Also known as RHG (Raussendorf-Harrington-Goyal) cluster-state code.

## Description

A three-dimensional cluster-state code defined on the bcc lattice (i.e., 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

Exhibits symmetry-protected self-correction [4]. The energy barrier for symmetry-preserving excitations outside of the code space grows linearly with the lattice width. When the system is coupled locally to a thermal bath respecting the symmetry and below a critical temperature, the memory time grows exponentially with the lattice width.

## Gates

The computation encoded in pre-determined fashion via topological features of the lattice, such as boundaries, defects, or twists. For example, qubits may be encoded in 2D defects along slices of the surface code, and Clifford gates are encoded by spatially braiding the defects along the 3rd dimension. Non-Clifford gates are performed by inserting non-Clifford states into particular singular qubits. To perform the computation, qubits along the extra dimension are measured, e.g., along one two-dimensional slice per time step. This effectively teleports the logical information into the remaining unmeasured portion of the cluster state.

## Decoding

MBQC syndrome extraction consists of single-qubit measurements and classical post-processing. The six \(X\)-measurements of qubits on the faces of a cube of the bcc lattice multiply to the product of the six cluster-state stabilizers whose vertices are on the faces of the cube. Such measurements, if done on a 2D slice, also yield \(Z\)-type syndromes on the next slice.Minimum weight perfect-matching (MWPM) [5,6] (based on work by Edmonds on finding a matching in a graph [7,8]).

## Threshold

\(0.75\%\) for preparation, gate, storage, and measurement errors [3].Concatenation of the RBH code with small codes such as the \([[2,1,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [9].

## Notes

Introduction to MBQC protocols with the RBH state [10].

## 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 [11; 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.
- Concatenated quantum code — Concatenation of the RBH code with small codes such as the \([[2,1,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [9].

## 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]
- Z. Li, I. Kim, and P. Hayden, “Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction”, Quantum 7, 1089 (2023) arXiv:2209.09390 DOI
- [10]
- K. Fujii, “Quantum Computation with Topological Codes: from qubit to topological fault-tolerance”, (2015) arXiv:1504.01444
- [11]
- S. Roberts and D. J. Williamson, “3-Fermion Topological Quantum Computation”, PRX Quantum 5, (2024) arXiv:2011.04693 DOI

## 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