\([[10,2,3]]\) rotated toric code

\([[10,2,3]]\) rotated toric code[1][2; Exam. 3]

Description

Rotated toric code that is the CSS form of the twisted XZZX toric code with parameters \(a=1\), \(b=3\) [3,4], related to the XZZX form by Hadamard on the \(B\)-sublattice. It is also the symplectic double (a.k.a. genus-one double cover) of the \([[5,1,3]]\) five-qubit perfect code [2,5], the symplectic double of the \([[5,1,2]]\) rotated surface code [5], and a BCC code [6].

A stabilizer tableau for this code is \begin{align} \begin{array}{cccccccccc} Z & Z & I & Z & Z & I & I & I & I & I \\ I & I & Z & Z & I & Z & Z & I & I & I \\ I & I & I & I & Z & Z & I & Z & Z & I \\ Z & I & I & I & I & I & Z & Z & I & Z \\ X & I & X & X & I & I & I & I & I & X \\ I & X & X & I & X & X & I & I & I & I \\ I & I & I & X & X & I & X & X & I & I \\ I & I & I & I & I & X & X & I & X & X \end{array}~. \tag*{(1)}\end{align}

Protection

Distance 3, correcting any single-qubit error. Achieves the lower bound \(n \geq d^2+1 = 10\) for weight-four GB codes of odd distance [7].

Transversal Gates

Transversal Hadamard-SWAP: qubit permutation \(m \mapsto -m\) (mod 10) paired with sublattice swap \(A \leftrightarrow B\), followed by Hadamard on all qubits [6].Logical Hadamard without SWAP via \(m \mapsto 3m\) (mod 10) followed by Hadamard [6].

Cousins

\([[5,1,3]]\) Five-qubit perfect code— The \([[10,2,3]]\) rotated toric code is the symplectic double (a.k.a. genus-one double cover) of the five-qubit perfect code [5][2; Exam. 3]. A non-CSS cyclic cluster code related to the \([[10,2,3]]\) rotated toric code yields the \([[5,1,3]]\) five-qubit perfect code for \(d=3\) [6].
\([[5,1,2]]\) rotated surface code— The \([[10,2,3]]\) rotated toric code is the symplectic double (a.k.a. genus-one double cover) of the \([[5,1,2]]\) rotated surface code [5].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Twist-defect surface codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

XZZX surface code

Twisted XZZX toric codeCyclic quantum QECC Quantum

Parents

Twisted XZZX toric code

This is the \(d=3\) instance of the \([[d^2+1,2,d]]\) family of twisted XZZX toric codes (parameters \(a=1\), \(b=3\)), presented in its CSS form [6].

Bipartite cyclic cluster (BCC) codeCSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based Cyclic quantum QECC Quantum

The \([[10,2,3]]\) rotated toric code is a \([[d^2+1,2,d]]\) BCC code for \(d=3\) [6]. A non-CSS cyclic cluster code related to the \([[10,2,3]]\) rotated toric code yields the \([[5,1,3]]\) five-qubit perfect code for \(d=3\) [6].

Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

\([[10,2,3]]\) rotated toric code

References

[1]: A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings 348 (2012) arXiv:1202.0928 DOI
[2]: A. A. Kovalev and L. P. Pryadko, “Quantum Kronecker sum-product low-density parity-check codes with finite rate”, Physical Review A 88, (2013) arXiv:1212.6703 DOI
[3]: A. A. Kovalev, I. Dumer, and L. P. Pryadko, “Design of additive quantum codes via the code-word-stabilized framework”, Physical Review A 84, (2011) arXiv:1108.5490 DOI
[4]: R. Sarkar and T. J. Yoder, “A graph-based formalism for surface codes and twists”, Quantum 8, 1416 (2024) arXiv:2101.09349 DOI
[5]: S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
[6]: M. B. Hastings, “A Class of Cyclic Quantum Codes”, (2025) arXiv:2509.06865
[7]: R. Wang and L. P. Pryadko, “Distance bounds for generalized bicycle codes”, (2022) arXiv:2203.17216

Page edit log

Victor V. Albert (2026-05-26) — most recent

Cite as:

“\([[10,2,3]]\) rotated toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/xzzx_10_2_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/10/xzzx_10_2_3.yml.