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On quantum algorithms and limitations for convex optimization and lattice problems

Y. Chen

Published Jan 1, 2024• 0 citations
OpenAlex

Abstract

Optimization is the process of selecting the best option from all possibilities, and problems related to finding the best option among many options are called optimization problems. A few examples of such problems are route picking, partial loading, crew scheduling, and portfolio optimization, and those problems naturally appear everywhere in society and in our daily lives. Most optimization problems could be solved by listing all possibilities and then searching for the best one, but we aim to find a systematic strategy (known as an algorithm) that finds the optimal solution in the least amount of time, as brute-force searching is often too time-consuming. That is, we would like to find optimal algorithms for finding optimal solutions. Quantum physics provides a more accurate explanation and prediction of nature in many cases than classical physics. Therefore, a quantum computer has the potential to be more powerful and useful than a classical one for solving optimization problems, though the extent of this advantage remains an active area of research. Moreover, designing algorithms for a quantum computer requires fundamentally different ideas and concepts than for a classical computer. In this thesis, we explore the uses of quantum computers in solving several fundamental optimization problems. We specifically consider solving linear regression, finding the top eigenvectors of a matrix, and finding the shortest nonzero vector of a lattice. More precisely, we provide near-optimal quantum algorithms that are asymptotically better than the best-possible classical algorithms for linear regression with $\ell_1$-norm constraint (known as Lasso) and for approximating the top eigenvectors of a matrix. Our algorithms for finding the shortest nonzero vector of a lattice are asymptotically better than the best known classical algorithms. On the other hand, there are some problems for which no useful computational advantage is possible, even when we use quantum computers. This situation, however, can sometimes be beneficial if we do not want quantum computers to solve specific problems fast - say, problems relevant to post-quantum cryptography. In such a situation, we would like evidence demonstrating the difficulty of solving these problems on a quantum computer. To this end, we propose variants of Boolean satisfiability problems that are believed to not be more efficiently solvable on a quantum computer, and then via reductions use those (conjectured) quantum lower bounds to study the quantum limitations of several popular optimization problems like lattice problems, quantum strong simulation, and hitting set problems.

