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arXiv:2610.23268

Unconditional Quantum PAC Learning Advantage in the Large Error Regime

Igor C. Oliveira

cs.CCquant-phSubmitted Oct 16, 2026• Announced Oct 19, 2026• 0 citations
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Abstract

We establish an unconditional super-polynomial separation between quantum and classical PAC learning of Boolean functions under the uniform distribution in the large error regime. For every advantage $λ(n) = n^{-ω(1)}$, we construct a concept class that a polynomial-time quantum learner learns to agreement $1/2+λ(n)$ with success probability at least $1-1/n^k$. The learner uses only quantum random examples and outputs a polynomial-size deterministic classical circuit. In contrast, no polynomial-time classical learner achieves agreement $1/2 + λ(n)/n^k$ with success probability greater than $1/n^k$, even with membership queries. To achieve this, we develop a nearly-optimal quantum learner that uses quantum random examples to produce a hypothesis achieving a prescribed correlation with an arbitrary Boolean function. The learner generalizes Fourier sampling from parity hypotheses to quadratic hypotheses. While standard Fourier sampling is insensitive to the sign of the correlation, we use interference to bias sampling toward positively correlated hypotheses. This is combined with a careful analysis of the resulting distribution to boost both the learner's success probability and its agreement with the target function. We believe that the resulting quantum learning algorithm might be of independent interest. Finally, although a robust quantum PAC learning advantage holds unconditionally in the large error regime, we note that an analogous separation between quantum and classical learning in the small error regime would imply that $\mathsf{NP} \nsubseteq \mathsf{BPP}$.

