$[[6,2,3,2]]$ BBS code

$[[6,2,3,2]]$ 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 (excluding 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].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Subsystem qubit codeSubsystem QECC Quantum

Subsystem qubit stabilizer codeSubsystem QECC Quantum

Subsystem qubit CSS codeSubsystem QECC Quantum

Bravyi-Bacon-Shor (BBS) code

Trapezoid subsystem codeLattice subsystem Subsystem QECC Quantum

Parents

Trapezoid subsystem code

The even-logical-qubit trapezoid family at $l=k=1$ reduces to the $[[6,2,3,2]]$ BBS code.

$[[6,2,3,2]]$ BBS code

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 (2026-06-08) — most recent
Victor V. Albert (2024-12-13)

Cite as:

“$[[6,2,3,2]]$ BBS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/bravyi_bacon_shor_6, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/subsystem/qldpc/bbs/bravyi_bacon_shor_6.yml.

\([[6,2,3,2]]\) BBS code [1]