# Honeycomb code[1]

## Description

Floquet code inspired by the Kitaev honeycomb model [2] whose logical qubits are generated through a particular sequence of measurements.

The code is defined on a hexagonal (honeycomb) lattice with a physical qubit located at each vertex. Edges are labeled \(x\), \(y\), and \(z\), such that one edge of each label meet at every vertex. Check operators are defined as \(XX\) acting on any two qubits joined by an \(x\) edge, and similarly for \(y\) and \(z\). The hexagonal lattice is 3-colorable, so the hexagons may be labeled 0, 1, 2 such that no two neighboring hexagons have the same label.

The code-generating measurement pattern consists of measuring the check operators located on all of the \(r\)-labeled edges in round \(r\) mod 3. The code space is the \(+1\) eigenspace of the instantaneous stabilizer group (ISG). The ISG specifies the state of the system as a Pauli stabilizer state at a particular round of measurement, and it evolves into a (potentially) different ISG depending on the check operators measured.

## Protection

## Encoding

## Gates

## Decoding

## Fault Tolerance

## Threshold

## Parent

- Floquet code — The honeycomb code is the first 2D Floquet code.

## Cousins

- Kitaev surface code — Measurement of each check operator involves two qubits and projects the state of the two qubits to a two-dimensional subspace, which we regard as an effective qubit. These effective qubits form a surface code on a hexagonal superlattice. Electric and magnetic operators on the embedded surface code correspond to outer logical operators of the Floquet code. In fact, outer logical operators transition back and forth from magnetic to electric surface code operators under the measurement dynamics. Inspired by this code, stabilizer measurement circuits consisting of two-body measurements have been designed for the surface code [9][10].
- Subsystem color code — Both honeycomb and subsystem color codes are generated via periodic sequences of measurements. However, any measurement sequence can be performed on the color code without destroying the logical qubits, while honeycomb codes can be maintained only with specific sequences. Honeycomb codes require a shorter measurement cycle and use fewer qubits at the given code distance [1].
- Majorana stabilizer code — The Honeycomb code admits a representation in terms of Majorana fermions. This leads to a possible physical realization of the code in terms of tetrons [11], where each physical qubit is composed of four Majorana modes.
- Matching code — Matching and honeycomb codes are both inspired by the Kitaev honeycomb model [2].
- Quantum low-density parity-check (QLDPC) code — The Floquet check operators are weight-two, and each qubit participates in one check each round.

## References

- [1]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021). DOI; 2107.02194
- [2]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006). DOI; cond-mat/0506438
- [3]
- J. Haah and M. B. Hastings, “Boundaries for the Honeycomb Code”, Quantum 6, 693 (2022). DOI; 2110.09545
- [4]
- Christophe Vuillot, “Planar Floquet Codes”. 2110.05348
- [5]
- D. Aasen, Z. Wang, and M. B. Hastings, “Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes”, Physical Review B 106, (2022). DOI; 2203.11137
- [6]
- C. Gidney et al., “A Fault-Tolerant Honeycomb Memory”, Quantum 5, 605 (2021). DOI; 2108.10457
- [7]
- Craig Gidney, Michael Newman, and Matt McEwen, “Benchmarking the Planar Honeycomb Code”. 2202.11845
- [8]
- Adam Paetznick et al., “Performance of planar Floquet codes with Majorana-based qubits”. 2202.11829
- [9]
- R. Chao et al., “Optimization of the surface code design for Majorana-based qubits”, Quantum 4, 352 (2020). DOI; 2007.00307
- [10]
- Craig Gidney, “A Pair Measurement Surface Code on Pentagons”. 2206.12780
- [11]
- T. Karzig et al., “Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes”, Physical Review B 95, (2017). DOI; 1610.05289

## Page edit log

- Victor V. Albert (2022-06-28) — most recent
- Victor V. Albert (2022-03-02)
- Chris Fechisin (2021-12-13)

## Zoo code information

## Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/honeycomb.yml.