Efficiently learning depth-3 circuits via quantum agnostic boosting
This work addresses the challenge of efficiently learning complex circuits in quantum machine learning, providing a novel solution that advances beyond classical limitations in the PAC model.
The paper tackles the problem of quantum agnostic learning of phase states for specific function classes, such as decision trees and DNF formulas, by introducing a quantum agnostic boosting protocol that converts weak learners into strong ones, achieving near-polynomial time algorithms for learning depth-3 circuits with quantum examples.
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 $n^{O(\log \log n \cdot \log(1/\varepsilon))}$-time algorithm for 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, which is near-polynomial time for constant $\varepsilon$. 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.