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arXiv:1903.02840•SIAM Journal on Computing•Jan 1, 2021•11 citations
We establish an unconditional super-polynomial separation between quantum and classical PAC learning of Boolean functions under the uniform distribution in the large error regime. For every advantage λ(n)=n−ω(1)λ(n) = n^{-ω(1)}λ(n)=n−ω(1), we construct a concept class that a polynomial-time quantum learner learns to agreement 1/2+λ(n)1/2+λ(n)1/2+λ(n) with success probability at least 1−1/nk1-1/n^k1−1/nk. The learner uses only quantum random examples and outputs a polynomial-size deterministic classical circuit. In contrast, no polynomial-time classical learner achieves agreement 1/2+λ(n)/nk1/2 + λ(n)/n^k1/2+λ(n)/nk with success probability greater than 1/nk1/n^k1/nk, even with membership queries.
To achieve this, we develop a nearly-optimal quantum learner that uses quantum random examples to produce a hypothesis achieving a prescribed correlation with an arbitrary Boolean function. The learner generalizes Fourier sampling from parity hypotheses to quadratic hypotheses. While standard Fourier sampling is insensitive to the sign of the correlation, we use interference to bias sampling toward positively correlated hypotheses. This is combined with a careful analysis of the resulting distribution to boost both the learner's success probability and its agreement with the target function. We believe that the resulting quantum learning algorithm might be of independent interest.
Finally, although a robust quantum PAC learning advantage holds unconditionally in the large error regime, we note that an analogous separation between quantum and classical learning in the small error regime would imply that NP⊈BPP\mathsf{NP} \nsubseteq \mathsf{BPP}NP⊈BPP.
We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let C\mathfrak{C}C be a class of polynomial-size concepts, and suppose that C\mathfrak{C}C can be PAC-learned with membership queries under the uniform distribution with error 1/2−γ1/2 - γ1/2−γ by a time TTT quantum algorithm. We prove that if γ2⋅T≪2n/nγ^2 \cdot T \ll 2^n/nγ2⋅T≪2n/n, then BQE⊈C\mathsf{BQE} \nsubseteq \mathfrak{C}BQE⊈C, where BQE=BQTIME[2O(n)]\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]BQE=BQTIME[2O(n)] is an exponential-time analogue of BQP\mathsf{BQP}BQP. This result is optimal in both γγγ and TTT, since it is not hard to learn any class C\mathfrak{C}C of functions in (classical) time T=2nT = 2^nT=2n (with no error), or in quantum time T=poly(n)T = \mathsf{poly}(n)T=poly(n) with error at most 1/2−Ω(2−n/2)1/2 - Ω(2^{-n/2})1/2−Ω(2−n/2) via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.
As quantum computing progresses toward fault-tolerant architectures, the question of which computational tasks admit provable quantum advantages and which do not has become increasingly central. Learning theory, and in particular learning from quantum examples, provides one of the few settings in which unconditional quantum-classical separations can be established. In distribution-free (i.e., worst-case) PAC learning, existing results show that quantum examples provide no asymptotic advantage in sample complexity. In contrast, under the uniform distribution, unbounded quantum-classical separations are known for learning Fourier-sparse Boolean functions. Together, these results reveal a striking dichotomy. However, this understanding has largely been developed in the context of learning Boolean functions in the batch setting, leaving open how these phenomena extend more broadly. This thesis develops the theory of learning with quantum examples beyond the batch Boolean setting along three directions: multiclass learning, online learning, and smoothed learning. In the multiclass PAC setting, we establish upper and lower bounds on quantum sample complexity in both the realizable and agnostic regimes, finding that quantum examples continue to yield no distribution-independent separation from classical examples, with learning rates governed by the Natarajan dimension up to logarithmic factors in the label-space size. We next study online learning, where no standard framework for learning with quantum examples existed prior to this work. We provide such a model by lifting the classical online framework to one in which the adversary provides distributions over labeled examples, and then by encoding these distributions as quantum examples. We establish expected regret guarantees for binary and multiclass classification in both the realizable and agnostic settings. The central finding is that unrestricted adversarial power permits highly concentrated distributions that dequantize the learning problem. Motivated by this dequantization phenomenon, we develop a smoothed learning framework that constrains distributions to be smooth, interpolating between the concentrated-distribution regime, in which no quantum advantage exists, and the uniform-distribution regime, in which unbounded separations are known. For the class of Fourier-sparse Boolean functions, we establish the robustness of quantum Fourier sampling throughout a nontrivial near-uniform regime in the batch setting and observe that these guarantees extend naturally to the online setting. We further analyze several approaches to recovering the target function once its Fourier support has been identified and, based on these observations, formulate conjectures supporting the persistence of the unbounded quantum advantage throughout this regime. Together, these results paint a coherent picture of learning with quantum examples beyond the batch Boolean setting, showing that quantum-classical separations depend on the interplay between hypothesis class structure, distributional assumptions, and the degree of adversarial control permitted in the learning process.
