# Topological Quantum Computing in Multiple Surface Codes

This is a Perspective on "A Hierarchy of Anyon Models Realised by Twists in Stacked Surface Codes" by T. R. Scruby and D. E. Browne, published in Quantum 4, 251 (2020).

By Paul Webster (Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia).

Continued functionality in the event of an error in one or more components — referred to as fault tolerance — is integral to an effective system. For quantum computing, fault tolerance is especially pressing, since all present and foreseeable architectures are afflicted by significant noise, which makes errors on a subset of the qubits inevitable over the course of running any useful quantum algorithm. In this context, topological quantum computing — in which quantum logic gates are implemented by braiding well-separated non-abelian anyons (an exotic type of quasiparticle) — has long attracted attention [1]. Its appeal is that its topological structure means that local errors have a trivial effect on the computation, and so it is naturally fault-tolerant. More recently, it has been discovered that the effects of non-abelian anyons can be reproduced even in models that only allow for more trivial quasiparticles by introducing and braiding defects [2]. This offers the promise of an approach to topological quantum computing without the challenges of realising highly exotic, non-abelian phases of matter.

Scruby and Browne’s work specifically explores the potential of topological quantum computation by braiding defects referred to as twists in a stack of surface codes. This is a particularly relevant problem since surface codes have attracted extensive attention for allowing impressive error protection properties (specifically, a high error-correcting threshold) while also requiring only local, low-weight entanglement of physical qubits and being flexible to a range of qubit architectures [3]. Finding the most efficient and feasible way to encode information and implement computations in surface codes is thus an extremely important question in the field of quantum computing. More generally, it is known that the much larger class of two-dimensional topological stabiliser codes, which includes other codes of interest such as the colour code, is equivalent to copies of surface codes and so this work has very broad applicability [4].

The main result builds on work by Bombin [2] — showing that the only non-trivial type of twist defect admitted by a surface code exhibits braiding properties equivalent to Ising anyons [2] — to analyse twist defects in stacks of multiple surface codes [2]. This is of great interest, since it allows for twist defects that entangle different surface codes, and so the range of twist defects grows significantly as more surface codes are added. For example, it is known that the colour code (equivalent to two surface codes) admits 72 twist defects [5]. The authors define a hierarchy of non-abelian anyon models (the extended Ising hierarchy) such that the $k$th level corresponds to the braiding properties of twist defects in a stack of $k$ surface codes. They identify the logic gates implied by these statistics, and thus provide the most complete classification to date of the set of logic gates implementable by braiding twist defects in two-dimensional topological stabiliser codes.

In addition to the technical result, however, this paper will be valuable to the interested reader for its clarity and perspective. It offers a highly accessible introduction to the mathematics of anyon models and their relationship with logic gates and twist defects. This is particularly valuable since the approach taken to analysing twist defects through anyon models provides a complementary perspective to that common in other recent works which focus on the action of braiding on logical Pauli operators. While the latter approach has proven effective for understanding a range of types of defects and their relationships [6] and for higher dimensional models [7,8], the former approach is especially promising for generalisation to models beyond topological stabiliser codes, especially those without transversal logical Pauli operators. This paper is thus especially highly recommended to readers with a background in topological quantum error correcting codes who wish to broaden their perspective through a better understanding of the mathematics of non-abelian anyon models and its relationship to topological defects.

### ► References

[1] M. H. Freedman, A. Kitaev, M. J. Larsen and Z. Wang, Topological quantum computation'', Bull. Amer. Math. Soc. , 40, 31–38, 2003. https:/​/​doi.org/​10.1090/​S0273-0979-02-00964-3.
https:/​/​doi.org/​10.1090/​S0273-0979-02-00964-3

[2] H. Bombin, Topological Order with a Twist: Ising Anyons from an Abelian Model'', Phys. Rev. Lett., 105, 030403, 2010. https:/​/​doi.org/​10.1103/​PhysRevLett.105.030403.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.030403

[3] A. G. Fowler, M. Mariantoni, J. M. Martinis and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation'', Phys. Rev. A, 86, 032324, 2012. https:/​/​doi.org/​10.1103/​PhysRevA.86.032324.
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[4] H. Bombin, G. Duclos-Cianci and D. Poulin, Universal topological phase of two-dimensional stabilizer codes'', New J. Phys, 14, 073048, 2012. https:/​/​doi.org/​10.1088/​1367-2630/​14/​7/​073048.
https:/​/​doi.org/​10.1088/​1367-2630/​14/​7/​073048

[5] M. S. Kesselring, F. Pastawski, J. Eisert and B. J. Brown, The boundaries and twist defects of the color code and their applications to topological quantum computation'', Quantum, 2, 101, 2018. https:/​/​doi.org/​10.22331/​q-2018-10-19-101.
https:/​/​doi.org/​10.22331/​q-2018-10-19-101

[6] B. J. Brown, K. Laubscher, M. S. Kesselring, J. R. Wootton, Poking holes and cutting corners to achieve Clifford gates with the surface code'', Phys. Rev. X, 7, 021029, 2017. https:/​/​doi.org/​10.1103/​PhysRevX.7.021029.
https:/​/​doi.org/​10.1103/​PhysRevX.7.021029

[7] P. Webster and S. D. Bartlett, Braiding defects in topological stabiliser codes of any dimension cannot be universal'', 2018. https:/​/​arxiv.org/​abs/​1811.11789.
arXiv:1811.11789

[8] P. Webster and S. D. Bartlett, Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes'', 2019. https:/​/​arxiv.org/​abs/​1906.01045.
arXiv:1906.01045

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