Alternative names: Raussendorf-Harrington-Goyal (RHG) 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
Various thresholds for optical quantum computing scheme with RBH codes [9,10].\(0.75\%\) for preparation, gate, storage, and measurement errors [3].\(24.9\%\) under erasure noise [11].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 [12].Notes
Introduction to MBQC protocols with the RBH state [13].Blind quantum computation is possible with the RBH state [14].Cousins
- Symmetry-protected self-correcting quantum code— The RBH code can exhibit self-correction protected by a certain symmetry.
- Symmetry-protected topological (SPT) code— In 3D, cluster states belong to SPT phases protected by higher-form symmetries [15] and enable universal fault-tolerant MBQC [16].
- 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— The RBH code is defined on the bcc lattice.
- Concatenated qubit 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 [12].
- \([[4,1,1,2]]\) Four-qubit subsystem 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 [12].
- \([[7,1,3]]\) Steane 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 [12].
- Hybrid cat code— Hybrid cat codes can be concatenated with RBH codes [17].
- Quantum parity code (QPC)— QPCs can be concatenated with RBH codes [18].
- 3D subsystem color code— Different stabilizer Hamiltonians of the 3D subsystem color code correspond to different SPTs, one of which describes the RBH model [19]. The RBH code for a certain boundary Hamiltonian is dual to the 3D subsystem color code [4; Sec. IV.C.1].
Primary Hierarchy
Parents
The Walker-Wang model code reduces to the RBH cluster-state code when the input category \(\mathcal{C}\) is that of the surface code [20; Sec. V.A].
Raussendorf-Bravyi-Harrington (RBH) cluster-state code
References
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- 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
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- 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
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- E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
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- A. G. Fowler, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average \(O(1)\) parallel time”, (2014) arXiv:1307.1740
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- J. Edmonds, “Paths, Trees, and Flowers”, Canadian Journal of Mathematics 17, 449 (1965) DOI
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- 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
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- C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise Thresholds for Optical Quantum Computers”, Physical Review Letters 96, (2006) arXiv:quant-ph/0509060 DOI
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- D. A. Herrera-Martí, A. G. Fowler, D. Jennings, and T. Rudolph, “Photonic implementation for the topological cluster-state quantum computer”, Physical Review A 82, (2010) arXiv:1005.2915 DOI
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- S. D. Barrett and T. M. Stace, “Fault Tolerant Quantum Computation with Very High Threshold for Loss Errors”, Physical Review Letters 105, (2010) arXiv:1005.2456 DOI
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- Z. Li, I. Kim, and P. Hayden, “Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction”, Quantum 7, 1089 (2023) arXiv:2209.09390 DOI
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- K. Fujii, “Quantum Computation with Topological Codes: from qubit to topological fault-tolerance”, (2015) arXiv:1504.01444
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- T. Morimae and K. Fujii, “Blind topological measurement-based quantum computation”, Nature Communications 3, (2012) arXiv:1110.5460 DOI
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- S. Roberts, B. Yoshida, A. Kubica, and S. D. Bartlett, “Symmetry-protected topological order at nonzero temperature”, Physical Review A 96, (2017) arXiv:1611.05450 DOI
- [16]
- R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation”, New Journal of Physics 9, 199 (2007) arXiv:quant-ph/0703143 DOI
- [17]
- J. Lee, N. Kang, S.-H. Lee, H. Jeong, L. Jiang, and S.-W. Lee, “Fault-Tolerant Quantum Computation by Hybrid Qubits with Bosonic Cat Code and Single Photons”, PRX Quantum 5, (2024) arXiv:2401.00450 DOI
- [18]
- S.-H. Lee, S. Omkar, Y. S. Teo, and H. Jeong, “Parity-encoding-based quantum computing with Bayesian error tracking”, npj Quantum Information 9, (2023) arXiv:2207.06805 DOI
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- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [20]
- 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