Tensor networks with a twist: Anyon-permuting domain walls and defects in projected entangled pair states

Abstract

We study the realization of anyon-permuting symmetries of topological phases on the lattice using tensor networks. Working on the virtual level of a projected entangled pair state, we find matrix product operators (MPOs) that realize all unitary topological symmetries for the toric and color codes. These operators act as domain walls that enact the symmetry transformation on anyons as they cross. By considering open boundary conditions for these domain wall MPOs, we show how to introduce symmetry twists and defect lines into the state. I. REVIEW: TOPOLOGICAL ORDER AND PEPSIn this section, we review some key concepts, notation and conventions required for the remainder of the paper.We begin with a discussion of topologically ordered phases, the kind of symmetries they support and the connections to fault-tolerant quantum computation. This motivates the discussion of locality preserving APS actions, domain walls, and defects. These topics form the primary objects of study in this paper.We introduce PEPS, the main tool used in this work, and a streamlined notation we use throughout the pa-arXiv:1708.08930v2 [quant-ph]

Fusing binary interface defects in topological phases: The Z/pZ case

“…For the special case q = l = m = 1, this recovers the known fusion rule of 'twists' with anyons [7,25,27]…”

Locality-preserving logical operators in topological stabilizer codes

“…Understanding the set of logical operators that can be implemented by braiding twists in a code is closely tied to this work, since twists exist as endpoints (or more generally, the boundaries) of domain walls [34,37]. Specifically, we may find a twist corresponding to the endpoint of a domain wall in a two dimensional code by allowing the domain wall to terminate in the bulk [38]. This necessitates altering the Hamiltonian at these endpoints so that all of the terms of the Hamiltonian continue to commute.…”

The boundaries and twist defects of the color code and their applications to topological quantum computation

“…Beyond this, the symmetries and twist defects with the color code model have been expressed using a tensor network formalism in Ref. [46]. This is complemented by Ref.…”

“…The color code model evidently has a very rich structure [45,46] that we may be able to exploit further to discover better and more resource efficient protocols to perform quantum-computational operations in a fault-tolerant manner. It is the primary goal of the present manuscript to systematically explore the different objects that can be realized with modifications to the color code lattice through a study of its underlying topological excitations.…”

“…Finally, we study how the new objects we introduce to the color code lattice can be represented on multiple copies of other topological phases through unfolding [26,33,46,[66][67][68]. In particular, in addition to representing the new objects we discover on two copies of the surface code, we also explore a lesser known [69] connection between the color code and two copies of the three-fermion model [70].…”