Gapped boundaries, group cohomology and fault-tolerant logical gates

TL;DR

This paper explores the relationship between gapped boundaries in topological phases, SPT phases, and fault-tolerant logical gates, using cocycle functions to construct models and gates with specific braiding and condensation properties.

Contribution

It introduces a method to construct gapped boundaries and fault-tolerant logical gates in quantum double models via $d$-cocycles, linking topological phases to quantum error correction.

Findings

Gapped boundaries support fluctuating charges with non-trivial braiding statistics.

Logical gates are constructed belonging to the $d$th Clifford hierarchy, outside the $(d-1)$th level.

Excitations with trivial braiding can condense into gapped boundaries.

Abstract

This paper attempts to establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error-correcting codes. We begin by presenting constructions of gapped boundaries for the $d$-dimensional quantum double model by using $d$-cocycles functions ($d\geq 2$). We point out that the system supports $m$-dimensional excitations ($m<d$), which we shall call fluctuating charges, that are superpositions of point-like electric charges characterized by $m$-dimensional bosonic SPT wavefunctions. There exist gapped boundaries where electric charges or magnetic fluxes may not condense by themselves, but may condense only when accompanied by fluctuating charges. Magnetic fluxes and codimension-$2$ fluctuating charges exhibit non-trivial multi-excitation braiding statistics, involving more than two excitations. The statistical angle can be computed by taking slant products of underlying cocycle functions sequentially. We find that excitations that may condense into a gapped boundary can be characterized by trivial multi-excitation braiding statistics, generalizing the notion of the Lagrangian subgroup. As an application, we construct fault-tolerantly implementable logical gates for the $d$-dimensional quantum double model by using $d$-cocycle functions. Namely, corresponding logical gates belong to the $d$th level of the Clifford hierarchy, but are outside of the $(d-1)$th level, if cocycle functions have non-trivial sequences of slant products.