | ψ ⟩ = U | 0n ⟩ prepared by a shallow quantum circuit U (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of | ψ ⟩. 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 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 QAC0^00 circuits -- constant-depth circuits with arbitrary single-qubit gates and polynomially many CZCZCZ 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, QAC0^00 circuits require polynomial depth to implement.
We give a sampling problem that is solvable by shallow quantum circuits, hard for polynomial-time classical algorithms under lattice-based assumptions, and efficiently verifiable by a classical computer. The quantum sampler admits two implementations: one uses log-logarithmic-depth quantum circuits with one- and two-qubit gates, i.e., QNC0[log⁡log⁡]\mathsf{QNC}^0[\log\log]QNC0[loglog] circuits, while the other uses constant-depth quantum circuits with unbounded fan-in gates, i.e., QAC0\mathsf{QAC}^0QAC0 circuits. Our construction can be seen as compiling the Learning with Errors (LWE)-based single-round proof of quantumness of Arabadjieva et al. (2025) to very low depth. The price paid for this compilation is the reliance on less standard, though well-motivated, assumptions: in addition to the lattice knowledge assumption used by Arabadjieva et al. (2025), we require a strengthened variant of the adaptive-hardcore-bit property of LWE, for which we provide supporting evidence. Unlike previous low-depth proofs of quantumness, the quantum computation here requires no mid-circuit measurements or feed-forward: it consists only of running a shallow circuit and sampling from its output distribution. This shows that shallow quantum circuits have sufficient structure to solve certain classically hard tasks whose solutions can be verified efficiently.
Recently, a polynomial time classical algorithm has been found for learning the shallow representation of a unitary UUU acting on nnn-qubits, by learning local inversions and then sewing them back together with ancilla qubits and SWAP gates. In this work, we bootstrap local inversion learning (LIL) to optimize quantum circuit depth by learning shallow representations for its sub-unitaries. We recursively cut circuits and apply the LIL algorithm to replace sub-circuits with their shallow representations, if it can be found by the algorithm. If not, we keep cutting until the optimization terminates, either by finding shallow representations or by reaching constant-depth sub-circuits. By replacing sub-circuits with their shallow representations, we hope to obtain some compression of the quantum circuit. Due to the binary search structure, the optimization algorithm has time complexity logarithmic in the depth of the original given circuit.
We give a polynomial time algorithm that, given copies of an unknown quantum state ∣ψ⟩=U∣0n⟩\vertψ\rangle=U\vert 0^n\rangle∣ψ⟩=U∣0n⟩ that is prepared by an unknown constant depth circuit UUU on a finite-dimensional lattice, learns a constant depth quantum circuit that prepares ∣ψ⟩\vertψ\rangle∣ψ⟩. The algorithm extends to the case when the depth of UUU is polylog(n)\mathrm{polylog}(n)polylog(n), with a quasi-polynomial run-time. The key new idea is a simple and general procedure that efficiently reconstructs the global state ∣ψ⟩\vertψ\rangle∣ψ⟩ 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.