Optimal Quantum-Classical Separations for Exact Learning

2026-09-29

AI Summary

This study investigates the relationship between quantum and classical query complexities within the exact learning framework, aiming to transcend the limitations of the Grover and Bernstein-Vazirani paradigms and resolve long-standing complexity separation conjectures. Methodologically, it integrates tools from theoretical computer science, quantum query model analysis, and constructive counterexample design to systematically explore the intrinsic connections among deterministic, randomized, and quantum query complexities for concept classes. The core contributions include providing the first proof that quantum speedups can surpass traditional paradigms, establishing the fundamental role of randomness in optimizing upper bounds, and constructing an Ω(Q³ log N) separation counterexample that matches known upper bounds. Collectively, these results delineate the optimal boundaries between quantum and classical query complexities.

Abstract

We study exact learning with membership queries for concept classes $\mathcal C\subseteq\{0,1\}^N$, focusing on the relationships among their deterministic, randomized, and quantum query complexities, denoted $\mathsf{D}(\mathcal C)$, $\mathsf{R}(\mathcal C)$, and $\mathsf{Q}(\mathcal C)$, respectively. The two canonical quantum speedups in this model are witnessed by Grover search and Bernstein-Vazirani, leading to the longstanding conjecture $$ \mathsf{R}(\mathcal C)=O(\mathsf{Q}(\mathcal C)^2+\mathsf{Q}(\mathcal C)\log N). $$ We first refute this conjecture by constructing concept classes $\mathcal C$ and $\mathcal C'$ satisfying \[ \mathsf{R}(\mathcal C)=Ω\!\left(\frac{\mathsf{Q}(\mathcal C)^3\log N}{\log \mathsf{Q}(\mathcal C)}\right) \qquad\text{and}\qquad \mathsf{D}(\mathcal C')=Ω(\mathsf{Q}(\mathcal C')^3\log N). \] The first bound matches the upper bound of Arunachalam et al.~[Quantum'21] up to constant factors, while the second matches the upper bound of Servedio and Gortler~[SICOMP'04]. In particular, this shows that the saving in the randomized upper bound of Arunachalam et al. fundamentally relies on randomness. Apart from characterizing the optimal relationship between classical and quantum query complexity, our results are the first to show that quantum speedups for learning can go beyond the Grover and Bernstein-Vazirani paradigms.

Problem

• exact learning
• query complexity
• quantum-classical separation
• membership queries
• quantum speedup

Innovation

• exact learning
• quantum query complexity
• membership queries
• concept classes
• quantum speedup