Year: 2026
Type: preprint
Abstract: Linear threshold functions (LTFs) are $\{-1,1\}$-valued functions $f_{w,θ}(x)=\text{sgn}(\langle w,x\rangle-θ)$, where $w\in\mathbb R^n$ has unit norm, $θ\in\mathbb R$, and $x\in\mathbb R^n$ or $\{-1,1\}^n$. If $θ=0$, the LTF is homogeneous, denoted $f_w=f_{w,0}$. LTFs are fundamental linear classifiers. We give three results: one for general LTFs and two for homogeneous LTFs. (1) With real-domain membership queries evaluating $f_{w,θ}(x)$ at any chosen $x\in\mathbb R^n$, we learn an LTF with classification error at most $\varepsilon$ under the standard Gaussian distribution using $O(\log(n/\varepsilon))$ queries and $\widetilde{O}(n)$ other gates. Classically, $Ω(n\log(1/\varepsilon))$ queries are required, giving an exponential quantum query speedup. (2) A homogeneous LTF $f_w$ over $\{-1,1\}^n$ whose $k$ nonzero weights are equal and positive is Majority on the support of $w$. We identify this hidden support exactly using $O(\log k)$ queries, exponentially improving Belovs' $O(k^{1/4})$ upper bound. Our algorithm uses $O(n\log n\log k)$ gates and has $O(\log n\log k)$ non-oracle depth. Classical algorithms require $Ω(k\log(n/k))$ queries. (3) Suppose a unitary $U$ prepares discretized Gaussian quantum examples, corresponding to $\int_x\sqrt{γ_n(x)}|x\rangle|f_w(x)\rangle\,dx$. This is weaker than membership queries. Using the efficient quantum Hermite transform circuit, we learn homogeneous LTFs with classification error at most $\varepsilon$ under the Gaussian distribution using $O(n^{1/4}/\sqrt{\varepsilon})$ applications of $U$ and $U^\dagger$ and $\widetilde{O}\!(\frac{n^{5/4}}{\sqrt{\varepsilon}}+\frac{n}{\varepsilon^2})$ other gates. Classically, $Ω(n/\varepsilon)$ examples are needed, giving a quartic quantum speedup in $n$. We show that the quantum query complexities in all three cases are optimal by proving matching lower bounds.
Source: arXiv (Cornell University)
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215120799