A local automaton for the 2D toric code

Shankar Balasubramanian; Margarita Davydova; Ethan Lake

doi:10.22331/q-2026-06-02-2125

A local automaton for the 2D toric code

Shankar Balasubramanian¹, Margarita Davydova^2,3, and Ethan Lake⁴

¹Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
²Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
³Walter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125
⁴Department of Physics, University of California Berkeley, Berkeley, CA 94720

Published:	2026-06-02, volume 10, page 2125
Editor:	Daniel Malz
Eprint:	arXiv:2412.19803v3
Doi:	https://doi.org/10.22331/q-2026-06-02-2125
Citation:	Quantum 10, 2125 (2026).

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Abstract

We construct a local decoder for the 2D toric code using ideas from the hierarchical classical cellular automata of Tsirelson and Gács. Our decoder is a circuit of strictly local quantum operations preserving a logical state for exponential time in the presence of circuit-level noise without the need for non-local classical computation or communication. Our construction is not translation invariant in spacetime, but can be made time-translation invariant in 3D with stacks of 2D toric codes. This solves the open problem of constructing a local topological quantum memory below four dimensions.

► BibTeX data

@article{Balasubramanian2026localautomatond,
  doi = {10.22331/q-2026-06-02-2125},
  url = {https://doi.org/10.22331/q-2026-06-02-2125},
  title = {A local automaton for the 2{D} toric code},
  author = {Balasubramanian, Shankar and Davydova, Margarita and Lake, Ethan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2125},
  month = jun,
  year = {2026}
}

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