Learning depth-3 circuits via quantum agnostic boosting
Srinivasan Arunachalam, Arkopal Dutt, Alexandru Gheorghiu, Michael de Oliveira
TL;DR
The paper advances quantum learning by introducing a quantum agnostic boosting framework that converts a weak agnostic learner into a strong one for phase-state representations. This umbrella approach yields efficient quantum algorithms for agnostic learning of poly$(n)$-sized depth-3 circuits, as well as for decision trees, juntas, and DNFs, with explicit runtimes that surpass classical bounds in several regimes. By integrating Fourier-analytic ideas with quantum subroutines (e.g., SWAP tests and stabilizer-based state preparations), the authors establish both state- and distributional-agnostic learning results and show a quantum advantage in PAC-learning depth-3 circuits. The results open avenues for broader quantum learnability of structured Boolean functions and suggest practical pathways for tomography-like state learning from quantum data. Overall, the work provides a principled framework and concrete algorithms that leverage quantum examples to achieve improved learning performance for nontrivial circuit classes in the agnostic setting.
Abstract
We initiate the study of quantum agnostic learning of phase states with respect to a function class $\mathsf{C}\subseteq \{c:\{0,1\}^n\rightarrow \{0,1\}\}$: given copies of an unknown $n$-qubit state $|ψ\rangle$ which has fidelity $\textsf{opt}$ with a phase state $|φ_c\rangle=\frac{1}{\sqrt{2^n}}\sum_{x\in \{0,1\}^n}(-1)^{c(x)}|x\rangle$ for some $c\in \mathsf{C}$, output $|φ\rangle$ which has fidelity $|\langle φ| ψ\rangle|^2 \geq \textsf{opt}-\varepsilon$. To this end, we give agnostic learning protocols for the following classes: (i) Size-$t$ decision trees which runs in time $\textsf{poly}(n,t,1/\varepsilon)$. This also implies $k$-juntas can be agnostically learned in time $\textsf{poly}(n,2^k,1/\varepsilon)$. (ii) $s$-term DNF formulas in time $\textsf{poly}(n,(s/\varepsilon)^{\log \log (s/\varepsilon) \cdot \log(1/\varepsilon)})$. Our main technical contribution is a quantum agnostic boosting protocol which converts a weak agnostic learner, which outputs a parity state $|φ\rangle$ such that $|\langle φ|ψ\rangle|^2\geq \textsf{opt}/\textsf{poly}(n)$, into a strong learner which outputs a superposition of parity states $|φ'\rangle$ such that $|\langle φ'|ψ\rangle|^2\geq \textsf{opt} - \varepsilon$. Using quantum agnostic boosting, we obtain a $n^{O(\log(n/\varepsilon) \cdot \log \log n)}$-time algorithm for $\varepsilon$-learning $\textsf{poly}(n)$-sized depth-$3$ circuits (consisting of $\textsf{AND}$, $\textsf{OR}$, $\textsf{NOT}$ gates) in the uniform $\textsf{PAC}$ model given quantum examples. Classically, obtaining an algorithm with a similar complexity has been an open question in the $\textsf{PAC}$ model and our work answers this given quantum examples.