Self-Correction from Higher-Form Symmetry Protection on a Boundary

Recent work has shown that a self-correcting quantum memory can exist in three spatial dimensions, provided that it is protected by a 1-form symmetry. Requiring that the dynamics of a system obey this type of symmetry is equivalent to enforcing a macroscopic number of symmetry terms throughout the bulk. In this paper, we show how to replace the explicit 1-form symmetry with an emergent 1-form symmetry in the bulk and an explicit 1-form symmetry on the boundary. To do so, we use the extended excitations of a three-dimensional (3D) toric code to confine anyons in a two-dimensional (2D) toric code on the boundary. The boundary anyons are bound to the bulk excitations by the explicit 1-form symmetry. Although the symmetry still has to be explicitly enforced on the boundary, this could conceivably be a more attainable constraint due to the accessibility of the boundary qubits. Furthermore, this only requires O(L2) terms for a system of linear size L, instead of O(L3) terms.4 MoreReceived 24 June 2022Revised 18 July 2023Accepted 6 September 2023DOI:https://doi.org/10.1103/PRXQuantum.4.030341Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum error correctionQuantum memoriesTopological orderQuantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics