Excursions at the Interface of Topological Phases of Matter and Quantum Error Correction
Dissertation Committee Chair: Professor Maissam Barkeshli
Professor Sankar Das Sarma
Professor Jay Deep Sau
Professor Michael Gullans
Professor Mohammad Hafezi
Abstract: Topological quantum error-correcting codes are a family of stabilizer codes that are built using a lattice of qubits. The stabilizers of the code are local, and logical information is encoded in the non-local degrees of freedom. From the condensed matter perspective, their code space corresponds to the ground state subspace of a local Hamiltonian belonging to a non-trivial topological phase of matter. Conversely, one can use fixed point Hamiltonian of a topological phase to define a topological quantum error-correcting code.
This close connection has motivated numerous studies which utilize insights from one viewpoint to address questions in the other. This thesis further explores the possibilities in this direction. In the first two parts, we present novel schemes to implement logical gates, which are motivated by the condensed matter perspective. In the third part, we show how the quantum error correction perspective could be used to realize robust topological entanglement phases in monitored random quantum circuits. And lastly, we explore the possibility of extending this connection beyond topological quantum error-correcting codes. In particular, we introduce an order parameter for detecting k-local non-trivial states, which can be thought of as a generalization of topological states that includes codewords of any quantum error-correcting code.
Location: ATL 4402