Abstract
The color code is both an interesting example of an exactly solved topologically ordered phase of matter and also among the most promising candidate models to realize fault-tolerant 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 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, we discover a number of interesting new applications of the cataloged objects for 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 error-correcting 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 bounded-weight 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.
Highlights
Topological phases of matter are of significant interest both from a condensed matter perspective [1] as well as for their potential application in fault-tolerant quantum computation [2,3,4,5,6,7,8,9]
Among the leading architectures that have been proposed for scalable quantum-information processing are topological quantum error-correcting codes. The physics of these phases is enriched further when their low-energy excitations are connected by symmetries, in which case they give rise to non-trivial domain walls and pointlike topological defects [10,11,12,13,14,15,16] that have been utilized in a number of instances [12, 17,18,19,20,21,22,23,24,25,26,27] to improve protocols for topological quantum computation and to reduce overhead requirements
We present our new family of stellated color codes, as well as our new four-qubit error-detecting code, in self-contained sections that can be understood with only prerequisite knowledge of the stabilizer formalism for the reader interested only in the quantum information applications of our work
Summary
Topological phases of matter are of significant interest both from a condensed matter perspective [1] as well as for their potential application in fault-tolerant quantum computation [2,3,4,5,6,7,8,9]. The color code model 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. In addition to representing the new objects we discover on two copies of the surface code, we explore a lesser known [69] connection between the color code and two copies of the three-fermion model [70]. D we present a new family of surface codes, namely, the stellated surface codes, which are related to the new color codes we present
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