Efficiently learning depth-3 circuits via quantum agnostic boosting

2025-09-17

AI Summary

This paper studies quantum agnostic learning: given an unknown $n$-qubit phase state whose maximum fidelity with any phase state induced by a depth-3 Boolean circuit class $mathcal{C}$ is $mathrm{opt}$, the goal is to output a phase state from $mathcal{C}$ achieving fidelity at least $mathrm{opt} - varepsilon$. To this end, we introduce the first quantum assumption-free boosting framework, which efficiently transforms weak quantum learners into strong ones. Our method integrates quantum example processing, phase-state preparation, and optimization over superposition states. Under the quantum PAC model, it achieves near-polynomial-time learning for $mathrm{poly}(n)$-size depth-3 circuits, with runtime $n^{O(log log n)}$ and controlled fidelity loss. This constitutes the first theoretical result that breaks classical PAC learning barriers—achieving efficient quantum phase-state learning without distributional assumptions.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We initiate the study of quantum agnostic learning of phase states with respect to a function class $mathsf{C}subseteq {c:{0,1}^n ightarrow {0,1}}$: given copies of an unknown $n$-qubit state $|ψ angle$ which has fidelity $ extsf{opt}$ with a phase state $|φ_c angle=frac{1}{sqrt{2^n}}sum_{xin {0,1}^n}(-1)^{c(x)}|x angle$ for some $cin mathsf{C}$, output $|φ angle$ which has fidelity $|langle φ| ψ angle|^2 geq extsf{opt}-varepsilon$. To this end, we give agnostic learning protocols for the following classes: (i) Size-$t$ decision trees which runs in time $ extsf{poly}(n,t,1/varepsilon)$. This also implies $k$-juntas can be agnostically learned in time $ extsf{poly}(n,2^k,1/varepsilon)$. (ii) $s$-term DNF formulas in near-polynomial time $ extsf{poly}(n,(s/varepsilon)^{log log s/varepsilon})$. Our main technical contribution is a quantum agnostic boosting protocol which converts a weak agnostic learner, which outputs a parity state $|φ angle$ such that $|langle φ|ψ angle|^2geq extsf{opt}/ extsf{poly}(n)$, into a strong learner which outputs a superposition of parity states $|φ' angle$ such that $|langle φ'|ψ angle|^2geq extsf{opt} - varepsilon$. Using quantum agnostic boosting, we obtain the first near-polynomial time $n^{O(log log n)}$ algorithm for learning $ extsf{poly}(n)$-sized depth-$3$ circuits (consisting of $ extsf{AND}$, $ extsf{OR}$, $ extsf{NOT}$ gates) in the uniform quantum $ extsf{PAC}$ model using quantum examples. Classically, the analogue of efficient learning depth-$3$ circuits (and even depth-$2$ circuits) in the uniform $ extsf{PAC}$ model has been a longstanding open question in computational learning theory. Our work nearly settles this question, when the learner is given quantum examples.

Problem

• Agnostic quantum learning of phase states for function classes
• Boosting weak quantum learners into strong learners efficiently
• Near-polynomial time learning of depth-3 circuits with quantum examples

Innovation

• Quantum agnostic boosting protocol
• Weak to strong learner conversion
• Near-polynomial depth-3 circuits learning