Topological color code and symmetry-protected topological phases

Yoshida, Beni

doi:10.1103/physrevb.91.245131

Cited by 77 publications

(141 citation statements)

References 56 publications

Supporting

Mentioning

135

Contrasting

Order By: Relevance

“…III A, we review the correspondence between LPLOs and gapped, transparent domain walls in two dimensional topological codes. This correspondence has previously been described by Yoshida [16], but we reproduce it here in our framework since it is central to the results we develop in the remainder of the paper. Where possible, we develop these ideas in a way that is independent of dimension, and so they will be relevant to our consideration of higher dimensional codes in subsequent sections.…”

Section: Classification For Two Dimensional Codesmentioning

confidence: 72%

Locality-preserving logical operators in topological stabilizer codes

Webster

Bartlett

2018

Phys. Rev. A

View full text Add to dashboard Cite

Locality-preserving logical operators in topological codes are naturally fault-tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure for finding all locality-preserving logical operators admitted by a large and important class of topological stabiliser codes. In particular, we focus on those equivalent to a stack of a finite number of surface codes of any spatial dimension, where our procedure fully specifies the group of locality-preserving logical operators. We also present examples of how our procedure applies to codes with different boundary conditions, including colour codes and toric codes, as well as more general codes such as abelian quantum double models and codes with fermionic excitations in more than two dimensions.

show abstract

Section: Classification For Two Dimensional Codesmentioning

confidence: 72%

Locality-preserving logical operators in topological stabilizer codes

Webster

Bartlett

2018

Phys. Rev. A

View full text Add to dashboard Cite

show abstract

“…We remark that there are only two possible types of defect lines in the toric code, one of which is trivial. However, in case of the color code, there are 72 different defect lines [53]. We encourage readers to explore [54] for an illuminating discussion of all the possible boundaries and defect lines in the 2D color code.…”

Section: Fig 2 Quasiparticle Excitations In the 2d Triangular Colormentioning

confidence: 99%

Advantages of versatile neural-network decoding for topological codes

2019

View full text Add to dashboard Cite

Finding optimal correction of errors in generic stabilizer codes is a computationally hard problem, even for simple noise models. While this task can be simplified for codes with some structure, such as topological stabilizer codes, developing good and efficient decoders still remains a challenge. In our work, we systematically study a very versatile class of decoders based on feedforward neural networks. To demonstrate adaptability, we apply neural decoders to the triangular color and toric codes under various noise models with realistic features, such as spatially-correlated errors. We report that neural decoders provide significant improvement over leading efficient decoders in terms of the error-correction threshold. Using neural networks simplifies the process of designing wellperforming decoders, and does not require prior knowledge of the underlying noise model.

show abstract

“…Recently SPT order has attracted a considerable amount of attention in quantum information community (See e.g. [11][12][13][14][15][16] ). It is often claimed that (symmetry-protected) topological order of the system is fundamentally related to its entanglement structure.…”

Section: Introductionmentioning

confidence: 99%

Symmetry-protected topological entanglement

Marvian

2017

Phys. Rev. B

View full text Add to dashboard Cite

We propose an order parameter for the Symmetry-Protected Topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the SPT-entanglement, is defined as the entanglement between A and B, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region C is measured and known, where C is a connected region surrounded by A, B and the boundaries of the system. In the case of 1-dimensional systems we prove that in the limit where A and B are large and far from each other compared to the correlation length, the SPT-entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the non-trivial phases. Moreover, we show that the SPT-entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPTentanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to a new algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.

show abstract

Topological color code and symmetry-protected topological phases

Cited by 77 publications

References 56 publications

Locality-preserving logical operators in topological stabilizer codes

Locality-preserving logical operators in topological stabilizer codes

Advantages of versatile neural-network decoding for topological codes

Symmetry-protected topological entanglement

Contact Info

Product

Resources

About