# 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

## Parents

- Compass code — The heavy-hex code can be viewed as a compass code if ancilla qubits are ignored [1].
- Lattice subsystem code

## Cousins

- Kitaev surface code — Surface code stabilizers are used to measure the Z-type stabilizers of the code. There are various ways to embed the surface code into the heavy-hex lattice [7].
- 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.
- \([[4,2,2]]\) CSS code — Magic states prepared using the \([[4,1,2]]\) subcode can be injected into the heavy-hex code [5,8].
- 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 [1].
- XY surface code — XY surface code can be adapted for a heavy-hexagonal lattice [9].
- XZZX surface code — XZZX surface code can be adapted for a heavy-hexagonal lattice [9].

## References

- [1]
- C. Chamberland et al., “Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits”, Physical Review X 10, (2020) arXiv:1907.09528 DOI
- [2]
- D. Bhoumik et al., “Efficient Machine-Learning-based decoder for Heavy Hexagonal QECC”, (2022) arXiv:2210.09730
- [3]
- B. Hall, S. Gicev, and M. Usman, “Artificial Neural Network Syndrome Decoding on IBM Quantum Processors”, (2023) arXiv:2311.15146
- [4]
- M. Takita et al., “Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits”, Physical Review Letters 119, (2017) arXiv:1705.09259 DOI
- [5]
- E. H. Chen et al., “Calibrated Decoders for Experimental Quantum Error Correction”, Physical Review Letters 128, (2022) arXiv:2110.04285 DOI
- [6]
- 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
- [7]
- C. Benito et al., “Comparative study of quantum error correction strategies for the heavy-hexagonal lattice”, (2024) arXiv:2402.02185
- [8]
- R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
- [9]
- Y. Kim, J. Kang, and Y. Kwon, “Design of Quantum error correcting code for biased error on heavy-hexagon structure”, (2022) arXiv:2211.14038

## Page edit log

- Christopher Chamberland (2022-03-17) — most recent
- Victor V. Albert (2022-03-17)

## Cite as:

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