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Cluster-state code[1]

Alternative names: Graph-state code.

Description

A code based on a cluster state and often used in measurement-based quantum computation (MBQC) [2,3] (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 [4]; see also Ref. [5]) or qubit stabilizer code [6]. The original MBQC scheme [7,8] uses the RBH cluster state on the bcc lattice (i.e., a cubic lattice with qubits on edges and faces).

Since they are stabilizer states, two cluster states are considered local-Clifford equivalent if they can be mapped into each other under a tensor product of arbitrary single-qubit Clifford operations. Such operations can sometimes be done via local complementation (LC) of the underlying graph [9,10]. More generally, any two local-unitary equivalent cluster states can be mapped into each other by operations at some level of the hierarchy [11], meaning that general tensor-product unitaries are not needed. Cluster states on \(\leq 19\) qubits are LC equivalent if and only if they are local-unitary equivalent [11], and there exists a cluster state on \(27\) qubits that is LC but not local-unitary equivalent [12–14]. Local complementation has been generalized to graphically characterize local unitary equivalence under local Clifford operations and, more generally, at other levels of the Clifford hierarchy [11].

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 [15]. Quantum weight enumerators of cluster state codes are known as sector weights [16–19].

Potential and limits of MBQC using probabilistic gates has been studied [20].

Encoding

Initialization of all qubits in the \(|+\rangle\) state and action of \(CZ\) gates along the edges of the graph.Optimal-depth preparation based on graph coloring [21].A cluster-like state, or a state that is in the same SPT phase as a cluster state, can be prepared in finite time [22].ZX calculus based encoder representation [23].

Gates

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 [6]. 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]. They can be done in any order, demonstrating parallelism [24].

Decoding

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

Fault Tolerance

There is a simple proof of a threshold for MBQC [25].Photonic architecture [26].Generalized foliation procedures exist for noise-bias preserving MBQC [27].Fault complexes yield fault-tolerance properties of cluster states that come from foliations of codes [28].

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 2.9\%\) under MWPM decoding [7]. 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 [29], calculated to be \(3.3 \%\) [30] (see also [31]).

Realizations

Polarizations of photons: quantum computation [32,33] and single-qubit error correction on an 8-qubit cluster state [34].

Notes

See Refs. [35,36] for a review of cluster states and their applications.Cluster states are useful for entanglement purification [37].

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 [4].
  • XP stabilizer code— XP stabilizer states are in one-to-one correspondence with weighted hypergraph states [38,39], which generalize both weighted graph states [35,40,41] and hypergraph states [42–44]. The latter can also be utilized in MBQC schemes [45,46].
  • Dynamically generated QECC— MBQC is done using a measurement-based dynamical process.
  • Symmetry-protected topological (SPT) code— Cluster states defined on various lattices are representatives of SPT phases, and states realizing these phases can be resources for MBQC. In 1D, cluster states are examples of SPT phases with global symmetries [47–51] and enable MBQC on a single qubit [2,3]. The square-lattice cluster state, which is the prototypical resource for universal MBQC [2,3], and other 2D cluster states [52–54] have SPT order protected by subsystem symmetries [52,55,56]. States like AKLT states and SPT fixed-point states can be efficiently converted into cluster states using local measurements and subsequently used as resources for MBQC [48,57–61]. In 3D, cluster states belong to SPT phases protected by higher-form symmetries [62] and enable universal fault-tolerant MBQC [63]. A cluster-like state, or a state that is in the same SPT phase as a cluster state, can be prepared in finite time [22]. Cluster states can be created on various lattices [64].
  • Lattice stabilizer code— Cluster states defined on various lattices are representatives of SPT phases, and states realizing these phases can be resources for MBQC. In 1D, cluster states are examples of SPT phases with global symmetries [47–51] and enable MBQC on a single qubit [2,3]. The square-lattice cluster state, which is the prototypical resource for universal MBQC [2,3], and other 2D cluster states [52–54] have SPT order protected by subsystem symmetries [52,55,56]. States like AKLT states and SPT fixed-point states can be efficiently converted into cluster states using local measurements and subsequently used as resources for MBQC [48,57–61]. In 3D, cluster states belong to SPT phases protected by higher-form symmetries [62] and enable universal fault-tolerant MBQC [63]. A cluster-like state, or a state that is in the same SPT phase as a cluster state, can be prepared in finite time [22]. Cluster states can be created on various lattices [64].
  • Kerdock code— Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group [65] that is a unitary 2-design [66]. As such, cluster states form complex projective 2-designs. These are useful in matrix-vector multiplication [67].
  • 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 [68].
  • Dual-rail quantum code— The KLM protocol can be combined with cluster states in various ways to yield MBQC protocols [69–71]; see review [72].
  • GKP CV-cluster-state code— GKP CV-cluster-state codes reduce to cluster-state codes concatenated with single-mode GKP codes [73] when all physical modes are initialized in GKP states.
  • Concatenated GKP code— GKP codes have been concatenated with cluster-state codes [73].
  • Topological code— There exist necessary and sufficient conditions for a family of cluster states to exhibit the TQO-1 property [15].
  • 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 [74–76].
  • Five-qubit perfect code— The five-qubit code admits a codeword that is the cluster state of the pentagon graph [68,77].
  • \([[9,1,3]]\) Shor code— The Shor code admits a codeword that is the cluster state of a particular nine-vertex graph [68,77].
  • Concatenated Steane code— The cluster state corresponding to the concatenated Steane code has been worked out [9].
  • XZZX surface code— XZZX surface code can be foliated for a noise-bias preserving MBQC [27] or FBQC [78] protocol; see also [79].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
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 [80] (see also [10,81]). As a corollary, any qubit stabilizer state is equivalent to a cluster state under a single-qubit Clifford circuit [10][82; Appx. A]. There are algorithms that determine whether two stabilizer states are equivalent that work by checking whether their corresponding cluster states are equivalent [83–85]. Any fault-tolerant scheme based on qubit stabilizer codes can be mapped into a cluster-state based MBQC protocol [6].
Modular-qudit cluster-state codeStabilizer Hamiltonian-based QECC Quantum
Modular-qudit cluster-state codes reduce to cluster-state codes for \(q=2\).
Cluster-state code
Children
Raussendorf-Bravyi-Harrington (RBH) cluster-state code
Square-lattice cluster-state code
Tree cluster-state code

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“Cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cluster_state
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@incollection{eczoo_cluster_state, title={Cluster-state code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/cluster_state} }
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