Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases

Andreas Bauer

doi:10.22331/q-2025-03-25-1673

Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases

Andreas Bauer

Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
MIT Department of Mechanical Engineering, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Published:	2025-03-25, volume 9, page 1673
Editor:	Alexander Dalzell
Eprint:	arXiv:2403.12119v3
Doi:	https://doi.org/10.22331/q-2025-03-25-1673
Citation:	Quantum 9, 1673 (2025).

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Abstract

We propose a family of explicit geometrically local circuits on a 2-dimensional planar grid of qudits, realizing any abelian non-chiral topological phase as an actively error-corrected fault-tolerant memory. These circuits are constructed from measuring 1-form symmetries in discrete fixed-point path integrals, which we express through cellular cohomology and higher-order cup products. The specific path integral we use is the abelian Dijkgraaf-Witten state sum on a 3-dimensional cellulation, which is a spacetime representation of the twisted quantum double model. The resulting circuits are based on a syndrome extraction circuit of the (qudit) stabilizer toric code, into which we insert non-Clifford phase gates that implement the “twist''. The overhead compared to the toric code is moderate, in contrast to known constructions for twisted abelian phases. We also show that other architectures for the (qudit) toric code phase, like measurement-based topological quantum computation or Floquet codes, can be enriched with phase gates to implement twisted quantum doubles instead of their untwisted versions. As a further result, we prove fault tolerance under arbitrary local (including non-Pauli) noise for a very general class of topological circuits that we call 1-form symmetric fixed-point circuits. This notion unifies the circuits in this paper as well as the stabilizer toric code, subsystem toric code, measurement-based topological quantum computation, or the (CSS) honeycomb Floquet code. We also demonstrate how our method can be adapted to construct fault-tolerant circuits for specific non-Abelian phases. In the appendix we present an explicit combinatorial procedure to define formulas for higher cup products on arbitrary cellulations, which might be interesting in its own right to the TQFT and topological-phases community.

Featured image: Quantum circuit implementing the double-semion topological phase.

Popular summary

Topological quantum error correction is a way of protecting quantum information from noise by encoding it in the space of ground states of a "topologically ordered" quantum spin system on a 2-dimensional grid. If we want to use topological quantum error correction in practice, we have to design a quantum circuit (which is local in 2+1 dimensions) which measures whether the current state of the system is still a ground state, and maps it back to its ground space if not. Such circuits have been extensively studied only for the simplest topological phase, namely the toric code. In this paper, we introduce a flexible method to construct such circuits also for other topological phases, in particular all abelian ones, where braiding the quasi-particles only gives rise to a phase factor. We also show an example for a non-Abelian topological phase which could be useful for universal quantum computing. The resulting circuits are surprisingly simple, and we prove the existence of a threshold noise rate, below which the encoded information can be protected arbitrarily well by making the system larger and larger.

► BibTeX data

@article{Bauer2025lowoverheadnon,
  doi = {10.22331/q-2025-03-25-1673},
  url = {https://doi.org/10.22331/q-2025-03-25-1673},
  title = {Low-overhead non-{C}lifford fault-tolerant circuits for all non-chiral abelian topological phases},
  author = {Bauer, Andreas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1673},
  month = mar,
  year = {2025}
}

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[3] Dongjin Lee and Beni Yoshida, "Chiral Color Code: Single-Shot Error Correction for Exotic Topological Order", PRX Quantum 7 3, 033015 (2026).

[4] Ryohei Kobayashi, Guanyu Zhu, and Po-Shen Hsin, "Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic States", Physical Review Letters 136 25, 250802 (2026).

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The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 14:36:17) and SAO/NASA ADS (last updated successfully 2026-08-13 14:36:18). The list may be incomplete as not all publishers provide suitable and complete citation data.

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