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Quantum algorithms and learning theory

Srinivasan Arunachalam

Published Apr 25, 2018• 1 citation
OpenAlex

Abstract

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.

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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.
Quantum computing has attracted a lot of attention in different research fields, such as mathematics, physics and computer science. Quantum algorithms can solve certain problems significantly faster than classical algorithms. There are many numerical problems, especially those arising from quantum systems, which are notoriously difficult to solve using classical computers, since the computational time required often scales exponentially with the size of the problem. However, quantum computers have the potential to solve these problems efficiently, which is also one of the founding ideas of the field of quantum computing. In this thesis, we explore five computational problems, designing innovative quantum algorithms and studying their computational complexity. First, we design an adiabatic quantum algorithm based on the Berry phases for the counting problem. For the running time, it is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result should be robust to decoherence and resilient to certain kinds of noise. Since the counting problem is the foundation of many other numerical problems, such as high-dimensional integration and path integration, our adiabatic algorithms can be directly generalized to these kinds of problems. In addition, we study the quantum PAC learning model, offering an improved lower bound on the query complexity. The lower bound is very close to the best lower bound on query complexity known for the classical PAC learning model. We also study the algorithms and the cost of simulating a system evolving under a given Hamiltonian. We consider high order splitting methods that are particularly applicable in quantum simulation and obtain bounds on the number of exponentials required. Moreover, we derive the optimal order of convergence given the required error bound. We compare our complexity estimates to previously known ones and show the resulting speedup. Furthermore, we consider randomized algorithms for Hamiltonian simulation. The evolution is simulated by a product of exponentials in a random sequence and random evolution times. Hence the final state of the system is approximated by a mixed quantum state. We provide a scheme to bound the error of the final quantum state in a randomized algorithm, and obtain randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement. We also apply Hamiltonian simulation in estimating the ground state energy of a multiparticle system, which is also known as the multivariate Sturm-Liouville eigenvalue problem. Since the cost of this problem grows exponentially with the number of particles using deterministic classical algorithms, it suffers from the curse of dimensionality. Quantum computers can vanquish the curse, and we exhibit a quantum algorithm whose total cost are linear in the number of particles.
This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.
We investigate learning models for implementing arbitrary unitary operations and perform quantum machine learning tasks. First, we discuss efficient constructions for building unitary transformations in U(d), thus providing upper bounds on the minimum time required to implement them. Second, we study the hardness of learning unitary operations in U(d) via gradient descent on the time parameters of an alternating operator sequence, and numerically find that, despite the nonconvexity of the loss landscape, gradient descent always converges to the target unitary when the sequence contains d2 or more parameters. Third, we introduce the first quantum boosting algorithm for converting a weak quantum learner into a strong accurate quantum learner for a Boolean concept class C; thereby achieving a quadratic quantum improvement over classical AdaBoost in terms of VC(C).
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.