Description
A rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. This is the smallest code that implements a fault-tolerant logical \(S\) gate using a diagonal depth-one Clifford circuit [1].
A stabilizer tableau for the code is given by [2; ID 18] \begin{align} \begin{array}{ccccc} Z & Z & I & Z & I \\ I & I & Z & Z & Z \\ I & X & X & X & I \\ X & I & I & X & X \end{array}~. \tag*{(1)}\end{align} The code is depicted in Fig. I.
Gates
In the qubit order of the stabilizer tableau above, the physical action \(S_0 S_2 S_3^{\dagger} CZ_{1,4}\) implements the logical gate \(\bar{S}\) [3; Fig. 1].Fault-tolerant implementation of the single-qubit Clifford group [3; Fig. 1].Fault Tolerance
Fault-tolerant implementation of the single-qubit Clifford group [3; Fig. 1].Cousins
- Twist-defect surface code— The \([[5,1,2]]\) rotated surface code is a genon code on a sphere, with the missing external \(Y\)-type stabilizer forming the back of the sphere. More generally, any surface code with a single boundary component can be interpreted this way [4].
- \([[7,1,3]]\) Steane code— The \([[5,1,2]]\) morphed Steane code is obtained by morphing the Steane code on a region whose child code is a \([[4,2,2]]\) code [3; Fig. 1].
- \([[4,2,2]]\) Four-qubit code— The \([[5,1,2]]\) morphed Steane code is obtained by morphing the Steane code on a region whose child code is a \([[4,2,2]]\) code [3; Fig. 1].
Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Kitaev surface codeCDSC Twist-defect surface QLDPC Qubit CSS Generalized homological-product Lattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Rotated surface codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
\([[2^r+r-1,1,2]]\) morphed simplex codeCSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
The \([[5,1,2]]\) code is a specific instance of the \([[2^r+r-1,1,2]]\) morphed simplex codes with \(r=2\) [3; Fig. 1].
Surface-code-fragment (SCF) holographic codeQubit CSS Stabilizer Hamiltonian-based HQECC QECC Quantum
The \([[5,1,2]]\) rotated surface code is the smallest SCF holographic code [5]. The encoding of more general SCF holographic codes is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a planar-perfect tensor.
The \([[5,1,2]]\) rotated surface code is the smallest SCF holographic code [5]. The encoding of more general SCF holographic codes is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a planar-perfect tensor.
\([[5,1,2]]\) rotated surface code
References
- [1]
- V. V. Albert, private communication, 2026
- [2]
- Qiskit Community, “Qiskit QEC framework”, URL
- [3]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
- [4]
- S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
- [5]
- R. J. Harris, E. Coupe, N. A. McMahon, G. K. Brennen, and T. M. Stace, “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020) arXiv:2008.10206 DOI
Page edit log
- Victor V. Albert (2024-07-01) — most recent
Cite as:
“\([[5,1,2]]\) rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_5_1_2