Learning depth-3 circuits via quantum agnostic boosting

Srinivasan Arunachalam, Arkopal Dutt, Alexandru Gheorghiu, Michael De Oliveira
Proceedings of Thirty Ninth Conference on Learning Theory, PMLR 336:371-426, 2026.

Abstract

We initiate the study of quantum agnostic learning of phase states with respect to a function class $C \subseteq {c:{0,1}^n\rightarrow {0,1}}$: given copies of an unknown $n$-qubit state $|\psi⟩$ which has fidelity $\textsf{opt}$ with a phase state $|\phi_c⟩=\frac{1}{\sqrt{2^n}}\sum_{x\in {0,1}^n}(-1)^{c(x)}|x⟩$ for some $c\in C$, output $|\phi⟩$ which has fidelity $|⟨\phi | \psi ⟩|^2 \geq \textsf{opt}-\varepsilon$. To this end, we give agnostic learning protocols for the following classes: 1. 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 $\textsf{poly}(n,2^k,1/\varepsilon)$. 2. $s$-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 $|\phi⟩$ such that $|⟨\phi|\psi⟩|^2\geq \textsf{opt}/\textsf{poly}(n)$, into a “strong” learner which outputs a superposition of parity states $|\phi’⟩$ such that $|⟨\phi’|\psi⟩|^2\geq \textsf{opt} - \varepsilon$. Using quantum agnostic boosting, we give 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v336-arunachalam26a, title = {Learning depth-3 circuits via quantum agnostic boosting}, author = {Arunachalam, Srinivasan and Dutt, Arkopal and Gheorghiu, Alexandru and De Oliveira, Michael}, booktitle = {Proceedings of Thirty Ninth Conference on Learning Theory}, pages = {371--426}, year = {2026}, editor = {Hanneke, Steve and Lattimore, Tor}, volume = {336}, series = {Proceedings of Machine Learning Research}, month = {29 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v336/main/assets/arunachalam26a/arunachalam26a.pdf}, url = {https://proceedings.mlr.press/v336/arunachalam26a.html}, abstract = {We initiate the study of quantum agnostic learning of phase states with respect to a function class $C \subseteq {c:{0,1}^n\rightarrow {0,1}}$: given copies of an unknown $n$-qubit state $|\psi⟩$ which has fidelity $\textsf{opt}$ with a phase state $|\phi_c⟩=\frac{1}{\sqrt{2^n}}\sum_{x\in {0,1}^n}(-1)^{c(x)}|x⟩$ for some $c\in C$, output $|\phi⟩$ which has fidelity $|⟨\phi | \psi ⟩|^2 \geq \textsf{opt}-\varepsilon$. To this end, we give agnostic learning protocols for the following classes: 1. 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 $\textsf{poly}(n,2^k,1/\varepsilon)$. 2. $s$-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 $|\phi⟩$ such that $|⟨\phi|\psi⟩|^2\geq \textsf{opt}/\textsf{poly}(n)$, into a “strong” learner which outputs a superposition of parity states $|\phi’⟩$ such that $|⟨\phi’|\psi⟩|^2\geq \textsf{opt} - \varepsilon$. Using quantum agnostic boosting, we give 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. } }
Endnote
%0 Conference Paper %T Learning depth-3 circuits via quantum agnostic boosting %A Srinivasan Arunachalam %A Arkopal Dutt %A Alexandru Gheorghiu %A Michael De Oliveira %B Proceedings of Thirty Ninth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Steve Hanneke %E Tor Lattimore %F pmlr-v336-arunachalam26a %I PMLR %P 371--426 %U https://proceedings.mlr.press/v336/arunachalam26a.html %V 336 %X We initiate the study of quantum agnostic learning of phase states with respect to a function class $C \subseteq {c:{0,1}^n\rightarrow {0,1}}$: given copies of an unknown $n$-qubit state $|\psi⟩$ which has fidelity $\textsf{opt}$ with a phase state $|\phi_c⟩=\frac{1}{\sqrt{2^n}}\sum_{x\in {0,1}^n}(-1)^{c(x)}|x⟩$ for some $c\in C$, output $|\phi⟩$ which has fidelity $|⟨\phi | \psi ⟩|^2 \geq \textsf{opt}-\varepsilon$. To this end, we give agnostic learning protocols for the following classes: 1. 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 $\textsf{poly}(n,2^k,1/\varepsilon)$. 2. $s$-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 $|\phi⟩$ such that $|⟨\phi|\psi⟩|^2\geq \textsf{opt}/\textsf{poly}(n)$, into a “strong” learner which outputs a superposition of parity states $|\phi’⟩$ such that $|⟨\phi’|\psi⟩|^2\geq \textsf{opt} - \varepsilon$. Using quantum agnostic boosting, we give 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.
APA
Arunachalam, S., Dutt, A., Gheorghiu, A. & De Oliveira, M.. (2026). Learning depth-3 circuits via quantum agnostic boosting. Proceedings of Thirty Ninth Conference on Learning Theory, in Proceedings of Machine Learning Research 336:371-426 Available from https://proceedings.mlr.press/v336/arunachalam26a.html.

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