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arXiv:2410.23618 · ACM Symposium on Theory of Computing (STOC)

Learning Quantum States Prepared by Shallow Circuits in Polynomial Time

Zeph A. Landau, Yunchao Liu

Published Jun 15, 2025• 5 citations
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Abstract

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.

DOI
10.1145/3717823.3718311

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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.
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 give an efficient algorithm for learning kkk-dimensional brickwork random quantum circuits using only copies of the output state obtained by applying UUU to the all-zero input. For a depth-ddd circuit on nnn sites with random 2ℓ2\ell2ℓ-qubit gates, the algorithm learns the original circuit UUU with high probability in poly(n,2ℓd)\text{poly}(n,2^{\ell d})poly(n,2ℓd) time for every constant dimensional lattice. In particular, the algorithm is polynomial time as long as ℓd=O(log⁡n)\ell d = O(\log n)ℓd=O(logn). In one dimension, this reaches the natural boundary suggested by pseudorandomness: pseudorandom states require ℓd=ω(log⁡n)\ell d=ω(\log n)ℓd=ω(logn), and structured cryptographic constructions suggest that this scale may be achievable from above. In higher dimensions, it remains plausible that the same ℓd=ω(log⁡n)\ell d=ω(\log n)ℓd=ω(logn) scale remains the threshold for ancilla-free pseudorandomness, and our results help clarify the conditions under which pseudorandomness can arise in this setting. The main idea behind our algorithm is a local correlation criterion that identifies gates in the final layer without learning their entire backward light cones, avoiding a bottleneck in the previous approaches. A key technical ingredient is a Carbery-Wright type anticoncentration inequality for low-degree polynomials of Haar random unitaries whose small-ball exponent is independent of the matrix dimension. The dimension-independent exponent is crucial for handling gates of growing locality. This anticoncentration result may also be of independent interest.
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.
Many-body ground state preparation is an important subroutine used in the simulation of physical systems. In this paper, we introduce a flexible and efficient framework for obtaining a state preparation circuit for a large class of many-body ground states. We introduce polynomial-time classical algorithms that take reduced density matrices over O(1)\mathcal{O}(1)O(1)-sized balls as inputs, and output a circuit that prepares the global state. We introduce algorithms applicable to (i) short-range entangled states (e.g., states prepared by shallow quantum circuits in any number of dimensions, and more generally, invertible states) and (ii) long-range entangled ground states (e.g., the toric code on a disk). Both algorithms can provably find a circuit whose depth is asymptotically optimal. Our approach uses a variant of the quantum Markov chain condition that remains robust against constant-depth circuits. The robustness of this condition makes our method applicable to a large class of states, whilst ensuring a classically tractable optimization landscape.