Foliated Cluster State Construction

• Foliated cluster state construction is a method that converts 2D CSS codes into a 3D resource by replacing each code qubit with a 1D teleportation chain, preserving stabilizer parity via local measurements.
• The protocol employs Tanner graph clusterization and alternating primal-dual sheet stacking to reproduce syndrome-extraction structures and maintain the code’s original distance.
• Generalizations such as bias-preserving foliation and non-foliated FTCS designs expand the framework, improving error thresholds and offering new architectures for fault-tolerant quantum computation.

Foliated cluster state construction is the procedure by which a lower-dimensional stabilizer code, most canonically a $2$-dimensional CSS code, is converted into a higher-dimensional cluster-state resource by replacing each code qubit with a $1$-dimensional teleportation chain and coupling adjacent chains so that stabilizer information is reproduced as local measurement parities. In the standard formulation, arbitrary CSS codes are first clusterized via the Tanner graph of their $Z$-check matrix and then stacked as alternating primal and dual sheets, yielding a $3$-dimensional measurement-based resource that generalizes the Raussendorf construction for the surface code (Bolt et al., 2016). Subsequent work has treated this construction both as a canonical route to fault-tolerant measurement-based quantum computation and as one member of a broader family of fault-tolerant cluster-state constructions, including bias-preserving generalized foliations, non-foliated FTCSs, and chain-complex formulations in which foliation is expressed as a tensoring operation with a $1$-dimensional repetition complex (Claes et al., 2022, Nickerson et al., 2018, Okuda et al., 2024).

1. Canonical construction from CSS codes

The standard CSS-based construction begins with a code whose stabilizer generators split into $Z$-type and $X$-type sets,

$$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$

For an $[[n,k,d]]$ CSS code, the clusterization step uses the $Z$-parity-check matrix $1$0. One forms a bipartite graph with one vertex for each code qubit and one ancilla vertex for each $1$1-type stabilizer, with edge rule

$1$2

This graph is the Tanner graph of $1$3, and its cluster-state stabilizers are

$1$4

Measuring each ancilla in the $1$5 basis projects the neighboring code qubits into eigenstates of the corresponding $1$6-checks, while products of code-qubit-centered cluster stabilizers reproduce the $1$7-type stabilizers because the ancilla $1$8-factors cancel whenever the corresponding $1$9 commutes with all $Z$0-checks. In this sense, a CSS code state is obtained from a larger bipartite cluster state by local measurements (Bolt et al., 2016).

Foliation proper stacks these clusterized sheets along a time-like direction. The sheets alternate between clusterized copies of the original CSS code and clusterized copies of its dual code. If $Z$1 labels sheets, then each code qubit in sheet $Z$2 is connected to the corresponding code qubits in adjacent sheets $Z$3. For an $Z$4-type stabilizer $Z$5 in sheet $Z$6, the product of the foliated cluster stabilizers centered on its support is

$Z$7

and multiplying by the neighboring-sheet ancilla stabilizers yields the foliated parity check

$Z$8

Because this parity check contains only $Z$9 operators, its value is inferred directly from single-qubit $3$0-measurement outcomes. The bulk of the foliated cluster is therefore measured in the $3$1 basis, and the two boundary sheets are left in an encoded Bell state. The construction inherits the distance of the underlying code rather than increasing it; foliation repackages the code into a measurement-based, time-extended resource (Bolt et al., 2016).

2. Geometric and syndrome-extraction structure

A complementary description treats a foliated cluster state as a $3$2-dimensional cell complex $3$3 with a dual complex $3$4. In the cell-complex picture used for the cubic FTCS and its generalizations, primal qubits live on faces and dual qubits on edges, or equivalently vice versa in the dual description. A general construction associates a dual qubit with every edge and a primal qubit with every face of the lattice, then connects by a graph-state bond the qubit at the center of each face with those around its boundary. This yields face stabilizers such as

$3$5

together with dual expressions obtained by exchanging primal and dual roles. Products of face stabilizers over closed surfaces eliminate the boundary $3$6-support and produce closed-cell stabilizers of the form

$3$7

These are exactly the stabilizers recoverable after all bulk qubits are measured in the $3$8 basis, because only $3$9-support survives the measurement pattern (Nickerson et al., 2018).

