# Non-Pauli topological stabilizer codes from twisted quantum doubles

Julio Carlos Magdalena de la Fuente1,2, Nicolas Tarantino2, and Jens Eisert2

1JARA Institute for Quantum Information, RWTH Aachen University, Aachen, Germany
2Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

### Abstract

It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.

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### Cited by

[1] Mark A. Webster, Benjamin J. Brown, and Stephen D. Bartlett, "The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases", Quantum 6, 815 (2022).

[2] Wilbur Shirley, Yu-An Chen, Arpit Dua, Tyler D. Ellison, Nathanan Tantivasadakarn, and Dominic J. Williamson, "Three-Dimensional Quantum Cellular Automata from Chiral Semion Surface Topological Order and beyond", PRX Quantum 3 3, 030326 (2022).

[3] Tyler D. Ellison, Yu-An Chen, Arpit Dua, Wilbur Shirley, Nathanan Tantivasadakarn, and Dominic J. Williamson, "Pauli Stabilizer Models of Twisted Quantum Doubles", PRX Quantum 3 1, 010353 (2022).

[4] S. Varona and M. A. Martin-Delgado, "Determination of the semion code threshold using neural decoders", Physical Review A 102 3, 032411 (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2023-01-27 08:42:41) and SAO/NASA ADS (last updated successfully 2023-01-27 08:42:42). The list may be incomplete as not all publishers provide suitable and complete citation data.