\([[5,1,2]]\) rotated surface code

\([[5,1,2]]\) rotated surface code[1; Ex. 5]

Rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it.

Admits generators \(\{ZZZII,IIZZZ,XIXXI,IXXIX\} \).

Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs [2].

Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs [2].

Rotated surface code
Surface-code-fragment (SCF) holographic code — The \([[5,1,2]]\) rotated surface code is the smallest SCF holographic code. 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.
Planar-perfect-tensor code — The \([[5,1,2]]\) rotated surface code is the smallest SCF holographic code. 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.
Small-distance block quantum code

\([[7,1,3]]\) Steane code — The \([[5,1,2]]\) rotated surface code can be obtained by morphing the Steane code [2].

[1]: A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1202.0928 DOI
[2]: M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI

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“\([[5,1,2]]\) rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_5_1_2