Cluster-state code[1] 

Also known as Graph-state code.


Code consisting of cluster states [1], which are stabilizer states defined on a graph. There is one stabilizer generator \(S_v\) per graph vertex \(v\) of the form \begin{align} S_v = X_{v} \prod_{w\in N(v)} Z_w~, \tag*{(1)}\end{align} where the neighborhood \(N(v)\) is the set of vertices which share an edge with \(v\).

Cluster-state codewords are used in measurement-based quantum computation (MBQC), which substitutes the temporal dimension necessary for decoding with a spatial dimension. This is done by encoding the computation into the topological features of the cluster state''s graph.

An MBQC scheme can be constructed out of any qubit CSS code (via foliation [2]) or qubit stabilizer code [3]. The original MBQC scheme [4,5] uses the RBH cluster state on the bcc lattice (equivalently, a cubic lattice with qubits on edges and faces).


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.


Initialization of all qubits in the \(|+\rangle\) state and action of \(CZ\) gates along the edges of the graph.


The computation encoded in pre-determined fashion via topological features of the cluster state's graph, such as boundaries, defects, or twists. Such features can be created using \(Z\)-type measurements, which effectively cut a qubit off from the cluster state. Non-Clifford gates are performed by inserting non-Clifford states into particular singular qubits. More generally, any gate protocol of a qubit stabilizer code yields an MBQC protocol [3]. To perform the computation, subsets qubits are measured, e.g., along one two-dimensional slice of a 3D lattice for each time step. This effectively teleports the logical information into the remaining unmeasured portion of the cluster state. The computation terminates after all qubits are measured. The entire cluster state does not need to be created at the start of the computation. Instead, the portion of the cluster state in the extra dimension can be initialized as the computation progresses.


MBQC syndrome extraction is performed by multiplying certain single-qubit \(X\)-type measurements, which yield syndrome values.

Fault Tolerance

Generalized foliation procedures exist for noise-bias preserving MBQC [6].

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 2.9\%\) under MWPM decoding [4]. The threshold under ML decoding corresponds to the value of critical point of the 3D random-plaquette \(\mathbb{Z}_2\) gauge theory (3D-RPGM) via the statistical mechanical mapping [7], calculated to be \(3.3 \%\) [8] (see also [9]).


See Ref. [10] for a review of cluster states and their applications.


  • Qubit stabilizer code — Cluster states are particular qubit stabilizer states defined on a graph. Any qubit stabilizer code is locally equivalent to a graph code [11] (see also [12]). As a corollary, any qubit stabilizer state is locally equivalent to a cluster state [13][14; Appx. A]. Any fault-tolerant scheme based on qubit stabilizer codes can be mapped into a cluster-state based MBQC protocol [3].



  • 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 [2].
  • GKP cluster-state code — The GKP cluster-state code is a concatenation of a cluster-state stabilizer code with a single-mode GKP code.
  • Dynamically-generated QECC — MBQC is done using a measurement-based dynamical process.
  • Dual-rail quantum code — MBQC can be achieved with dual-rail codes using linear optical elements and photon detectors [15].
  • Group GKP code — Cluster states can be generalized to finite groups [16].
  • XZZX surface code — XZZX surface code can be foliated for a noise-bias preserving MBQC [6] or FBQC [17] protocol.


H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles”, Physical Review Letters 86, 910 (2001) arXiv:quant-ph/0004051 DOI
A. Bolt et al., “Foliated Quantum Error-Correcting Codes”, Physical Review Letters 117, (2016) arXiv:1607.02579 DOI
B. J. Brown and S. Roberts, “Universal fault-tolerant measurement-based quantum computation”, Physical Review Research 2, (2020) arXiv:1811.11780 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
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
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
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
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
M. Hein et al., “Entanglement in Graph States and its Applications”, (2006) arXiv:quant-ph/0602096
D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
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
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
J. Bausch and F. Leditzky, “Error Thresholds for Arbitrary Pauli Noise”, SIAM Journal on Computing 50, 1410 (2021) arXiv:1910.00471 DOI
D. E. Browne and T. Rudolph, “Resource-Efficient Linear Optical Quantum Computation”, Physical Review Letters 95, (2005) arXiv:quant-ph/0405157 DOI
C. G. Brell, “Generalized cluster states based on finite groups”, New Journal of Physics 17, 023029 (2015) arXiv:1408.6237 DOI
H. Bombín et al., “Increasing error tolerance in quantum computers with dynamic bias arrangement”, (2023) arXiv:2303.16122
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Zoo Code ID: cluster_state

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“Cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_cluster_state, title={Cluster-state code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.