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.
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 k nonzero Fourier coefficients. We give a learner with near-optimal O(k) sample complexity independent of the input length n. The learner uses only separate measurements of individual examples. This improves the previous 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) by combining known sparse-recovery and lower-bound results.
Arunachalam and de Wolf (2018) \cite{arunachalam2018optimal} showed that the sample complexity of quantum batch learning of boolean functions, in the realizable and agnostic settings, has the same form and order as the corresponding classical sample complexities. In this paper, we extend this, ostensibly surprising, message to batch multiclass learning, online boolean learning, and online multiclass learning. For our online learning results, we first consider an adaptive adversary variant of the classical model of Dawid and Tewari (2022) \cite{dawid2022learnability}. Then, we introduce the first (to the best of our knowledge) model of online learning with quantum examples.
Abstract The problem of learning Boolean linear functions from quantum examples w.r.t. the uniform distribution can be solved on a quantum computer using the Bernstein–Vazirani algorithm (Bernstein and Vazirani, in: Kosaraju (ed) Proceedings of the twenty-fifth annual ACM symposium on theory of computing, ACM, New York, 1993. 10.1145/167088.167097 ). A similar strategy can be applied in the case of noisy quantum training data, as was observed in Grilo et al. (Learning with errors is easy with quantum samples, 2017). However, extensions of these learning algorithms beyond the uniform distribution have not yet been studied. We employ the biased quantum Fourier transform introduced in Kanade et al. (Learning dnfs under product distributions via μ μ -biased quantum Fourier sampling, 2018) to develop efficient quantum algorithms for learning Boolean linear functions on n bits from quantum examples w.r.t. a biased product distribution. Our first procedure is applicable to any (except full) bias and requires O(ln(n)) O(ln(n)) quantum examples. The number of quantum examples used by our second algorithm is independent of n, but the strategy is applicable only for small bias. Moreover, we show that the second procedure is stable w.r.t. noisy training data and w.r.t. faulty quantum gates. This also enables us to solve a version of the learning problem in which the underlying distribution is not known in advance. Finally, we prove lower bounds on the classical and quantum sample complexities of the learning problem. Whereas classically, Ω(n) Ω(n) examples are necessary independently of the bias, we are able to establish a quantum sample complexity lower bound of Ω(ln(n)) Ω(ln(n)) only under an assumption of large bias. Nevertheless, this allows for a discussion of the performance of our suggested learning algorithms w.r.t. sample complexity. With our analysis, we contribute to a more quantitative understanding of the power and limitations of quantum training data for learning classical functions.
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/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)/nk with success probability greater than 1/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.
This dissertation explores results at the intersection of two important branches of theoretical computer science: quantum computation, which studies the power of devices based on quantum physical phenomena, and computational theory, which studies the foundations of machine algorithms. We refer to the area of research at this intersection as quantum computational theory. Like its classical counterpart, quantum computational theory aims to understand the computational and information theoretic requirements of problems and algorithms. However unlike the classical models, the models for quantum learners generally involve quantum sources of information and quantum gates employing quantum physical phenomena such as superposition and entanglement, which do not have classical equivalents.
Our results can be summarized as follows: (1) In Chapter 3, we study the information theoretic requirements of quantum learning, partition and Probably Approximately Correct (PAC) learning. First, we develop a new general quantum algorithm, resolving a conjecture by Hunziker et al. [HMP+03]. Next, we present positive and negative results towards the rather general problem of partition of which both exact learning and computing a binary property are special cases. Finally, we derive an improved lower bound on the number of quantum examples required for PAC learning. (2) In Chapter 4, we consider the problem of unions of high dimensional rectangles over the domain [b]n using membership queries and uniform quantum examples. These classes oral generalizations of classes of DNF formulae over {0,1}n that have been extensively studied in the theory literature. (3) In Chapter 5, we develop quantum algorithms for and testing juntas (Boolean functions that depend on a few variables) and sparse functions. These algorithms are based on the quantum subroutine due to Bshouty and Jackson [BJ99] that enables sampling from the Fourier spectrum.
Online learning is a framework for interfacing with environments whose structure is revealed through interactions; quantum information theory is a means of representing data and its processing consistent with modern understanding of physics. Their union is symbiotic---sequential processing is light on precious quantum resources, and quantum algorithms can surpass known bounds of classical algorithms---their combination being an area of independent interest. This thesis aims to exemplify the above point via five case studies on tomography, change detection, and function evaluation. In the process of deriving efficient tomography procedures, prior work has shown that quantum states can be learned in online adversarial environments. We extend this notion to subsets of positive semidefinite operators, including general quantum objects such as states, measurements, channels, strategies, co-strategies, and Gram matrices. We tailor a regularized follow-the-leader algorithm and achieve sublinear regret. As a byproduct, we show a generalization of Pinsker's inequality that accounts for differences in trace, its proof method more broadly applicable to general divergences. We further extend online learnability to objects in arbitrary physical theories. We derive mappings between Euclidean Jordan algebras and generalized probabilistic theories, using methods for the former to show that states in the latter are learnable. Specifically, we develop a projective version of the symmetric cone multiplicative weights update algorithm that achieves sublinear regret when learning states of generalized probabilistic theories. We consider the task of detecting changes in sequences of unknown quantum states with a constraint of no false positives, developing efficient and optimal online quantum algorithms. These include online algorithms with small amounts of quantum memory based on the swap test, and optimal algorithms projecting onto the symmetric subspace. We find circuit constructions for these algorithms, analyze their complexity, and show their equivalence under certain memory limitations. The proposed algorithms are applicable to quality control and anomaly detection in quantum devices, and may be of independent interest for identity testing and optimal cloning. We elaborate on the quality control aspect of changepoint methods by developing procedures for aiding the calibration of near-term quantum devices. These consist of algorithms for certification and changepoint detection in Hamiltonian dynamics, which enable continuous monitoring and trigger recalibration procedures. Using a cumulative sum procedure, we attain an asymptotically-optimal scaling of the average times to detection versus false positive, dependent primarily on Hamiltonian norm bounds. Lastly, we quantify the effect of memory limits on Boolean function evaluation of quantum sequences---a task applicable to distributed sensing and quantum reading of classical memories, among others. We show that minimum-error function evaluation in the memoryless regime is equivalent to the so-called pretty good measurement and hence shares its performance guarantee. Additionally, we characterize the precise set of functions on which this strategy performs optimally---affine functions. The above studies reveal insights about learnability as a universal property and the tradeoffs between performance and resource requirements in online settings, informing the choice of quantum settings best-suited for online learning. We conclude by summarizing these takeaways, along with listing remaining open questions.