Algorithms, Security and Complexity for Quantum Computers

As progress is made in implementing quantum computers, the question is looming: What will we do with them?This proposal is concerned with the theoretical computer science aspects of this question. Part of this question is concerned with quantum algorithms (WP1). We know of several examples of quantum algorithms with large speedups over the best known classical algorithms, such as Shor's poly-time integer factorization algorithm. While this is evidence that quantum computers will be useful once built, it does not tell us what quantum computers will be used for in practice (probably not much factoring). To ensure that quantum computer users are best able to make use of them, we will focus on developing general techniques for the design of quantum algorithms that can be easily applied by subject-matter experts in different fields to the problems that interest them.We will also consider the constraints of early quantum computers in our algorithm design. First, we would like to understand how the limited memory of early quantum computers will impact what they can do. Some of the most important techniques for designing quantum algorithms are already well-suited to the study of space-bounded computation, and we will generalize and improve these in WP1. To complement this, we will study lower bounds and complexity (WP2), focusing on space-bounded complexity classes, which have many relationships with other complexity classes. Second, since most early users will have to delegate their quantum computations, we would like to understand which quantum algorithms can still be used in various delegated or multiparty settings where some type of security is a consideration. We take the novel approach of using a quantum algorithmic model called span programs to design secure quantum computing protocols (WP3). It turns out that space-bounded models and secure quantum computation are very much related, and understanding this relationship is what ties this proposal together.