Quantum multivariate estimation and span program algorithms
Arjan Cornelissen
Abstract:
Over the past half century, the advent of the computer has increased our ability to perform computations tremendously. Consequently, we can now solve computational problems much more efficiently than ever before, the effects of which have revolutionized many aspects of society, ranging from how government keeps track of tax records, to how we use routing software to navigate to our destination.
However, over the last decade it has become increasingly apparent that the development of conventional computational hardware is reaching its physical limits. In an attempt to overcome this barrier, new research areas have focused on rethinking the very physical principles based on which we perform computations.
Quantum computing has emerged among the most promising of these new areas, and as such has experienced an explosive increase of interest over the past twenty years. Quantum computing loosely speaking encapsulates the idea of performing computations based on the principles of quantum mechanics, and as such describes a computational model that is fundamentally different from its conventional counterpart. The central question we address in this thesis is how powerful this new computational model is, i.e., we take computational problems and assess how efficiently they can be solved on a quantum computer. Oftentimes, we compare the obtained results to those in the conventional setting, to quantify the computational advantage that quantum computers provide us with on a theoretical level.
The thesis is divided in two parts. In Part I, we develop quantum algorithms that solve estimation problems. Specifically, we investigate three problems separately, namely mean estimation, state tomography, and partition function estimation. For the first two we generalize our findings to the multivariate setting, and we augment our results with matching lower bound proofs, providing a precise characterization of the problemsâ€™ quantum computational complexity. Surprisingly, we conclude that for the mean estimation problem quantum computers provide a computational advantage over conventional ones, if and only if the number of quantum samples used exceeds the dimension of the estimated quantity.
In Part II, we investigate the span program formalism, which is a framework for designing quantum algorithms. We provide a self-contained overview of the formalism with emphasis on visual and intuitive interpretations of the relevant objects. We elaborate on how the framework relates to the adversary bound, and simplify the exposition over previous works. Moreover, we arrive at three new results. First, we show how this framework can be used to turn classical decision trees into quantum algorithms. Second, we improve the best-known algorithm that evaluates approximate span programs. Finally, we show that quantum algorithms of a particular type correspond to span programs that preserve time-efficiency.