# Symmetry-protected self-correcting quantum code[1]

## Description

A code which admits a restricted notion of thermal stability against symmetric perturbations, i.e., perturbations that commute with a set of operators forming a group \(G\) called the symmetry group.

Given a symmetry group \(G\) and its unitary representation \(S\) on the \(n\)-site physical Hilbert space (in this case, a lattice), an operator \(O\) is \(G\)-symmetric (a.k.a. respects the \(G\) symmetry) if \([S(g),O]=0\) for all \(g\in G\). A symmetry-protected self-correcting memory is a ground-state encoding of an \(n\)-body \(G\)-symmetric geometrically local Hamiltonian whose logical information is recoverable for arbitrary long times in the \(n\to\infty\) limit after a \(G\)-symmetric interaction with a thermal environment at sufficiently low temperature.

Tensor-product symmetries of the form \(S(g)=u(g)^{\otimes n}\), where \(u\) is a unitary representation of \(G\ni g\) on a site, cannot support symmetry-protected self-correction. One can instead use one-form symmetries, i.e., symmetries generated by operators of the form \begin{align} S_{\mathcal{M}}(g)=\bigotimes_{\text{sites}\in\mathcal{M}}u(g), \tag*{(1)}\end{align} where \(\mathcal{M}\) runs over all closed codimension-one submanifolds of the lattice. Recent work further relaxed the requirement so that symmetries need only be enforced on the system's boundaries [2].

## Protection

The code is intended to be used as a self-correcting quantum memory when the symmetry is enforced, and protection is characterized by the scaling of the memory time \(\tau\) in the system size.

Another characterization of the protection property is the symmetric version of the energy barrier \(\Delta\), defined as follows. For a given logical operator and a given decomposition into a product of local operators, we consider the maximal energy attained when implementing this logical operator stepwise with this decomposition. Then, \(\Delta\) is defined by minimizing this quantity over all logical operators and over those decompositions for which each local operator respects the symmetry. For some models [1], the linear growth of \(\Delta\) with system size \(n\) implies the exponential growth of \(\tau\) below a critical temperature.

## Parent

## Child

- Self-correcting quantum code — A self-correcting quantum memory does not require symmetry for self correction.

## Cousins

- Symmetry-protected topological (SPT) code — Symmetry-protected self-correcting quantum codes can realize symmetry-protected topological phases [1]. Metastable subspaces of certain many-body Lindbladians can arise due to symmetry relations in the low-lying excitations [3].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH code can exhibit self-correction protected by a certain symmetry.
- 3D subsystem color code — A particular gauge-fixed version of a subsystem code on a 3D lattice yields a self-correcting memory protected by one-form symmetries [4][1; Sec. IV D]. The symmetric energy barrier grows linearly with the length of a side of the lattice. When the system is coupled locally to a thermal bath respecting the symmetry and below a critical temperature, the memory time grows exponentially with the side length. The subsystem color code is not a self-correcting quantum memory if symmetry protection is removed [5].

## References

- [1]
- S. Roberts and S. D. Bartlett, “Symmetry-Protected Self-Correcting Quantum Memories”, Physical Review X 10, (2020) arXiv:1805.01474 DOI
- [2]
- C. Stahl, “Self-Correction from Higher-Form Symmetry Protection on a Boundary”, PRX Quantum 4, (2023) arXiv:2206.05294 DOI
- [3]
- M. J. Kastoryano et al., “A little bit of self-correction”, (2024) arXiv:2408.14970
- [4]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [5]
- Y. Li et al., “Phase diagram of the three-dimensional subsystem toric code”, (2023) arXiv:2305.06389

## Page edit log

- Victor V. Albert (2022-05-18) — most recent
- Yi-Ting (Rick) Tu (2022-05-02)

## Cite as:

“Symmetry-protected self-correcting quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/symmetry_protected_self_correct