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 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.
In this work we make progress in understanding the relationship between learning models with access to entangled, separable and statistical measurements in the quantum statistical query (QSQ) model. To this end, we show the following results. Entangled versus separable measurements. The goal here is to learn an unknown f from the concept class C⊆{f:{0,1}n→[k]} given copies of 2n1∑x∣x,f(x)⟩. We show that, if T copies suffice to learn f using entangled measurements, then O(nT2) copies suffice to learn f using just separable measurements. Entangled versus statistical measurements The goal here is to learn a function f∈C given access to separable measurements and statistical measurements. We exhibit a class C that gives an exponential separation between QSQ learning and quantum learning with entangled measurements (even in the presence of noise). This proves the "quantum analogue" of the seminal result of Blum et al. [BKW'03]. that separates classical SQ and PAC learning with classification noise. QSQ lower bounds for learning states. We introduce a quantum statistical query dimension (QSD), which we use to give lower bounds on the QSQ learning. With this we prove superpolynomial QSQ lower bounds for testing purity, shadow tomography, Abelian hidden subgroup problem, degree-2 functions, planted bi-clique states and output states of Clifford circuits of depth polylog(n). Further applications. We give and unconditional separation between weak and strong error mitigation and prove lower bounds for learning distributions in the QSQ model. Prior works by Quek et al. [QFK+'22], Hinsche et al. [HIN+'22], and Nietner et al. [NIS+'23] proved the analogous results assuming diagonal measurements and our work removes this assumption.
Kearns' statistical query (SQ) oracle (STOC'93) lends a unifying perspective for most classical machine learning algorithms. This ceases to be true in quantum learning, where many settings do not admit, neither an SQ analog nor a quantum statistical query (QSQ) analog. In this work, we take inspiration from Kearns' SQ oracle and Valiant's weak evaluation oracle (TOCT'14) and establish a unified perspective bridging the statistical and parametrized learning paradigms in a novel way. We explore the problem of learning from an evaluation oracle, which provides an estimate of function values, and introduce an extensive yet intuitive framework that yields unconditional lower bounds for learning from evaluation queries and characterizes the query complexity for learning linear function classes. The framework is directly applicable to the QSQ setting and virtually all algorithms based on loss function optimization. Our first application is to extend prior results on the learnability of output distributions of quantum circuits and Clifford unitaries from the SQ to the (multi-copy) QSQ setting, implying exponential separations between learning stabilizer states from (multi-copy) QSQs versus from quantum samples. Our second application is to analyze some popular quantum machine learning (QML) settings. We gain an intuitive picture of the hardness of many QML tasks which goes beyond existing methods such as barren plateaus and the statistical dimension, and contains crucial setting-dependent implications. Our framework not only unifies the perspective of cost concentration with that of the statistical dimension in a unified language but exposes their connectedness and similarity.
In learning theory, the VC dimension of a concept class C is the most common way to measure its “richness.” A fundamental result says that the number of examples needed to learn an unknown target concept c∈C under an unknown distribution D, is tightly determined by the VC dimension d of the concept class C. Specifically, in the PAC model
Θ(dϵ+log(1/δ)ϵ)
examples are necessary and sufficient for a learner to output, with probability 1−δ, a hypothesis h that is ϵ-close to the target concept c (measured under D). In the related agnostic model, where the samples need not come from a c∈C, we know that
Θ(dϵ2+log(1/δ)ϵ2)
examples are necessary and sufficient to output an hypothesis h∈C whose error is at most ϵ worse than the error of the best concept in C. Here we analyze quantum sample complexity, where each example is a coherent quantum state. This model was introduced by Bshouty and Jackson (1999), who showed that quantum examples are more powerful than classical examples in some fixed-distribution settings. However, Atici and Servedio (2005), improved by Zhang (2010), showed that in the PAC setting (where the learner has to succeed for every distribution), quantum examples cannot be much more powerful: the required number of quantum examples is
Ω(d1−ηϵ+d+log(1/δ)ϵ) for arbitrarily small constant η>0.
Our main result is that quantum and classical sample complexity are in fact equal up to constant factors in both the PAC and agnostic models. We give two proof approaches. The first is a fairly simple information-theoretic argument that yields the above two classical bounds and yields the same bounds for quantum sample complexity up to a log(d/ϵ) factor. We then give a second approach that avoids the log-factor loss, based on analyzing the behavior of the “Pretty Good Measurement” on the quantum state-identification problems that correspond to learning. This shows classical and quantum sample complexity are equal up to constant factors for every concept class C.
$ \newcommand{\eps}{\varepsilon} $In learning theory, the VC dimension of a concept class C is the most common way to measure its "richness." In the PAC model $$ Θ\Big(\frac{d}{\eps} + \frac{\log(1/δ)}{\eps}\Big) $$ examples are necessary and sufficient for a learner to output, with probability 1−δ, a hypothesis h that is $\eps$-close to the target concept c. In the related agnostic model, where the samples need not come from a c∈C, we know that $$ Θ\Big(\frac{d}{\eps^2} + \frac{\log(1/δ)}{\eps^2}\Big) $$ examples are necessary and sufficient to output an hypothesis h∈C whose error is at most $\eps$ worse than the best concept in C. Here we analyze quantum sample complexity, where each example is a coherent quantum state. This model was introduced by Bshouty and Jackson, who showed that quantum examples are more powerful than classical examples in some fixed-distribution settings. However, Atici and Servedio, improved by Zhang, showed that in the PAC setting, quantum examples cannot be much more powerful: the required number of quantum examples is $$ Ω\Big(\frac{d^{1-η}}{\eps} + d + \frac{\log(1/δ)}{\eps}\Big)\mbox{ for all }η> 0. $$ Our main result is that quantum and classical sample complexity are in fact equal up to constant factors in both the PAC and agnostic models. We give two approaches. The first is a fairly simple information-theoretic argument that yields the above two classical bounds and yields the same bounds for quantum sample complexity up to a $\log(d/\eps)$ factor. We then give a second approach that avoids the log-factor loss, based on analyzing the behavior of the "Pretty Good Measurement" on the quantum state identification problems that correspond to learning. This shows classical and quantum sample complexity are equal up to constant factors.
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.
We propose several algorithms for learning unitary operators from quantum statistical queries with respect to their Choi-Jamiolkowski state. Quantum statistical queries capture the capabilities of a learner with limited quantum resources, which receives as input only noisy estimates of expected values of measurements. Our approach leverages quantum statistical queries to estimate the Fourier mass of a unitary on a subset of Pauli strings, generalizing previous techniques developed for uniform quantum examples. Specifically, we show that the celebrated quantum Goldreich-Levin algorithm can be implemented with quantum statistical queries, whereas the prior version of the algorithm involves oracle access to the unitary and its inverse. As an application, we prove that quantum Boolean functions with constant total influence or with constant degree are efficiently learnable in our model. Moreover, we prove that O(logn) -juntas are efficiently learnable and constant-depth circuits are learnable query-efficiently with quantum statistical queries. On the other hand, all previous algorithms for these tasks demand significantly greater resources, such as oracle access to the unitary or direct access to the Choi-Jamiolkowski state. We also demonstrate that, despite these positive results, quantum statistical queries lead to an exponentially larger query complexity for certain tasks, compared to separable measurements to the Choi-Jamiolkowski state. In particular, we show an exponential lower bound for learning a class of phase-oracle unitaries and a double exponential lower bound for testing the unitarity of channels. Taken together, our results indicate that quantum statistical queries offer a unified framework for various unitary learning tasks, with potential applications in quantum machine learning, many-body physics and benchmarking of near-term devices.