This paper surveys quantum learning theory: the theoretical aspects of machine learning using quantum computers. We describe the main results known for three models of learning: exact learning from membership queries, and Probably Approximately Correct (PAC) and agnostic learning from classical or quantum examples.
This thesis studies strengths and weaknesses of quantum computers. In the first part we present three contributions to quantum algorithms. 1) consider a search space of N elements. One of these elements is "marked" and our goal is to find this. We describe a quantum algorithm to solve this problem using essentially sqrt{N} queries and other operations, improving over the gate count of Grover's algorithm. 2) We give a succinct characterization of quantum algorithms in terms of polynomials, and develop a new technique for showing upper and lower bounds on quantum query complexity based on this. 3) One generic technique used to compute the minimum of a given function is "gradient descent". We present a quantum gradient-calculation algorithm and a gradient-optimization algorithm that is quadratically faster than classical algorithms. In the second part of this thesis we look at quantum learning theory. 1) We survey quantum learning theory, describing the main results known for three models of learning, using classical as well as quantum data: exact learning from membership queries, the probably approximately correct (PAC) learning model, and the agnostic learning model. 2) We then consider if a quantum learner can PAC-learn a given concept class from fewer quantum examples. We give a negative answer by showing that quantum examples are not more powerful than classical random examples, both for PAC learning and for agnostic learning.
This paper surveys various results in the field of Quantum Learning theory, specifically focusing on learning quantum-encoded classical concepts in the Probably Approximately Correct (PAC) framework. The cornerstone of this work is the emphasis on query, sample, and time complexity separations between classical and quantum learning that emerge under learning with query access to different labeling oracles. This paper aims to consolidate all known results in the area under the above umbrella and underscore the limits of our understanding by leaving the reader with 23 open problems.
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.
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.
Quantum machine learning combines quantum computing with machine learning to solve complex computational problems more efficiently than classical approaches. This survey provides an introduction to the foundations, algorithms, frameworks, data and applications of quantum machine learning, serving as a resource for researchers and practitioners. We begin by reviewing existing surveys to identify gaps that this work addresses, followed by a detailed discussion of the foundational principles of quantum mechanics and machine learning essential for quantum machine learning. Key algorithms are examined, highlighting their mechanisms, advantages, and applications across various domains. Current frameworks and platforms for implementing quantum machine learning algorithms are explored, emphasizing their unique features and suitability for different contexts. Existing quantum datasets for practical usage are also reported and commented on. This survey also reviews over 135 articles, categorized into theoretical and practical contributions, to identify key advances, limitations, and application areas within quantum machine learning. Critical challenges such as hardware limitations, error rates, and scalability are analyzed to detect the obstacles that must be addressed for practical deployment. By synthesizing these elements into a structured overview, this survey aims at serving as both an introduction and a guide for advancing research and development in this disruptive field.