# Cluster-state code[1]

## Description

A code based on a cluster state and often used in measurement-based quantum computation (MBQC) [2] (a.k.a. one-way quantum processing), which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. This is done by encoding the computation into the features of the cluster state''s graph.

Cluster states are stabilizer states defined on a graph. There is one stabilizer generator \(S_j\) per graph vertex \(j\) of the form \begin{align} S_j = X_{j} \prod_{k\in N(j)} Z_k~, \tag*{(1)}\end{align} where the neighborhood \(N(j)\) is the set of vertices which share an edge with \(j\).

An MBQC scheme can be constructed out of any qubit CSS code (via foliation [3]) or qubit stabilizer code [4]. The original MBQC scheme [5,6] uses the RBH cluster state on the bcc lattice (i.e., a cubic lattice with qubits on edges and faces).

## Protection

Protection is related to the stabilizer code underlying the cluster state. There is no physical error correction, and decoding output is simply used to update the Pauli frame.

There exist necessary and sufficient conditions for a family of cluster states to exhibit TQO-1 [7]. Quantum weight enumerators of cluster state codes are known as sector weights [8–11].

## Encoding

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Realizations

## Notes

## Parents

- Qubit stabilizer code — Cluster-state codes are particular qubit stabilizer codes. Any qubit stabilizer code is equivalent to a graph quantum code via a single-qubit Clifford circuit [27] (see also [13,28]). As a corollary, any qubit stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [13][29; Appx. A]. Any fault-tolerant scheme based on qubit stabilizer codes can be mapped into a cluster-state based MBQC protocol [4].
- Modular-qudit cluster-state code — Modular-qudit cluster-state codes reduce to cluster-state codes for \(q=2\).

## Children

## Cousins

- Fusion-based quantum computing (FBQC) code — FBQC and MBQC are both computational models in which computation is done by measuring resource states (which are qubit stabilizer states). The difference between the two is in how the states are constructed. FBQC is based exclusively on two-qubit measurements tailored to photonic platforms. These measurements require a foliation with more qubits but one which can be built by fusing smaller modules.
- Qubit CSS code — A resource cluster state can be constructed out of any qubit CSS code via foliation. Conversely, CSS codes can be constructed out of cluster states [3].
- XP stabilizer code — XP stabilizer states are in one-to-one correspondence with weighted hypergraph states [30,31], which generalize both weighted graph states [25,32,33] and hypergraph states [34–36]. The latter can also be utilized in MBQC schemes [37,38].
- Dynamically-generated QECC — MBQC is done using a measurement-based dynamical process.
- Dual-rail quantum code — The KLM protocol can be combined with cluster states in various ways to yield MBQC protocols [39–41].
- Symmetry-protected topological (SPT) code — States realizing various SPT phases are universal resources for MBQC [16,17,42–47]. A cluster-like state, or a state that is in the same SPT phase as a cluster state, can be prepared in finite time [12].
- Kerdock code — Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group [48] that is a unitary two-design [49]. As such, cluster states form complex projective two-designs. These are useful in matrix-vector multiplication [50].
- Graph quantum code — A graph quantum code for \(G=\mathbb{Z}_2\) contains a cluster state as one of its codewords and reduces to a cluster state when its logical dimension is one [51].
- Dual-rail quantum code — The KLM protocol can be combined with cluster states in various ways to yield MBQC protocols [39–41]; see review [52].
- GKP CV-cluster-state code — GKP CV-cluster-state codes reduce to cluster-state codes concatenated with single-mode GKP codes [53] when all physical modes are initialized in GKP states.
- Concatenated GKP code — GKP codes have been concatenated with cluster-state codes [53].
- Topological code — There exist necessary and sufficient conditions for a family of cluster states to exhibit the TQO-1 property [7].
- Codeword stabilized (CWS) code — A single cluster-state codeword is used to construct a CWS code.
- Quantum repetition code — GHZ states can be used as resource states for MBQC protocols [54–56].
- \([[9,1,3]]\) Shor code — The Shor code admits a codeword that is the cluster state of a particular nine-vertex graph [51,57].
- Five-qubit perfect code — The five-qubit code admits a codeword that is the cluster state of the pentagon graph [51,57].
- XZZX surface code — XZZX surface code can be foliated for a noise-bias preserving MBQC [19] or FBQC [58] protocol; see also [59].

