This paper studies quantum supervised learning for classical inference from quantum states. In this model, a learner has access to a set of labeled quantum samples as the training set. The objective is to find a quantum measurement that predicts the label of the unseen samples. The hardness of learning is measured via sample complexity under a quantum counterpart of the well-known probably approximately correct (PAC). Quantum sample complexity is expected to be higher than classical one, because of the measurement incompatibility and state collapse. Recent efforts showed that the sample complexity of learning a finite quantum concept class$\mathcal{C}$scales as$O(\vert \mathcal{C}\vert)$. This is significantly higher than the classical sample complexity that grows logarithmically with the class size. This work improves the sample complexity bound to$O\left(V_{\mathcal{C}^*} \log \left\vert\mathcal{C}^*\right\vert\right)$, where$\mathcal{C}^*$is the set of extreme points of the convex closure of$\mathcal{C}$and$V_{\mathcal{C}^*}$is the shadow-norm of this set. We show the tightness of our bound for the class of bounded Hilbert-Schmidt norm, scaling as$O\left(\log \left\vert\mathcal{C}^*\right\vert\right)$. Our approach is based on a new quantum empirical risk minimization (ERM) algorithm equipped with a shadow tomography method.
This paper studies quantum supervised learning for classical inference from quantum states. In this model, a learner has access to a set of labeled quantum samples as the training set. The objective is to find a quantum measurement that predicts the label of the unseen samples. The hardness of learning is measured via sample complexity under a quantum counterpart of the well-known probably approximately correct (PAC). Quantum sample complexity is expected to be higher than classical one, because of the measurement incompatibility and state collapse. Recent efforts showed that the sample complexity of learning a finite quantum concept classCscales asO(∣C∣). This is significantly higher than the classical sample complexity that grows logarithmically with the class size. This work improves the sample complexity bound toO(VC∗log∣C∗∣), whereC∗is the set of extreme points of the convex closure ofCandVC∗is the shadow-norm of this set. We show the tightness of our bound for the class of bounded Hilbert-Schmidt norm, scaling asO(log∣C∗∣). Our approach is based on a new quantum empirical risk minimization (ERM) algorithm equipped with a shadow tomography method.
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.
Over decades traditional information theory of source and channel coding advances toward learning and effective extraction of information from data. We propose to go one step further and offer a theoretical foundation for learning classical patterns from quantum data. However, there are several roadblocks to lay the groundwork for such a generalization. First, classical data must be replaced by a density operator over a Hilbert space. Hence, deviated from problems such as state tomography, our samples are i.i.d density operators. The second challenge is even more profound since we must realize that our only interaction with a quantum state is through a measurement which - due to no-cloning quantum postulate - loses information after measuring it. With this in mind, we present a quantum counterpart of the well-known probably approximately correct (PAC) framework. Based on that, we propose a quantum analogous of the Empirical Risk Minimization (ERM) algorithm for learning measurement hypothesis classes. Then, we establish upper bounds on the quantum sample complexity quantum concept classes.
We continue the study of the learnability of quantum measurement classes in the setting where the learner is given access only to prepared quantum states, aiming for necessary and sufficient conditions for PAC learnability, along with corresponding sample complexity bounds. In the quantum setting, in contrast with the classical probabilistically observed case, sampled states are perturbed when a quantum measurement is applied, according to the Born rule, so that distinct samples in the training data cannot be arbitrarily reused. We first probe the results from previous works on this setting. We show that the empirical risk defined in previous works and matching the definition in the classical theory can fail to satisfy the uniform convergence property enjoyed in the classical learning setting for classes that we can show to be PAC learnable. Moreover, we show that VC dimension generalization upper bounds in previous work are in many cases infinite, even for measurement classes defined on a finite-dimensional Hilbert space. We then show that, nonetheless, every measurement class defined on a finite-dimensional Hilbert space is PAC learnable via a modification of the ERM rule.
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.
A fundamental question in PAC learning is determining the number of labeled examples required to learn a concept class to a desired accuracy and confidence. In classical learning theory, this quantity is characterized by the VC-dimension, while several quantum generalizations have established analogous results when examples are provided in quantum superposition. In this work, we study a distinct quantum PAC-learning model in which concepts are functions acting on quantum states. We demonstrate that the VC-dimension, although still relevant, fails to fully capture the sample complexity of this model. To further characterize this setting, we develop a new lower bound on the required number of samples and establish an upper bound when the states in the domain are linearly independent. Remarkably, this upper bound has a form similar to the classical PAC-learning bound. We further examine a setting in which the learner receives more informative data and show that the limitations of the VC-dimension persist in this extended model.
We generalize the PAC (probably approximately correct) learning model to the quantum world by generalizing the concepts from classical functions to quantum processes, defining the problem of \emph{PAC learning quantum process}, and study its sample complexity. In the problem of PAC learning quantum process, we want to learn an ϵ-approximate of an unknown quantum process c∗ from a known finite concept class C with probability 1−δ using samples {(x1,c∗(x1)),(x2,c∗(x2)),…}, where {x1,x2,…} are computational basis states sampled from an unknown distribution D and {c∗(x1),c∗(x2),…} are the (possibly mixed) quantum states outputted by c∗. The special case of PAC-learning quantum process under constant input reduces to a natural problem which we named as approximate state discrimination, where we are given copies of an unknown quantum state c∗ from an known finite set C, and we want to learn with probability 1−δ an ϵ-approximate of c∗ with as few copies of c∗ as possible. We show that the problem of PAC learning quantum process can be solved with O(ϵ2log∣C∣+log(1/δ)) samples when the outputs are pure states and O(ϵ2log3∣C∣(log∣C∣+log(1/δ))) samples if the outputs can be mixed. Some implications of our results are that we can PAC-learn a polynomial sized quantum circuit in polynomial samples and approximate state discrimination can be solved in polynomial samples even when concept class size ∣C∣ is exponential in the number of qubits, an exponentially improvement over a full state tomography.