Description
A member of a family of BBS codes with weight-two (two-body) gauge generators designed to suppress errors in adiabatic quantum computation.
The family consists of odd-\(m\) codes with \(m=2k+1\) and parameters \([[4k+2l,2k,2k+2l-2,2]]\), together with even-\(m\) codes with \(m=2k\) and parameters \([[4k+2l-2,2k-1,2k+2l-3,2]]\), where \(1 \leq l \leq k\) [2].
Protection
These are distance-two subsystem codes, so they detect arbitrary single-qubit errors. In the energy-penalty setting of Hamiltonian quantum computation, the \(l=1\) subfamily maximizes the code rate and has the largest penalty gap within the trapezoid family [2].Gates
Single-qubit dressed logical operators are two-local, and products of two dressed logical operators of the same Pauli type can also be implemented using two-local physical interactions up to gauge operators [1,2].Cousins
- \([[2m,2m-2,2]]\) error-detecting code— The trapezoid code family can be obtained from the \([[2m,2m-2,2]]\) error-detecting code by using some logical qubits as gauge qubits and imposing a two-dimensional qubit geometry [2].
- \([[6r,2r,2]]\) Ganti-Onunkwo-Young code— The odd-\(m\) trapezoid family at \(l=k\) has parameters \([[6k,2k,4k,2]]\) and reproduces the two-local subsystem construction used for universal Hamiltonian quantum computation in [1]; this is a subsystem analogue of the \([[6k,2k,2]]\) Ganti-Onunkwo-Young family.
- Small-distance qubit stabilizer code
Primary Hierarchy
Parents
Trapezoid subsystem code
Children
The even-logical-qubit trapezoid family at \(l=k=1\) reduces to the \([[6,2,3,2]]\) BBS code.
References
- [1]
- M. Marvian and S. Lloyd, “Robust universal Hamiltonian quantum computing using two-body interactions”, (2019) arXiv:1911.01354
- [2]
- P. Singkanipa, Z. Xia, and D. A. Lidar, “Families of d=2 2D subsystem stabilizer codes for universal Hamiltonian quantum computation with two-body interactions”, Quantum 9, 1821 (2025) arXiv:2412.06744 DOI
Page edit log
- Victor V. Albert (2024-12-13) — most recent
Cite as:
“Trapezoid subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/trapezoid