Cluster-state code[1] 

Also known as Graph-state code.


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 \(S_v\) 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 \(v\).

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 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 the TQO-1 property [7]. Quantum weight enumerators of cluster state codes are known as sector weights [8,9].


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 [4]. 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.Clifford operations can be realized as operations acting on graphs underlying a cluster state [10].1D cluster states are resources for universal MBQC [2,1113].


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 [14].

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 2.9\%\) under MWPM decoding [5]. The threshold under ML decoding corresponds to the value of a critical point of the 3D random-plaquette \(\mathbb{Z}_2\) gauge theory (3D-RPGM) via the statistical mechanical mapping [15], calculated to be \(3.3 \%\) [16] (see also [17]).


Quantum compututation with cluster states has been realized in the polarizations of photons [18].


See Ref. [19] for a review of cluster states and their applications.


  • 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 [20] (see also [10,21]). As a corollary, any qubit stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [10][22; 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 \(q=2\).



  • 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 [23,24], which generalize both weighted graph states [19,25,26] and hypergraph states [2729]. The latter can also be utilized in MBQC schemes [30,31].
  • 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 [32].
  • Symmetry-protected topological (SPT) code — States realizing various SPT phases are universal resources for MBQC [12,13,3338].
  • Kerdock code — Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group [39] that is a unitary two-design [40].
  • 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 [41].
  • Dual-rail quantum code — The KLM protocol can be combined cluster states in various ways [32,42,43].
  • 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.
  • \([[9,1,3]]\) Shor code — The Shor code admits a codeword that is the cluster state of a particular nine-vertex graph [41,44].
  • Five-qubit perfect code — The five-qubit code admits a codeword that is the cluster state of the pentagon graph [41,44].
  • XZZX surface code — XZZX surface code can be foliated for a noise-bias preserving MBQC [14] or FBQC [45] protocol; see also [46].


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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.