## References

- [1]
- H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles”, Physical Review Letters 86, 910 (2001) arXiv:quant-ph/0004051 DOI
- [2]
- R. Raussendorf, D. Browne, and H. Briegel, “The one-way quantum computer--a non-network model of quantum computation”, Journal of Modern Optics 49, 1299 (2002) arXiv:quant-ph/0108118 DOI
- [3]
- A. Bolt et al., “Foliated Quantum Error-Correcting Codes”, Physical Review Letters 117, (2016) arXiv:1607.02579 DOI
- [4]
- B. J. Brown and S. Roberts, “Universal fault-tolerant measurement-based quantum computation”, Physical Review Research 2, (2020) arXiv:1811.11780 DOI
- [5]
- 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
- [6]
- 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
- [7]
- P. Liao, B. C. Sanders, and D. L. Feder, “Topological graph states and quantum error-correction codes”, Physical Review A 105, (2022) arXiv:2112.02502 DOI
- [8]
- N. Wyderka and O. Gühne, “Characterizing quantum states via sector lengths”, Journal of Physics A: Mathematical and Theoretical 53, 345302 (2020) arXiv:1905.06928 DOI
- [9]
- C. Eltschka and J. Siewert, “MaximumN-body correlations do not in general imply genuine multipartite entanglement”, Quantum 4, 229 (2020) arXiv:1908.04220 DOI
- [10]
- M. Miller and D. Miller, “GraphStateVis: Interactive Visual Analysis of Qubit Graph States and their Stabilizer Groups”, 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) (2021) arXiv:2105.12752 DOI
- [11]
- D. Miller et al., “Shor–Laflamme distributions of graph states and noise robustness of entanglement”, Journal of Physics A: Mathematical and Theoretical 56, 335303 (2023) arXiv:2207.07665 DOI
- [12]
- N. Tantivasadakarn and A. Vishwanath, “Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps”, Physical Review Letters 129, (2022) arXiv:2107.04019 DOI
- [13]
- M. Van den Nest, J. Dehaene, and B. De Moor, “Graphical description of the action of local Clifford transformations on graph states”, Physical Review A 69, (2004) arXiv:quant-ph/0308151 DOI
- [14]
- R. Raussendorf and H. Briegel, “Computational model underlying the one-way quantum computer”, (2002) arXiv:quant-ph/0108067
- [15]
- R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states”, Physical Review A 68, (2003) arXiv:quant-ph/0301052 DOI
- [16]
- W. Son, L. Amico, and V. Vedral, “Topological order in 1D Cluster state protected by symmetry”, Quantum Information Processing 11, 1961 (2011) arXiv:1111.7173 DOI
- [17]
- X. Chen et al., “Symmetry protected topological orders in interacting bosonic systems”, (2013) arXiv:1301.0861
- [18]
- S. J. Devitt et al., “Architectural design for a topological cluster state quantum computer”, New Journal of Physics 11, 083032 (2009) arXiv:0808.1782 DOI
- [19]
- J. Claes, J. E. Bourassa, and S. Puri, “Tailored cluster states with high threshold under biased noise”, npj Quantum Information 9, (2023) arXiv:2201.10566 DOI
- [20]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [21]
- T. Ohno et al., “Phase structure of the random-plaquette gauge model: accuracy threshold for a toric quantum memory”, Nuclear Physics B 697, 462 (2004) arXiv:quant-ph/0401101 DOI
- [22]
- K. Takeda, T. Sasamoto, and H. Nishimori, “Exact location of the multicritical point for finite-dimensional spin glasses: a conjecture”, Journal of Physics A: Mathematical and General 38, 3751 (2005) arXiv:cond-mat/0501372 DOI
- [23]
- P. Walther et al., “Experimental one-way quantum computing”, Nature 434, 169 (2005) arXiv:quant-ph/0503126 DOI
- [24]
- R. Ceccarelli et al., “Experimental Entanglement and Nonlocality of a Two-Photon Six-Qubit Cluster State”, Physical Review Letters 103, (2009) arXiv:0906.2233 DOI
- [25]
- M. Hein et al., “Entanglement in Graph States and its Applications”, (2006) arXiv:quant-ph/0602096
- [26]
- A. Furusawa and P. van Loock, Quantum Teleportation and Entanglement (Wiley, 2011) DOI
- [27]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [28]
- M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI
- [29]
- J. Bausch and F. Leditzky, “Error Thresholds for Arbitrary Pauli Noise”, SIAM Journal on Computing 50, 1410 (2021) arXiv:1910.00471 DOI
- [30]
- M. A. Webster, B. J. Brown, and S. D. Bartlett, “The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases”, Quantum 6, 815 (2022) arXiv:2203.00103 DOI
- [31]
- Webster, Mark. The XP Stabilizer Formalism. Dissertation, University of Sydney, 2023.
- [32]
