## Description

Also called an RHG (Raussendorf-Harrington-Goyal) cluster-state code. A three-dimensional cluster-state code defined on the bcc lattice (equivalently, a cubic lattice with qubits on edges and faces).

The MBQC version of the code is defined as the unique ground state of a certain code Hamiltonian. This state is the resource state used in the first MBQC scheme [2,3]. It encodes the temporal gate operations on the surface code into a third spatial dimension.

Addition of certain boundary Hamiltonians yields a degenerate ground-state space that serves as an example of a symmetry-protected self-correcting memory [4].

## Protection

## Gates

## Decoding

## Parents

- Cluster-state code
- 3D lattice stabilizer code
- Walker-Wang model code — The Walker-Wang model code reduces to the RBH cluster-state code when the input category \(\mathcal{C}\) is that of the surface code [9; Sec. V.A].

## Cousins

- Symmetry-protected self-correcting quantum code — The RBH code can exhibit self-correction protected by a certain symmetry.
- Subsystem color code — The RBH code for a certain boundary Hamiltonian is dual to the gauge color code [4; Sec. IV.C.1].
- Kitaev surface code — The RBH state encodes the temporal gate operations on the surface code into a third spatial dimension [2,3]. In addition, one of possible 2D boundaries of the RBH code is effectively a 2D toric code.
- Body-centered cubic (bcc) lattice code — The RBH code is defined on the bcc lattice.

## References

- [1]
- R. Raussendorf, S. Bravyi, and J. Harrington, “Long-range quantum entanglement in noisy cluster states”, Physical Review A 71, (2005) arXiv:quant-ph/0407255 DOI
- [2]
- R. Raussendorf, J. Harrington, and K. Goyal, “A fault-tolerant one-way quantum computer”, Annals of Physics 321, 2242 (2006) arXiv:quant-ph/0510135 DOI
- [3]
- R. Raussendorf and J. Harrington, “Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions”, Physical Review Letters 98, (2007) arXiv:quant-ph/0610082 DOI
- [4]
- S. Roberts and S. D. Bartlett, “Symmetry-Protected Self-Correcting Quantum Memories”, Physical Review X 10, (2020) arXiv:1805.01474 DOI
- [5]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [6]
- A. G. Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average \(O(1)\) parallel time”, (2014) arXiv:1307.1740
- [7]
- J. Edmonds, “Paths, Trees, and Flowers”, Canadian Journal of Mathematics 17, 449 (1965) DOI
- [8]
- J. Edmonds, “Maximum matching and a polyhedron with 0,1-vertices”, Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics 69B, 125 (1965) DOI
- [9]
- S. Roberts and D. J. Williamson, “3-Fermion topological quantum computation”, (2020) arXiv:2011.04693

## Page edit log

- Victor V. Albert (2022-05-18) — most recent
- Yi-Ting (Rick) Tu (2022-04-23)

## Cite as:

“Raussendorf-Bravyi-Harrington (RBH) cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/rbh

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/mbqc/rbh.yml.