Low-Overhead Quantum Fault Tolerance

Fault tolerance is an essential property of future quantum computers where a quantum computation is mapped to a new one that is resilient to operational errors. This resilience comes at an additional time and space overhead. In this thesis, we study schemes that asymptotically reduce the overhead of quantum fault tolerance in various models of computation. We construct a scheme for fault-tolerant quantum computation that requires nearly-logarithmic spacetime overhead assuming access to noiseless classical computation. In the second half, we construct a quantum memory with a memory threshold that uses local quantum operations on a 2D lattice.