We study the number of uniform quantum examples needed to exactly learn a Boolean function with at most kkk nonzero Fourier coefficients. We give a learner with near-optimal O~(k)\widetilde{O}(k)O(k) sample complexity independent of the input length nnn. The learner uses only separate measurements of individual examples. This improves the previous O(k3/2log⁡2k)O(k^{3/2}\log^2 k)O(k3/2log2k) upper bound (Arunachalam, Chakraborty, Lee, Paraashar, and de Wolf, 2021). The main ingredient is a candidate set formed by symmetric differences of subsets from independent groups of Fourier samples. A multilinear majority polynomial combines unbiased random approximations into an accurate witness whose Fourier support lies in this set. Granularity then forces the set to contain the entire Fourier support, including subsets never directly sampled. For comparison, we derive the tight classical sample complexity Θ(nk)Θ(nk)Θ(nk) by combining known sparse-recovery and lower-bound results.
We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, letC\mathfrak{C}Cbe a class of polynomial-size concepts, and suppose thatC\mathfrak{C}Ccan be PAC-learned with membership queries under the uniform distribution with error1/2−γ1/2 -\gamma1/2−γby a timeTTTquantum algorithm. We prove that ifγ2⋅T≪2n/n\gamma^{2}\cdot T \ll 2^{n} /nγ2⋅T≪2n/n, thenBQE⊄C\mathsf{BQE}\not\subset \mathfrak{C}BQE⊂C, whereBQE=BQTIME[2O(n)]\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]BQE=BQTIME[2O(n)]is an exponential-time analogue ofBQP\mathsf{BQP}BQP. This result is optimal in bothγ\gammaγandTTT, since it is not hard to learn any classC\mathfrak{C}Cof functions in (classical) timeT=2nT=2^{n}T=2n(with no error), or in quantum timeT=poly(n)T= \mathsf{poly}(n)T=poly(n)with error at most1/2−Ω(2−n/2)1/2-\Omega(2^{-n/2})1/2−Ω(2−n/2)via Fourier sampling. In other words, even a marginal quantum speedup over these generic learning algorithms would lead to major consequences in complexity lower bounds. As a consequence, our result shows that the study of quantum learning speedups is intimately connected to fundamental open problems about algorithms, quantum computing, and complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges, since extracting computational hardness from a quantum computation is inherently more complicated. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.
The agnostic setting is the hardest generalization of the PAC model since it is akin to learning with adversarial noise. In this paper, we give a poly(n,t,1ε)(n,t,{\frac{1}{\varepsilon}})(n,t,ε1​) quantum algorithm for learning size ttt decision trees with uniform marginal over instances, in the agnostic setting, without membership queries. Our algorithm is the first algorithm (classical or quantum) for learning decision trees in polynomial time without membership queries. We show how to construct a quantum agnostic weak learner by designing a quantum version of the classical Goldreich-Levin algorithm that works with strongly biased function oracles. We show how to quantize the agnostic boosting algorithm by Kalai and Kanade (NIPS 2009) to obtain the first efficient quantum agnostic boosting algorithm. Our quantum boosting algorithm has a polynomial improvement in the dependence of the bias of the weak learner over all adaptive quantum boosting algorithms while retaining the standard speedup in the VC dimension over classical boosting algorithms. We then use our quantum boosting algorithm to boost the weak quantum learner we obtained in the previous step to obtain a quantum agnostic learner for decision trees. Using the above framework, we also give quantum decision tree learning algorithms for both the realizable setting and random classification noise model, again without membership queries.
In this paper, we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of probably approximately correct (PAC) learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results. 1. Hardness of PAC learning AC0{AC}^0AC0 and TC0{TC}^0TC0 under the uniform distribution. Our first result concerns the concept class TC0{TC}^0TC0 (resp., AC0{AC}^0AC0), the class of constant-depth, polynomial-sized circuits with unbounded fan-in majority gates (resp., AND,OR,NOT{AND}, {OR}, {NOT}AND,OR,NOT gates). We show the following: if there exists no quantum (quasi-)polynomial-time algorithm to solve the ring-learning with errors (RLWE{RLWE}RLWE) problem, then there exists no (quasi-)polynomial-time quantum learning algorithm for TC0{TC}^0TC0; and if there exists no 2O(d1/η)2^{O(d^{1/\eta})}2O(d1/η)-time quantum algorithm to solve RLWE{RLWE}RLWE with dimension d=O(polylogn)d = O(polylog n)d=O(polylogn) (for every constant η>2\eta > 2η>2), then there exists no O(nlog⁡νn)O(n^{ \log^{\nu} n} )O(nlogνn)-time quantum learning algorithm for poly(n)poly(n)poly(n)-sized AC0{AC}^0AC0 circuits (for a constant ν>0\nu>0ν>0), matching the classical upper bound of Linial, Mansour and Nisan [J. ACM, 40 (1993), pp. 607--620], where the learning algorithms are under the uniform distribution (even with access to quantum membership queries). The main technique in these results uses an explicit family of pseudorandom functions that are believed to be quantum-secure to construct concept classes that are hard to learn quantumly under the uniform distribution. 2. Hardness of learning TC20{TC}^0_2TC20​ in the PAC setting. Our second result shows that if there exists no quantum polynomial-time algorithm for the LWE{LWE}LWE problem, then there exists no polynomial-time quantum-PAC learning algorithm for the class TC20{TC}^0_2TC20​, i.e., depth-2 TC0{TC}^0TC0 circuits. The main technique in this result is to establish a connection between the quantum security of public-key encryption schemes and the learnability of a concept class that consists of decryption functions of the cryptosystem. Our results show that quantum resources do not give an exponential improvement to learning constant-depth polynomial-sized neural networks. This also gives a strong (conditional) negative answer to one of the “Ten Semi-Grand Challenges for Quantum Computing Theory" raised by Aaronson https://www.scottaaronson.com/writings/qchallenge.html, 2005.