Detecting topological order with ribbon operators

We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express t...

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Bibliographic Details
Published in:Physical review. B 2016-11, Vol.94 (20), Article 205123
Main Authors: Bridgeman, Jacob C., Flammia, Steven T., Poulin, David
Format: Article
Language:English
Subjects:
Coding
Error correcting codes
Error correction
Honeycomb construction
Ising model
Numerical methods
Operators (mathematics)
Qubits (quantum computing)
Test procedures
Two dimensional models
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Summary:We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in a matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with Abelian topological order by finding ribbon operators for Zd quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the Abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting non-Abelian topological order.
ISSN:2469-9950
2469-9969
DOI:10.1103/PhysRevB.94.205123