Learning depth-3 circuits via quantum agnostic boosting
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| Format: | Preprint |
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2025
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| _version_ | 1866910024873279488 |
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| author | Arunachalam, Srinivasan Dutt, Arkopal Gheorghiu, Alexandru de Oliveira, Michael |
| author_facet | Arunachalam, Srinivasan Dutt, Arkopal Gheorghiu, Alexandru de Oliveira, Michael |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_14461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Learning depth-3 circuits via quantum agnostic boosting Arunachalam, Srinivasan Dutt, Arkopal Gheorghiu, Alexandru de Oliveira, Michael Quantum Physics Computational Complexity Machine Learning 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. |
| title | Learning depth-3 circuits via quantum agnostic boosting |
| topic | Quantum Physics Computational Complexity Machine Learning |
| url | https://arxiv.org/abs/2509.14461 |