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arXiv:quant-ph/0202066

Quantum DNF Learnability Revisited

Jeffrey C. Jackson, Christino Tamon, Tomoyuki Yamakami

Published Feb 12, 2002• 0 citations
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Abstract

We describe a quantum PAC learning algorithm for DNF formulae under the uniform distribution with a query complexity of $\tilde{O}(s^{3}/ε+ s^{2}/ε^{2})$, where $s$ is the size of DNF formula and $ε$ is the PAC error accuracy. If $s$ and $1/ε$ are comparable, this gives a modest improvement over a previously known classical query complexity of $\tilde{O}(ns^{2}/ε^{2})$. We also show a lower bound of $Ω(s\log n/n)$ on the query complexity of any quantum PAC algorithm for learning a DNF of size $s$ with $n$ inputs under the uniform distribution.

DOI
10.48550/arxiv.quant-ph/0202066

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We describe a quantum PAC learning algorithm for DNF formulae under the uniform distribution with a query complexity of O~(s3/ε+s2/ε2)\tilde{O}(s^{3}/ε+ s^{2}/ε^{2})O~(s3/ε+s2/ε2), where sss is the size of DNF formula and εεε is the PAC error accuracy. If sss and 1/ε1/ε1/ε are comparable, this gives a modest improvement over a previously known classical query complexity of O~(ns2/ε2)\tilde{O}(ns^{2}/ε^{2})O~(ns2/ε2). We also show a lower bound of Ω(slog⁡n/n)Ω(s\log n/n)Ω(slogn/n) on the query complexity of any quantum PAC algorithm for learning a DNF of size sss with nnn inputs under the uniform distribution.
We propose a learning model called the quantum statistical learning QSQ model, which extends the SQ learning model introduced by Kearns to the quantum setting. Our model can be also seen as a restriction of the quantum PAC learning model: here, the learner does not have direct access to quantum examples, but can only obtain estimates of measurement statistics on them. Theoretically, this model provides a simple yet expressive setting to explore the power of quantum examples in machine learning. From a practical perspective, since simpler operations are required, learning algorithms in the QSQ model are more feasible for implementation on near-term quantum devices. We prove a number of results about the QSQ learning model. We first show that parity functions, (log n)-juntas and polynomial-sized DNF formulas are efficiently learnable in the QSQ model, in contrast to the classical setting where these problems are provably hard. This implies that many of the advantages of quantum PAC learning can be realized even in the more restricted quantum SQ learning model. It is well-known that weak statistical query dimension, denoted by WSQDIM(C), characterizes the complexity of learning a concept class C in the classical SQ model. We show that log(WSQDIM(C)) is a lower bound on the complexity of QSQ learning, and furthermore it is tight for certain concept classes C. Additionally, we show that this quantity provides strong lower bounds for the small-bias quantum communication model under product distributions. Finally, we introduce the notion of private quantum PAC learning, in which a quantum PAC learner is required to be differentially private. We show that learnability in the QSQ model implies learnability in the quantum private PAC model. Additionally, we show that in the private PAC learning setting, the classical and quantum sample complexities are equal, up to constant factors.
We generalize the notion of probably approximately correct (PAC) learning from an example oracle to a notion of efficient learning on a quantum computer using a quantum example oracle. This quantum example oracle is a natural extension of the traditional PAC example oracle, and it immediately follows that all PAC-learnable function classes are learnable in the quantum model. Furthermore, we obtain positive quantum learning results for classes that are not known to be PAC learnable. Specifically, we show that disjunctive normal form (DNF) is efficiently learnable with respect to the uniform distribution by a quantum algorithm using a quantum example oracle. While it was already known that DNF is uniform-learnable using a membership oracle, we prove that a quantum example oracle with respect to uniform is less powerful than a membership oracle.
Whether quantum computation can reduce the amount of data required for conventional machine learning is a central problem in quantum machine learning. In probably approximately correct (PAC) learning, a learner receives training examples sampled from an unknown probability distribution. Quantum PAC learning extends this framework by allowing access to quantum examples, which are quantum states that coherently encode this distribution, or to unitaries that prepare quantum examples, with or without inverse access. When only copies of quantum examples are available, the optimal worst-case sample complexity asymptotically matches that of classical learning. In contrast, access to both a preparation unitary and its inverse can improve the accuracy dependence of query complexity. However, it has remained open whether access to the preparation unitary alone suffices for this improvement. In this work, we resolve this question by determining the optimal query complexity with forward-only access, taking the worst case over compatible preparation unitaries and their dimensions. In both realizable and agnostic PAC learning, this complexity matches, up to constant factors, the optimal sample complexity achievable with classical examples or copies of quantum examples. To prove these results, we construct, for any algorithm with forward-only access, a simulator that approximates the algorithm's Haar-averaged output state using as many copies of the prepared state as the algorithm makes queries. These results show that inverse access is essential for the known improvement in accuracy dependence in this worst-case setting. Beyond PAC learning, our reduction provides a general tool for transferring lower bounds for state copies to query lower bounds for forward-only access to the corresponding preparation unitaries.
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.