# Heavy-hexagon code[1]

## Description

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.

## Protection

## Rate

## Encoding

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Threshold

## Realizations

## Parent

## Cousins

- 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 \([[4,1,2]]\) code.

## Zoo code information

## References

- [1]
- C. Chamberland et al., “Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits”, Physical Review X 10, (2020). DOI; 1907.09528
- [2]
- M. Takita et al., “Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits”, Physical Review Letters 119, (2017). DOI; 1705.09259
- [3]
- Edward H. Chen et al., “Calibrated decoders for experimental quantum error correction”. 2110.04285
- [4]
- Neereja Sundaresan et al., “Matching and maximum likelihood decoding of a multi-round subsystem quantum error correction experiment”. 2203.07205

## Cite as:

“Heavy-hexagon code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/heavy_hex