Mirror code[1]
Description
Stabilizer code defined by a finite group \(G\) and two subsets \(A,B\subseteq G\), with one qubit per group element [1]. Each stabilizer generator acts as \(Z\) on a translate of \(A\) and as \(X\) on a translate of \(B\), so the check weight is at most \(|A|+|B|\). Mirror codes contain all qubit Abelian 2BGA codes up to qubit permutations and Hadamard gates, and are not CSS in general.
The symmetric mirror code has stabilizer generators \begin{align} S(g) = Z(Ag)\,X(Bg^{-1})~,\qquad g\in G~, \tag*{(1)}\end{align} where \(Ag=\{ag \mid a\in A\}\) and \(Z(T)\), \(X(T)\) denote products of \(Z\), \(X\) over the qubits in \(T\). The asymmetric mirror code uses \(X(g^{-1}B)\) instead, and neither family contains the other [1; Thm. B.2]. For Abelian \(G\), the two constructions coincide and any \(A,B\) yield commuting generators. An Abelian mirror code is equivalent to a CSS code via Hadamard gates iff \(A\) and \(B\) lie in different cosets of an index-two subgroup of \(G\) [1; Thm. 3.15].
Protection
Weight-six mirror codes include \([[60,4,10]]\), \([[36,6,6]]\), \([[48,8,6]]\), and \([[85,8,9]]\) codes, none of which is equivalent to a CSS code via Hadamard gates [1; Table 1]. Weight-seven mirror codes with \(kd>n\) exist, e.g., a \([[48,10,6]]\) code [1; Table 1].Fault Tolerance
Superdense syndrome extraction with one ancilla per check, adapted from the color-code circuit of Ref. [2], pairs generators so that each flags faults on the other [1].The CSS-FT6 circuit uses three ancillas per check and is fault tolerant for all CSS codes with check weight at most six [1].The FT6 circuit uses six ancillas per check and is fault tolerant for all stabilizer codes with check weight at most six [1].Threshold
SI1000 circuit-level noise: pseudo-threshold of order \(0.2\%\) [1].Notes
See Ref. [1; Table 1] for a table of Abelian mirror codes.Cousin
- Two-block group-algebra (2BGA) codes— Any qubit 2BGA code with normal subsets \(A,B\) is equivalent to a mirror code on \(\mathbb{Z}_2\times G\) via Hadamard gates on one block and a qubit permutation [1; Prop. 3.21]. Neither family contains the other [1; Prop. 3.22][1; Thm. B.2].
Primary Hierarchy
References
- [1]
- A. B. Khesin and J. Z. Lu, “Mirror codes: High-threshold quantum LDPC codes beyond the CSS regime”, (2026) arXiv:2603.05496
- [2]
- C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
Page edit log
- Victor V. Albert (2026-09-24) — most recent
Cite as:
“Mirror code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/mirror, arXiv:2606.11484