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

Markus S. Kesselring1, Fernando Pastawski1, Jens Eisert1, and Benjamin J. Brown2,3

1Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
2Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia
3Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark

### 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.

A scalable quantum computer must be able to function even if its individual components may become subject to errors. To this end, we encode the logical qubits used for quantum computing in quantum error-correcting codes which are composed of many physical qubits. Encoded qubits can maintain their coherence arbitrarily well provided the physical qubits of the quantum error-correcting code experience errors at a rate that is suitably low. To reduce the demands on laboratory resources, we seek to find designs for quantum error-correcting codes with low dimensionality that use as few physical qubits as possible.

Promising models for fault-tolerant quantum computation are based on topological phases of matter, and of particular interest is the color code model, due to its numerous symmetries. By exploring the remarkable quasiparticle excitations of the color code, we describe and classify all of its boundaries and topological defects. These various features that we examine can be used to design better quantum error-correcting codes and, moreover, offer new ways for us to perform logical gates on encoded quantum information. Our work reveals new two-dimensional codes with better encoding rates than other proposals with the same dimensionality. It gives further substance to the idea that a deeper examination of condensed matter theory will lead to improved designs of quantum error-correcting codes in the future.

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[6] Michael E Beverland, Benjamin J Brown, Michael J Kastoryano, and Quentin Marolleau, "The role of entropy in topological quantum error correction", Journal of Statistical Mechanics: Theory and Experiment 2019 7, 073404 (2019).

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[11] Jacob C Bridgeman and Daniel Barter, "Computing defects associated to bounded domain wall structures: the $\mathbb{Z}/p\mathbb{Z}$ case", Journal of Physics A: Mathematical and Theoretical 53 23, 235206 (2020).

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[21] Tsuf Lichtman, Ryan Thorngren, Netanel H. Lindner, Ady Stern, and Erez Berg, "Bulk Anyons as Edge Symmetries: Boundary Phase Diagrams of Topologically Ordered States", arXiv:2003.04328.

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