Raussendorf-Bravyi-Harrington (RBH) cluster-state code[13] 

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


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].


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.


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.


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]).


\(0.75\%\) 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].


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




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
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
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
S. Roberts and S. D. Bartlett, “Symmetry-Protected Self-Correcting Quantum Memories”, Physical Review X 10, (2020) arXiv:1805.01474 DOI
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
A. G. Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average \(O(1)\) parallel time”, (2014) arXiv:1307.1740
J. Edmonds, “Paths, Trees, and Flowers”, Canadian Journal of Mathematics 17, 449 (1965) DOI
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
Z. Li, I. Kim, and P. Hayden, “Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction”, Quantum 7, 1089 (2023) arXiv:2209.09390 DOI
K. Fujii, “Quantum Computation with Topological Codes: from qubit to topological fault-tolerance”, (2015) arXiv:1504.01444
S. Roberts and D. J. Williamson, “3-Fermion Topological Quantum Computation”, PRX Quantum 5, (2024) arXiv:2011.04693 DOI
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Zoo Code ID: rbh

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“Raussendorf-Bravyi-Harrington (RBH) cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/rbh
@incollection{eczoo_rbh, title={Raussendorf-Bravyi-Harrington (RBH) cluster-state code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/rbh} }
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“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/stabilizer/mbqc/rbh.yml.