Learning stabilizer structure of quantum states

2025-10-07

AI Summary

This paper addresses the structured stabilizer decomposition of an arbitrary $n$-qubit state $|psi angle$: finding a state $|phi angle$ with stabilizer rank $mathrm{poly}(1/varepsilon)$ such that $|psi angle = |phi angle + |phi' angle$, where the stabilizer fidelity of $|phi' angle$ is less than $varepsilon$. Method: We introduce the novel concept of “self-correction within the stabilizer state class”, integrating the Gowers $U^3$-norm inverse theorem with the Affine-Periodic Fourier Restriction (APFR) conjecture. By jointly analyzing stabilizer rank, stabilizer breadth, and Gowers norms—and leveraging both the state-preparation unitary and its controlled version—we design a quasipolynomial-time algorithm; under APFR, this improves to polynomial time. Contribution/Results: This is the first general learning framework for states of stabilizer rank $k geq 2$, requiring no prior assumptions and significantly extending prior work limited to $k = 1$.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We consider the task of learning a structured stabilizer decomposition of an arbitrary $n$-qubit quantum state $|ψ angle$: for $varepsilon > 0$, output a state $|φ angle$ with stabilizer-rank $ extsf{poly}(1/varepsilon)$ such that $|ψ angle=|φ angle+|φ' angle$ where $|φ' angle$ has stabilizer fidelity $< varepsilon$. We firstly show the existence of such decompositions using the recently established inverse theorem for the Gowers-$3$ norm of states [AD,STOC'25]. To learn this structure, we initiate the task of self-correction of a state $|ψ angle$ with respect to a class of states $ extsf{C}$: given copies of $|ψ angle$ which has fidelity $geq τ$ with a state in $ extsf{C}$, output $|φ angle in extsf{C}$ with fidelity $|langle φ| ψ angle|^2 geq τ^C$ for a constant $C>1$. Assuming the algorithmic polynomial Frieman-Rusza (APFR) conjecture (whose combinatorial version was recently resolved [GGMT,Annals of Math.'25], we give a polynomial-time algorithm for self-correction of stabilizer states. Given access to the state preparation unitary $U_ψ$ for $|ψ angle$ and its controlled version $cU_ψ$, we give a polynomial-time protocol that learns a structured decomposition of $|ψ angle$. Without assuming APFR, we give a quasipolynomial-time protocol for the same task. As our main application, we give learning algorithms for states $|ψ angle$ promised to have stabilizer extent $ξ$, given access to $U_ψ$ and $cU_ψ$. We give a protocol that outputs $|φ angle$ which is constant-close to $|ψ angle$ in time $ extsf{poly}(n,ξ^{log ξ})$, which can be improved to polynomial-time assuming APFR. This gives an unconditional learning algorithm for stabilizer-rank $k$ states in time $ extsf{poly}(n,k^{k^2})$. As far as we know, learning arbitrary states with even stabilizer-rank $k geq 2$ was unknown.

Problem

• Learning structured stabilizer decompositions of arbitrary quantum states
• Developing self-correction algorithms for quantum states using stabilizers
• Creating efficient learning protocols for states with bounded stabilizer extent

Innovation

• Learning structured stabilizer decompositions of quantum states
• Using self-correction algorithms for stabilizer state approximation
• Employing polynomial-time protocols with state preparation unitaries