Topics in quantum information theory and quantum many-body physics

Abstract

In this thesis we present two results relating to the intersection between quantum information theory and quantum many-body physics. The first pertains to quantum algorithms, where few computational problems are believed to exhibit exponential separation between quantum and classical performance. For those that are, natural generalizations remain elusive. One speedup that has especially resisted generalization is the use of quantum walks to traverse the welded tree graph, due to Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman. We show how to generalize this to a large class of hierarchical graphs in which the vertices are grouped into “supervertices” that are arranged according to a d-dimensional lattice. Supervertices can have different sizes, and edges between supervertices correspond to random connections between their constituent vertices. The traversal time of quantum walks on these graphs are related to (a) the existence of small subspaces within which the quantum walk evolves and (b) the localization properties of the quantum walk within these subspaces. We find examples of hierarchical graphs that yield provable speedups over classical algorithms ranging from superpolynomial to exponential, depending on the underlying dimension and the random graph model. We also discuss how to relax criterion (a) to the existence of a small and approximate subspace by using techniques from graph sparsification. The second result pertains to fault-tolerant quantum memories. Storing a qubit in a noisy environment is crucial for developing full-scale quantum computers. While constructions of fault-tolerant quantum memories exist, they often assume that quantum operations are not local and assisting classical computation operates instantaneously and noislessly. In particular, constructing a topological quantum memory below four dimensions with local quantum and classical operations that is fault-tolerant under both quantum and classical noise is an open problem. We construct a local quantum memory for the 2D toric code using ideas from the classical cellular automata of Tsirelson and Gács. Our memory preserves a logical state for exponential time in the presence of both classical and quantum noise below a constant noise rate. While our 2D quantum memory is built from operations that depend on space and time, we construct a fault-tolerant quantum memory in 3D using stacks of 2D toric codes that can be built with time-independent operations.