- W. Dür et al., “Entanglement in Spin Chains and Lattices with Long-Range Ising-Type Interactions”, Physical Review Letters 94, (2005) arXiv:quant-ph/0407075 DOI
- [33]
- S. Anders et al., “Ground-State Approximation for Strongly Interacting Spin Systems in Arbitrary Spatial Dimension”, Physical Review Letters 97, (2006) arXiv:quant-ph/0602230 DOI
- [34]
- M. Rossi et al., “Quantum hypergraph states”, New Journal of Physics 15, 113022 (2013) arXiv:1211.5554 DOI
- [35]
- O. Gühne et al., “Entanglement and nonclassical properties of hypergraph states”, Journal of Physics A: Mathematical and Theoretical 47, 335303 (2014) arXiv:1404.6492 DOI
- [36]
- D. W. Lyons et al., “Local unitary symmetries of hypergraph states”, Journal of Physics A: Mathematical and Theoretical 48, 095301 (2015) arXiv:1410.3904 DOI
- [37]
- M. Gachechiladze, O. Gühne, and A. Miyake, “Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states”, Physical Review A 99, (2019) arXiv:1805.12093 DOI
- [38]
- Y. Takeuchi, T. Morimae, and M. Hayashi, “Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements”, Scientific Reports 9, (2019) DOI
- [39]
- N. Yoran and B. Reznik, “Deterministic Linear Optics Quantum Computation with Single Photon Qubits”, Physical Review Letters 91, (2003) arXiv:quant-ph/0303008 DOI
- [40]
- M. A. Nielsen, “Optical Quantum Computation Using Cluster States”, Physical Review Letters 93, (2004) arXiv:quant-ph/0402005 DOI
- [41]
- D. E. Browne and T. Rudolph, “Resource-Efficient Linear Optical Quantum Computation”, Physical Review Letters 95, (2005) arXiv:quant-ph/0405157 DOI
- [42]
- A. Miyake, “Quantum computational capability of a 2D valence bond solid phase”, Annals of Physics 326, 1656 (2011) arXiv:1009.3491 DOI
- [43]
- T.-C. Wei, I. Affleck, and R. Raussendorf, “Two-dimensional Affleck-Kennedy-Lieb-Tasaki state on the honeycomb lattice is a universal resource for quantum computation”, Physical Review A 86, (2012) arXiv:1009.2840 DOI
- [44]
- T.-C. Wei, I. Affleck, and R. Raussendorf, “Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource”, Physical Review Letters 106, (2011) arXiv:1102.5064 DOI
- [45]
- D. V. Else et al., “Symmetry-Protected Phases for Measurement-Based Quantum Computation”, Physical Review Letters 108, (2012) arXiv:1201.4877 DOI
- [46]
- T.-C. Wei, P. Haghnegahdar, and R. Raussendorf, “Hybrid valence-bond states for universal quantum computation”, Physical Review A 90, (2014) arXiv:1310.5100 DOI
- [47]
- H. P. Nautrup and T.-C. Wei, “Symmetry-protected topologically ordered states for universal quantum computation”, Physical Review A 92, (2015) arXiv:1509.02947 DOI
- [48]
- A. Calderbank et al., “Z\({}_{\text{4}}\) -Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets”, Proceedings of the London Mathematical Society 75, 436 (1997) DOI
- [49]
- T. Can et al., “Kerdock Codes Determine Unitary 2-Designs”, IEEE Transactions on Information Theory 66, 6104 (2020) arXiv:1904.07842 DOI
- [50]
- T. Fuchs et al., “Sketching with Kerdock’s crayons: Fast sparsifying transforms for arbitrary linear maps”, (2021) arXiv:2105.05879
- [51]
- Y. Hwang and J. Heo, “On the relation between a graph code and a graph state”, (2015) arXiv:1511.05647
- [52]
- S. Slussarenko and G. J. Pryde, “Photonic quantum information processing: A concise review”, Applied Physics Reviews 6, (2019) arXiv:1907.06331 DOI
- [53]
- K. Fukui et al., “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
- [54]
- M. Gimeno-Segovia et al., “From Three-Photon Greenberger-Horne-Zeilinger States to Ballistic Universal Quantum Computation”, Physical Review Letters 115, (2015) arXiv:1410.3720 DOI
- [55]
- S.-W. Lee et al., “Nearly Deterministic Bell Measurement for Multiphoton Qubits and its Application to Quantum Information Processing”, Physical Review Letters 114, (2015) arXiv:1502.07437 DOI
- [56]
- S. Omkar et al., “All-Photonic Architecture for Scalable Quantum Computing with Greenberger-Horne-Zeilinger States”, PRX Quantum 3, (2022) arXiv:2109.12280 DOI
- [57]
- Griffiths, Robert B. "Graph states and graph codes."
- [58]
- H. Bombín et al., “Increasing error tolerance in quantum computers with dynamic bias arrangement”, (2023) arXiv:2303.16122
- [59]
- A. M. Stephens, W. J. Munro, and K. Nemoto, “High-threshold topological quantum error correction against biased noise”, Physical Review A 88, (2013) arXiv:1308.4776 DOI

## Page edit log

- Victor V. Albert (2023-03-01) — most recent
- Yaron Jarach (2023-03-01)
- Victor V. Albert (2023-03-01)

## Cite as:

“Cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/cluster_state