Michael de Oliveira

We initiate the study of \emph{quantum agnostic learning} of phase states with respect to a function class $\mathcal{C}\subseteq \{c:\{0,1\}^n\rightarrow \{0,1\}$: given copies of an unknown $n$-qubit state $|\psi\rangle$ which has fidelity $\textsf{opt}$ with a phase state $|\phi_c\rangle=\frac{1}{\sqrt{2^n}}\sum_{x\in \{0,1\}^n}(-1)^{c(x)}|x\rangle$ for some $c\in \mathcal{C}$, output $|\phi\rangle$ which has fidelity $|\langle \phi | \psi \rangle|^2 \geq \textsf{opt}-\varepsilon$. To this end, we give agnostic learning protocols for the following classes: \begin{enumerate} \item 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 $\textsf{poly}(n,2^k,1/\varepsilon)$. \item $s$-term DNF formulas in near-polynomial time $\textsf{poly}(n,(s/\varepsilon)^{\log \log s/\varepsilon})$. \end{enumerate} Our main technical contribution is a \emph{quantum agnostic boosting} protocol which converts a ``weak'' agnostic learner (which outputs a \emph{parity state} $|\phi\rangle$ such that $|\langle \phi|\psi\rangle|^2\geq \textsf{opt}/\textsf{poly}(n)$) into a ``strong'' learner (which outputs a sum of parity states $|\phi'\rangle$ such that $|\langle \phi'|\psi\rangle|^2\geq \textsf{opt} - \varepsilon$). Using quantum agnostic boosting, we obtain the first ``near'' polynomial-time $n^{O(\log \log n)}$ algorithm for learning $\textsf{poly}(n)$-sized depth-$3$ circuits (consisting of $\textsf{AND}$, $\textsf{OR}$, $\textsf{NOT}$ gates) in the uniform quantum $\textsf{PAC}$ model using quantum examples. Classically, the analogue of efficient learning depth-$3$ circuits (and even depth-$2$ circuits) in the uniform $\textsf{PAC}$ model has been a longstanding open question in computational learning theory. Our work nearly settles this question, when the learner is given quantum examples.