This dissertation investigates the fundamental computational capabilities of quantum algorithms with respect to combinatorial optimization and distribution learning. We establish rigorous quantum advantages over classical algorithms for such problems under both widely accepted cryptographic assumptions and unconditionally. First, we demonstrate a super-polynomial quantum advantage for approximating solutions to combinatorial optimization problems, explicitly constructing advantage-bearing instances of Integer Linear Programming. Our proofs build upon approximation hardness results that instantiate a connection between the representation-independent hardness of learning and approximation hardness, as captured by Occam’s razor within computational learning theory. We show that the instances we construct can be efficiently approximated by quantum algorithms utilizing Shor’s factoring algorithm, yet remain classically intractable unless standard cryptographic assumptions fail. Second, we fill a previously open gap by proving a super-polynomial quantum advantage for learning the density function of probability distributions, complementing prior results established for generative distribution learning. We show that the hardness of density modeling can be established under weaker cryptographic assumptions involving pseudorandom functions. The quantum advantage for density modeling fundamentally arises from Shor’s algorithm for computing discrete logarithms. Moreover, we explicitly articulate known connections between Boolean function learning hardness and distribution learning hardness using established constructions. Third, addressing practical constraints of near-term quantum devices, we establish an unconditional quantum advantage for generative distribution learning using shallow, constant-depth quantum circuits. We find that shallow quantum circuits pose a superior class of hypotheses compared to classical counterparts. This result underscores the potential of near-term quantum hardware by highlighting the unique computational capabilities enabled by quantum entanglement and non-local correlations, offered even by constrained devices. Collectively, these contributions deepen our understanding of quantum computational advantages for combinatorial optimization and distribution learning, clarifying their foundations and pointing toward promising avenues for future exploration.
Combinatorial optimization - a field of research addressing problems that feature strongly in a wealth of scientific and industrial contexts - has been identified as one of the core potential fields of applicability of quantum computers. It is still unclear, however, to what extent quantum algorithms can actually outperform classical algorithms for this type of problems. In this work, by resorting to computational learning theory and cryptographic notions, we prove that quantum computers feature an in-principle super-polynomial advantage over classical computers in approximating solutions to combinatorial optimization problems. Specifically, building on seminal work by Kearns and Valiant and introducing a new reduction, we identify special types of problems that are hard for classical computers to approximate up to polynomial factors. At the same time, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The core of the quantum advantage discovered in this work is ultimately borrowed from Shor's quantum algorithm for factoring. Concretely, we prove a super-polynomial advantage for approximating special instances of the so-called integer programming problem. In doing so, we provide an explicit end-to-end construction for advantage bearing instances. This result shows that quantum devices have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms. Our results also give clear guidance on how to construct such advantage-bearing problem instances.
It is unclear to what extent quantum algorithms can outperform classical algorithms for problems of combinatorial optimization. In this work, by resorting to computational learning theory and cryptographic notions, we give a fully constructive proof that quantum computers feature a super-polynomial advantage over classical computers in approximating combinatorial optimization problems. Specifically, by building on seminal work by Kearns and Valiant, we provide special instances that are hard for classical computers to approximate up to polynomial factors. Simultaneously, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The quantum advantage in this work is ultimately borrowed from Shor's quantum algorithm for factoring. We introduce an explicit and comprehensive end-to-end construction for the advantage bearing instances. For these instances, quantum computers have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms.
The most general examples of quantum learning advantages involve data labeled by cryptographic or intrinsically quantum functions, where classical learners are limited by the infeasibility of evaluating the labeling functions using polynomial-sized classical circuits. While broad in scope, such results reveal little about advantages arising from the learning process itself. In cryptographic settings, further insight is possible via random-generatability - the ability to classically generate labeled data - enabling hardness proofs for identification tasks, where the goal is to identify the labeling function from a dataset, even when evaluation is classically intractable. These tasks are particularly relevant in quantum contexts, including Hamiltonian learning and identifying physically meaningful order parameters. However, for quantum functions, random-generatability is conjectured not to hold, leaving no known identification advantages in genuinely quantum regimes. In this work, we give the first proofs of quantum identification learning advantages under standard complexity assumptions. We confirm that quantum-hard functions are not random-generatable unless BQP is contained in the second level of the polynomial hierarchy, ruling out cryptographic-style data generation strategies. We then introduce a new approach: we show that verifiable identification - solving the identification task for valid datasets while rejecting invalid ones - is classically hard for quantum labeling functions unless BQP is in the polynomial hierarchy. Finally, we show that, for a broad class of tasks, solving the identification problem implies verifiable identification in the polynomial hierarchy. This yields our main result: a natural class of quantum identification tasks solvable by quantum learners but hard for classical learners unless BQP is in the polynomial hierarchy.
An overarching milestone of quantum machine learning (QML) is to demonstrate the advantage of QML over all possible classical learning methods in accelerating a common type of learning task, as represented by supervised learning from classical data. However, the provable advantages of QML in supervised learning have been known so far only for the learning tasks designed for using the advantage of specific quantum algorithms, i.e., Shor’s algorithms. Here, we explicitly construct an unprecedentedly broader family of supervised learning tasks with classical data to offer the provable advantage of QML based on general quantum computational advantages, progressing beyond Shor’s algorithms. Our learning task is feasibly achievable by executing a general class of functions that can be computed efficiently in polynomial time for a large fraction of inputs by arbitrary quantum algorithms, but not by any classical algorithm. We prove the hardness of achieving this learning task for any possible polynomial-time classical learning method. We also clarify protocols for preparing the classical data to demonstrate this learning task in experiments. These results provide a way to exploit a variety of quantum advantages in computing functions for the realization of provably advantageous QML.
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 and
TC0 under the uniform distribution. Our first result concerns the concept class
TC0 (resp.,
AC0), the class of constant-depth, polynomial-sized circuits with unbounded fan-in majority gates (resp.,
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) problem, then there exists no (quasi-)polynomial-time quantum learning algorithm for
TC0; and if there exists no
2O(d1/η)-time quantum algorithm to solve
RLWE with dimension
d=O(polylogn) (for every constant
η>2), then there exists no
O(nlogνn)-time quantum learning algorithm for
poly(n)-sized
AC0 circuits (for a constant
ν>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 in the PAC setting. Our second result shows that if there exists no quantum polynomial-time algorithm for the
LWE problem, then there exists no polynomial-time quantum-PAC learning algorithm for the class
TC20, i.e., depth-2
TC0 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.
Abstract Quantum computers are believed to bring computational advantages in simulating quantum many-body systems. However, recent works have shown that classical machine learning algorithms are able to predict numerous properties of quantum systems with classical data. Despite examples of learning tasks with provable quantum advantages being proposed, they all involve cryptographic functions and do not represent any physical scenarios encountered in laboratory settings. In this paper, we prove quantum advantages for the physically relevant task of learning quantum observables from classical (measured-out) data. We consider two types of observables: first, we prove a learning advantage for linear combinations of Pauli strings, then we extend our results to a broader case of unitarily parametrized observables. For each case, we delineate sharp boundaries separating physically relevant tasks that admit efficient classical learning from those for which a quantum computer remains necessary for data analysis. Unlike previous works, our classical hardness results rely only on the weaker assumption that
BQP B Q P hard processes cannot be simulated by polynomial-size classical circuits, and we also provide a nontrivial quantum learning algorithm. Our results clarify when quantum resources are useful for learning problems in quantum many-body physics, and suggest practical directions in which quantum learning improvements may emerge.