\([[2^r+r-1,1,2]]\) morphed simplex code[1]
Alternative Names: \([[2^r+r-1,1,2]]\) morphed quantum RM code.
Description
A member of a family of codes obtained by morphing the \([[2^{r+1}-1,1,3]]\) simplex codes on a region whose child code is a \([[2^r,r,2]]\) hypercube code [1; Appx. C]. The morphing process replaces a subset of qubits with their logical qubits, yielding a code with parameters \([[2^r+r-1,1,2]]\) that inherits a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy from the parent code.Gates
Each code implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy using transversal operations and \(C^{r}Z\) gates [1; Appx. C].Cousins
- \([[2^r-1,1,3]]\) simplex code— The \([[2^r+r-1,1,2]]\) morphed simplex code is obtained by morphing the \([[2^{r+1}-1,1,3]]\) simplex code on a region whose child code is a \([[2^r,r,2]]\) hypercube code [1; Appx. C].
- \([[2^D,D,2]]\) hypercube quantum code— The \([[2^r+r-1,1,2]]\) morphed simplex code is obtained by morphing the \([[2^{r+1}-1,1,3]]\) simplex code on a region whose child code is a \([[2^r,r,2]]\) hypercube code [1; Appx. C].
Member of code lists
Primary Hierarchy
Parents
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[2^r+r-1,1,2]]\) morphed simplex code
Children
The \([[10,1,2]]\) code is a specific instance of the \([[2^r+r-1,1,2]]\) morphed simplex codes with \(r=3\) [1; Appx. C].
The \([[5,1,2]]\) code is a specific instance of the \([[2^r+r-1,1,2]]\) morphed simplex codes with \(r=2\) [1; Fig. 1].
References
- [1]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Michael Vasmer (2024-06-05)
- Victor V. Albert (2026-05-20)
Cite as:
“\([[2^r+r-1,1,2]]\) morphed simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/morphed_diagonal_clifford, arXiv:2606.11484