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Optimization is the process of selecting the best option from all possibilities, and problems related to finding the best option among many options are called optimization problems. A few examples of such problems are route picking, partial loading, crew scheduling, and portfolio optimization, and those problems naturally appear everywhere in society and in our daily lives. Most optimization problems could be solved by listing all possibilities and then searching for the best one, but we aim to find a systematic strategy (known as an algorithm) that finds the optimal solution in the least amount of time, as brute-force searching is often too time-consuming. That is, we would like to find optimal algorithms for finding optimal solutions. Quantum physics provides a more accurate explanation and prediction of nature in many cases than classical physics. Therefore, a quantum computer has the potential to be more powerful and useful than a classical one for solving optimization problems, though the extent of this advantage remains an active area of research. Moreover, designing algorithms for a quantum computer requires fundamentally different ideas and concepts than for a classical computer. In this thesis, we explore the uses of quantum computers in solving several fundamental optimization problems. We specifically consider solving linear regression, finding the top eigenvectors of a matrix, and finding the shortest nonzero vector of a lattice. More precisely, we provide near-optimal quantum algorithms that are asymptotically better than the best-possible classical algorithms for linear regression with ℓ1\ell_1ℓ1​-norm constraint (known as Lasso) and for approximating the top eigenvectors of a matrix. Our algorithms for finding the shortest nonzero vector of a lattice are asymptotically better than the best known classical algorithms. On the other hand, there are some problems for which no useful computational advantage is possible, even when we use quantum computers. This situation, however, can sometimes be beneficial if we do not want quantum computers to solve specific problems fast - say, problems relevant to post-quantum cryptography. In such a situation, we would like evidence demonstrating the difficulty of solving these problems on a quantum computer. To this end, we propose variants of Boolean satisfiability problems that are believed to not be more efficiently solvable on a quantum computer, and then via reductions use those (conjectured) quantum lower bounds to study the quantum limitations of several popular optimization problems like lattice problems, quantum strong simulation, and hitting set problems.
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.
Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector θ∈Rdθ\in\mathbb{R}^dθ∈Rd of coefficients is constrained in either ℓ1\ell_1ℓ1​-norm (for Lasso) or in ℓ2\ell_2ℓ2​-norm (for Ridge). We study the complexity of quantum algorithms for finding ε\varepsilonε-minimizers for these minimization problems. We show that for Lasso we can get a quadratic quantum speedup in terms of ddd by speeding up the cost-per-iteration of the Frank-Wolfe algorithm, while for Ridge the best quantum algorithms are linear in ddd, as are the best classical algorithms. As a byproduct of our quantum lower bound for Lasso, we also prove the first classical lower bound for Lasso that is tight up to polylog-factors.
Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector θ ∈ R^d of coefficients is constrained in either ℓ1-norm (for Lasso) or in ℓ2-norm (for Ridge). We study the complexity of quantum algorithms for finding ε-minimizers for these minimization problems. We show that for Lasso we can get a quadratic quantum speedup in terms of d by speeding up the cost-per-iteration of the Frank-Wolfe algorithm, while for Ridge the best quantum algorithms are linear in d, as are the best classical algorithms. As a byproduct of our quantum lower bound for Lasso, we also prove the first classical lower bound for Lasso that is tight up to polylog-factors.
Machine learning is a branch of Artificial Intelligence that seeks to make machines learn from data. It is being applied for solving real world problems with huge amount of data. Though, Machine Learning is receiving wide acceptance, however, execution time is one of the major concerns in practical implementations of Machine Learning techniques. It largely comprises of a set of techniques that trains a model by reducing the error between the desired or actual outcome and an estimated or predicted outcome, which is often called as loss function. Thus, training in machine learning techniques often requires solving a difficult optimization problem, which is the most expensive step in the entire model-building process and its applications. One of the possible solutions in near future for reducing execution time of training process in Machine learning techniques is to implement them on quantum computers instead of classical computers. It is conjectured that quantum computers may be exponentially faster than classical computers for solving problems which involve matrix operations. Some of the machine learning techniques like support vector machines make extensive use of matrices, which can be made faster by implementing them on quantum computers. However, their efficient implementation is non-trivial and requires existence of quantum memories. Thus, another possible solution in near term is to use a hybrid of Classical Quantum approach, where a machine learning model is implemented in classical computer but the optimization of loss function during training is performed on quantum computer instead of classical computer. Several Quantum optimization algorithms have been proposed in recent years, which can be classified as gradient based and gradient free optimization techniques. Gradient based techniques require the nature of optimization problem being solved to be convex, continuous and differentiable otherwise if the problem is non-convex then they can find local optima only whereas gradient free optimization techniques work well even with non-continuous, non-linear and nonconvex optimization problems. This chapter discusses a global optimization technique based on Adiabatic Quantum Computation (AQC) to solve minimization of loss function without any restriction on its structure and the underlying model, which is being learned. Further, it is also shown that in the proposed framework, AQC based approach would be superior to circuit-based approach in solving global optimization problems.
All the data stored and processed by our computers is encoded as sequences of zeros and ones, called bits. Quantum computers offer an alternative to this traditional way of encoding and manipulating information. The information in a collection of qubits can be a combination, also called a superposition, over many possible sequences of zeros and ones. Where classical computation consists of flipping zeros and ones, quantum computation consists of manipulating high-dimensional vectors called quantum states. Researchers have found ways to use this to engineer quantum algorithms that solve some tasks much faster than any classical computer would. These are called quantum advantages, and are proven for specific tasks, such as breaking some encryption schemes and solving certain large linear systems. The goal of this thesis was to find new tasks for which quantum computers provably have an advantage over classical computers. In this thesis, I present two such advantages and associated results. The first one revolves around a physically relevant learning task. In other words, I explore how quantum computers can be used to help solve specialized machine learning tasks. The second one addresses the simulation of classical systems. One of the first high-potential and natural uses of quantum computers was the simulation of quantum systems. A lot of real-world problems, however, are ruled by classical physics. It turns out that quantum computers are well suited for the simulation of some exponentially large classical systems. In this thesis, I add a new problem to this list.
Recently, several researchers proposed portfolio optimization as a potential use case for quantum optimization. However, the literature is lacking an extensive benchmark quantifying the potential of quantum computers for portfolio optimization. In this work, we contribute to closing this gap. We provide a computational study, comparing quantum approaches against state-of-the-art classical methods on a meaningful, real-world instance set. In particular, we compare quantum annealing and the quantum approximate optimization algorithm against classical mixed-integer programming, simulated annealing, steepest descent local search, tabu search and a problem-tailored heuristic. We consider a variance-minimizing variant of portfolio optimization which we show to be more difficult to solve for classical optimizers than return-maximizing or multi-objective formulations. Our benchmark data set comprises 270 instances with up to 1,000 assets from actual stock data. Due to hardware limitation, quantum methods could only be tested for instances with at most 30 assets. The results show that all instances can be solved to proven optimality by mixed-integer programming in the order of seconds. Moreover, the problem-tailored heuristic consistently outperforms quantum approaches in terms of solution quality for fixed runtime. Thus, we conclude that there is only very limited room for a potential quantum advantage for the considered variant of portfolio optimization.