Subsystem stabilizer code on the heavy-hexagonal lattice that combines Bacon-Shor and surface-code stabilizers. Encodes one logical qubit into \(n=(5d^2-2d-1)/2\) physical qubits with distance \(d\). The heavy-hexagonal lattice allows for low degree (at most 3) connectivity between all the data and ancilla qubits, which is suitable for fixed-frequency transom qubits subject to frequency collision errors.
Data qubits and ancillas of the code are placed on a heavy-hexagonal lattice (vertices and edges of a tilling of hexagons). A subset of the ancilla qubits are flag qubits used for detecting high-weight errors arising from fewer faults. The code stabilizers for detecting \(X\)-type errors are measured by measuring weight-two \(Z\)-type gauge operators whose product produces stabilizers of the surface code. \(X\)-type stabilizers are column operators corresponding to stabilizers of the Bacon-Shor code, which are measured by taking products of weight-four and weight-two \(X\)-type gauge operators.
- Kitaev surface code — Surface code stabilizers are used to measure the Z-type stabilizers of the code.
- Bacon-Shor code — Bacon-Shor stabilizers are used to measure the X-type stabilizers of the code.
- \([[4,2,2]]\) CSS code — The \(d=2\) heavy-hexagonal code is closely related to the \([[4,1,2]]\) subcode.
- Rotated surface code — A rotated surface code can be mapped onto a heavy square lattice, resulting in a code similar to the heavy-hexagon code .
- XY surface code — XY surface code can be adapted for a heavy-hexagonal lattice .
- XZZX surface code — XZZX surface code can be adapted for a heavy-hexagonal lattice .
- C. Chamberland et al., “Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits”, Physical Review X 10, (2020) arXiv:1907.09528 DOI
- D. Bhoumik et al., “Efficient Machine-Learning-based decoder for Heavy Hexagonal QECC”, (2022) arXiv:2210.09730
- M. Takita et al., “Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits”, Physical Review Letters 119, (2017) arXiv:1705.09259 DOI
- E. H. Chen et al., “Calibrated Decoders for Experimental Quantum Error Correction”, Physical Review Letters 128, (2022) arXiv:2110.04285 DOI
- N. Sundaresan et al., “Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders”, Nature Communications 14, (2023) arXiv:2203.07205 DOI
- Y. Kim, J. Kang, and Y. Kwon, “Design of Quantum error correcting code for biased error on heavy-hexagon structure”, (2022) arXiv:2211.14038
“Heavy-hexagon code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/heavy_hex