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arXiv:2609.38073

Optimal Quantum-Classical Separations for Exact Learning

Srinivasan Arunachalam, Amin Shiraz Gilani, Nikhil S. Mande

quant-phcs.CCcs.LGSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
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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.

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We study exact learning with membership queries for concept classes C⊆{0,1}N\mathcal C\subseteq\{0,1\}^NC⊆{0,1}N, focusing on the relationships among their deterministic, randomized, and quantum query complexities, denoted D(C)\mathsf{D}(\mathcal C)D(C), R(C)\mathsf{R}(\mathcal C)R(C), and Q(C)\mathsf{Q}(\mathcal C)Q(C), respectively. The two canonical quantum speedups in this model are witnessed by Grover search and Bernstein-Vazirani, leading to the longstanding conjecture R(C)=O(Q(C)2+Q(C)log⁡N).\mathsf{R}(\mathcal C)=O(\mathsf{Q}(\mathcal C)^2+\mathsf{Q}(\mathcal C)\log N).R(C)=O(Q(C)2+Q(C)logN). We first refute this conjecture by constructing concept classes C\mathcal CC and C′\mathcal C'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.
We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k -Fourier-sparse n -bit Boolean function from O(k1.5(log⁡k)2) uniform quantum examples for that function. This improves over the bound of Θ~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our O~(k1.5) upper bound by proving an improvement of Chang's lemma for k -Fourier-sparse Boolean functions. Second, we show that if a concept class C can be exactly learned using Q quantum membership queries, then it can also be learned using O(Q2log⁡Qlog⁡|C|) classical membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a log⁡Q -factor.
We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k^{1.5}(log k)^2) uniform quantum examples for that function. This improves over the bound of Theta~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang’s lemma for sparse Boolean functions. Second, we show that if a concept class {C} can be exactly learned using Q quantum membership queries, then it can also be learned using O ({Q^2}/{log Q} * log|C|) classical membership queries. This improves the previous-best simulation result (Servedio-Gortler, SICOMP'04) by a log Q-factor.
We study how large the certificate complexity C(f){C}(f)C(f) of a total Boolean function can be relative to its randomized and quantum query complexities. We construct a total Boolean function fff whose randomized query complexity with one-sided error satisfies R1(f)=Θ(C(f)){R}_1(f) = Θ(\sqrt{{C}(f)})R1​(f)=Θ(C(f)​). This separation is optimal even when two-sided error is allowed. From this construction, we obtain another total Boolean function FFF with bounded-error quantum query complexity Q(F)=O~(C(F)1/4){Q}(F) = \widetilde O({C}(F)^{1/4})Q(F)=O(C(F)1/4), attaining the general quartic bound up to polylogarithmic factors. For the same function, both exact and zero-error quantum query complexities are O~(C(F))\widetilde O(\sqrt{{C}(F)})O(C(F)​). We also prove matching lower bounds for these two measures on FFF.
Quantum query complexity underpins much of our understanding of quantum speedups, yet nearly all existing lower and upper bounds are derived for simple Boolean oracles. Many practical tasks-noisy database search, stochastic optimization, adaptive learning, and time-varying systems-are best modeled by richer oracle classes. This paper systematically extends quantum query complexity theory to four non-standard oracle models: (1) stochastic oracles that return random outputs drawn from a distribution; (2) multi-solution oracles encoding a solution set of size k; (3) time-dependent oracles whose internal state evolves between queries; and (4) relational oracles encoding structured constraints rather than simple membership. Using generalized adversary methods, polynomial method techniques, and hybrid arguments, we derive tight (up to polylog factors) lower bounds for each model and construct matching algorithms where possible. For instance, we show that searching for any one of k marked items in an unstructured database still admits a Grover-like quadratic speedup, but with optimal complexity Θ(N/k), and that stochastic noise in oracle outputs degrades the advantage gracefully as a function of the noise parameter. For certain time-dependent and relational models, we prove that no better-than-classical speedup is possible, clarifying the limits of quantum advantage in adaptive settings. These results provide a more realistic foundation for quantum algorithms targeting noisy search, stochastic optimization, and constraint satisfaction, and they highlight the precise structural features that enable or preclude query-based quantum speedups.
We initiate a systematic study of pseudo-deterministic quantum algorithms. These are quantum algorithms that, for any input, output a canonical solution with high probability. Focusing on the query complexity model, our main contributions include the following complexity separations, which require new lower bound techniques specifically tailored to pseudo-determinism: - We exhibit a problem, Avoid One Encrypted String (AOES), whose classical randomized query complexity is O(1)O(1)O(1) but is maximally hard for pseudo-deterministic quantum algorithms (Ω(N)Ω(N)Ω(N) query complexity). - We exhibit a problem, Quantum-Locked Estimation (QL-Estimation), for which pseudo-deterministic quantum algorithms admit an exponential speed-up over classical pseudo-deterministic algorithms (O(log⁡(N))O(\log(N))O(log(N)) vs. Θ(N)Θ(\sqrt{N})Θ(N​)), while the randomized query complexity is O(1)O(1)O(1). Complementing these separations, we show that for any total problem RRR, pseudo-deterministic quantum algorithms admit at most a quintic advantage over deterministic algorithms, i.e., D(R)=O~(psQ(R)5)D(R) = \tilde O(psQ(R)^5)D(R)=O~(psQ(R)5). On the algorithmic side, we identify a class of quantum search problems that can be made pseudo-deterministic with small overhead, including Grover search, element distinctness, triangle finding, kkk-sum, and graph collision.
We construct a total Boolean function fff satisfying R(f)=Ω~(Q(f)5/2)R(f)=\tildeΩ(Q(f)^{5/2})R(f)=Ω~(Q(f)5/2), refuting the long-standing conjecture that R(f)=O(Q(f)2)R(f)=O(Q(f)^2)R(f)=O(Q(f)2) for all total Boolean functions. Assuming a conjecture of Aaronson and Ambainis about optimal quantum speedups for partial functions, we improve this to R(f)=Ω~(Q(f)3)R(f)=\tildeΩ(Q(f)^3)R(f)=Ω~(Q(f)3). Our construction is motivated by the Göös-Pitassi-Watson function but does not use it.