\([[6,2,3,2]]\) BBS code[1]
Description
Error-detecting six-qubit BBS subsystem code with parameters \([[6,2,3,2]]\) that can suppress errors in adiabatic quantum computation [2].
The code is defined by the \(3\times 3\) binary matrix \begin{align} A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}, \tag*{(1)}\end{align} with physical qubits placed at the six nonzero entries of \(A\) and labeled \(1\)–\(6\) in row-major order [1]. The gauge generators (exluding phases) are \begin{align} \begin{array}{cccccc} X & X & I & I & I & I \\ I & I & X & X & I & I \\ I & I & I & I & X & X \\ Z & I & I & I & Z & I \\ I & Z & Z & I & I & I \\ I & I & I & Z & I & Z \end{array}~, \tag*{(2)}\end{align} where each \(XX\) row corresponds to a same-row pair in \(A\) and each \(ZZ\) row to a same-column pair.
Protection
This code detects arbitrary single-qubit errors and is used as an error-suppressing encoding with non-commuting two-local Hamiltonian terms built from gauge generators [2]. For the symmetric choice of coefficients considered in [2], the resulting two-local suppression Hamiltonian has energy separation \(4-2\sqrt{3}\).Gates
When both logical qubits are encoded in the same block, \(X_{L1}X_{L2}\) and \(Z_{L1}Z_{L2}\) are implementable using two-local physical interactions up to stabilizers [2].Cousins
- Small-distance qubit stabilizer code
- \([[4,1,1,2]]\) Four-qubit subsystem code— Both the \([[6,2,3,2]]\) BBS code and the four-qubit subsystem code can be used to suppress errors in adiabatic quantum computation [2].
Primary Hierarchy
References
- [1]
- S. Bravyi, “Subsystem codes with spatially local generators”, Physical Review A 83, (2011) arXiv:1008.1029 DOI
- [2]
- Z. Jiang and E. G. Rieffel, “Non-commuting two-local Hamiltonians for quantum error suppression”, Quantum Information Processing 16, (2017) arXiv:1511.01997 DOI
Page edit log
- Victor V. Albert (2024-12-13) — most recent
Cite as:
“\([[6,2,3,2]]\) BBS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/bravyi_bacon_shor_6