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

Abstract

The color code is both an interesting example of an exactly solvable topologically ordered phase of matter and also among the most promising candidate models to realize faulttolerant quantum computation with minimal resource overhead. The contributions of this work are threefold. First of all, we build upon the abstract theory of boundaries and domain walls of topological phases of matter to comprehensively catalog the objects as they are realizable in color codes. Together with our classification we also provide lattice representations of these objects which include three new types of boundaries as well as a generating set for all 72 color code twist defects. Our work thus provides an explicit toy model that will help to better understand the abstract theory of domain walls. Secondly, building upon the established framework, we discover a number of interesting new applications of the cataloged objects for devising quantum information protocols. These include improved methods for performing quantum computations by code deformation, a new four-qubit error-detecting code, as well as families of new quantum errorcorrecting codes we call stellated color codes, which encode logical qubits at the same distance as the next best color code, but using approximately half the number of physical qubits. To the best of our knowledge, our new topological codes have the highest encoding rate of local stabilizer codes with boundedweight stabilizers in two dimensions. Finally, we show how the boundaries and twist defects of the color code are represented by multiple copies of other phases. Indeed, in addition to the well studied comparison between the color code and two copies of the surface code, we also compare the color code to two copies of the three-fermion model. In particular, we find that this analogy offers a very clear lens through which we can view the symmetries of the color code which gives rise to its multitude of domain walls.

A Hierarchy of Anyon Models Realised by Twists in Stacked Surface Codes

“…These anyons are all self-inverse and Abelian. Two anyons in a row rx gx bx ry gy by rz gz bz Table I: The nine non-trivial bosonic anyons of the 2d colour code arranged as in [6]. rx implies an X error on a red plaquette and so on.…”

“…The fact that models from the second level of the hierarchy can be realised in the 2d colour code is readily apparent from [6]. For example the twist B which exchanges the r and g columns of Table I has In general if we consider equivalence up to global phases then there are only two Rmatrices for k = 2:…”

“…Recall that the first of these matrices belongs to a model with four bosonic Abelian charges while the second belongs to a model with two bosonic and two fermionic charges. In the notation of Kesselring et al [6] the models containing four bosons are those associated with a twist from conjugacy class B while those containing two bosons and two fermions are associated with twists from conjugacy class C. The twists in conjugacy class G have only two localisable charges and correspond to models from the first level of the hierarchy.…”

“…In order to do this we require that if we map x i → x j then we must also map x i → x j and if we map x i → x i x j then we must map x j → x i x j , where x i can be either e i or m i and x i x i = i . In other words all symmetries of the model can be generated by the transformations which are simply the generators of all colour code symmetries generalised to act on a stack of more than two surface codes [6]. A simple way to obtain a twist corresponding to a β k anyon is simply to combine twists associated with symmetry (25) on k different levels.…”

“…A recent paper by M. Kesselring et al [6] fully categorises the twists of the 2d colour code, sorting them into nine conjugacy classes. * Electronic address: thomas.scruby.17@ucl.ac.uk…”