QM Research Seminar: Path-integral perspective on fault-tolerant topological quantum computation – from circuits to chomology

Speaker: Julio C. Magdalena de la Fuente (Freie Universität Berlin).

Abstract: 
​​​​​​​Topological quantum error correction encodes quantum information in the ground space of a topologically ordered lattice system or in the fusion space of topological defects. In this talk I will focus on the former. To use such codes for reliable quantum computation, one must design low-overhead circuits that protect and manipulate the encoded information in a fault-tolerant way.I will start with presenting simple protocols based on two-dimensional topological codes that are composed of 2D local gates. Representing these protocols as tensor networks local in a 3D spacetime lattice leads to a useful viewpoint. We identify a topological QEC circuit with an imaginary-time topological path integral of a topological gauge theory. Within this picture, both physical errors and non-trivial measurement outcomes appear as certain defects in the path integral and their combinatorial structure defines the associated classical decoding problem.I will go through the construction of logically non-trivial fault-tolerant gates using topological boundaries and domain walls between topological phases. In order to perform universal computation a non-Abelian state must be stabilized during the computation. The path-integral perspective allows to calculate the logical action and extract the resulting causality constraints on the classical decoder that is needed to make the protocols resilient to arbitrary errors. It also provides an efficient way to simulate the success of a given decoding strategy using Monte-Carlo sampling based on third-order cohomology invariants of the underlying spacetime complex. I want to introduce the necessary ingredients to construct and analyze such QEC protocols for their performance and argue about their reliable implementation in the presence of errors.