This geometric language explains the operational role of foliation. The resulting $1$0-dimensional cluster state is a fault-tolerant channel between input and output boundary codes. Syndrome extraction is encoded statically into the resource rather than produced dynamically by repeated rounds of ancilla interaction. The syndrome graph is the $1$1-skeleton of the relevant dual cell complex: vertices represent closed stabilizers, edges represent measured qubits, and a single qubit error flips the parity at the two adjacent syndrome vertices, so errors appear as chains with syndrome endpoints. Logical information is carried by membrane-like correlation surfaces extending through the $1$2-dimensional complex. In the standard foliated setting, the structure is layered and prismatic: foliation of a square-lattice surface code gives the cubic FTCS, and more generally foliation of an arbitrary $1$3-dimensional surface code gives a $1$4-dimensional FTCS made of layers of prismatic cells, with a natural input layer and output layer carrying the same $1$5-dimensional surface code (Nickerson et al., 2018).

3. Bias-preserving generalized foliation

The standard all-$1$6, all-$1$7, all-$1$8-measurement foliation does not preserve a physical bias toward dephasing errors. In the ordinary $1$9-dimensional teleportation chain, the teleported logical operators mix physical $Z$0-measurement outcomes into the logical $Z$1 bookkeeping, so a physical $Z$2 fault on a measured qubit can act as a logical $Z$3 error. A bias-preserving construction avoids this by enlarging the cluster-state formalism to two species of qubits. $Z$4-type qubits are initialized in $Z$5 and measured in the $Z$6 basis; $Z$7-type qubits are initialized in $Z$8 and measured in the $Z$9 basis. Entangling gates are chosen accordingly: $X$0 between two $X$1-type qubits and $X$2, with the $X$3-type qubit as control and the $X$4-type qubit as target, between $X$5-type and $X$6-type qubits. The resulting $X$7-start and $X$8-start teleportation chains have the crucial property that the teleported logical $X$9 depends only on physical $$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$0 measurements and the teleported logical $$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$1 depends only on physical $$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$2 measurements. This preserves the distinction between dominant $$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$3-type noise and subdominant bit-flip-causing faults (Claes et al., 2022).

The flagship example is the XZZX cluster state, obtained by foliating the XZZX surface code with an alternating grid of $$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$4-start and $$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$5-start chains and $$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$6-type ancillas. Its local cell stabilizer is mixed rather than all-$$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$7: the product of the $$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$8 operators on the $$S_Z \in \{I,Z\}^{\otimes n}, \qquad S_X \in \{I,X\}^{\otimes n}.$$9-type qubits and the $[[n,k,d]]$0 operators on the $[[n,k,d]]$1-type qubits is a stabilizer of the state. The construction retains degree-$[[n,k,d]]$2 connectivity, but dominant $[[n,k,d]]$3 errors generate syndrome strings restricted to disconnected $[[n,k,d]]$4-dimensional planes rather than arbitrary $[[n,k,d]]$5-dimensional paths. Under the circuit-level biased Pauli noise model used in that work, the threshold exceeds $[[n,k,d]]$6 for large bias $[[n,k,d]]$7, whereas the usual RHG cluster state under $[[n,k,d]]$8-biased noise remains below $[[n,k,d]]$9. This is the sense in which the generalized construction is bias-preserving rather than merely code-equivalent under local Clifford deformation (Claes et al., 2022).

4. Construction beyond conventional foliation

Foliation is a powerful but restricted subclass of fault-tolerant cluster-state construction. One line of work characterizes conventional foliated FTCSs by their layered prism decomposition and then exhibits non-foliated FTCSs that still support syndrome extraction and logical correlation surfaces but cannot be interpreted as a single $Z$0-dimensional code undergoing multiple rounds of stabilizer measurement. In that formulation, any space-filling $Z$1-dimensional cell complex defines a FTCS. The constructive operation called splitting acts on the syndrome graph by replacing a vertex with multiple vertices joined by new edges; in the dual cell-complex picture, this inserts new faces that cut cells into multiple cells. Dual and primal splits commute, so the two sectors can be modified independently. The diamond FTCS, triamond FTCS, and doubled-edge cubic FTCS are presented as explicit non-foliated examples, and the triamond FTCS reaches degree $Z$2 in the syndrome graph while remaining self-dual (Nickerson et al., 2018).

A more systematic generalization treats a fault-tolerant cluster state itself as the primary object and builds it from a length-$Z$3 chain complex

$Z$4

For geometric constructions, $Z$5 are the $Z$6-cells, faces, edges, and vertices of a $Z$7-dimensional tiling. The underlying graph state is bipartite with parts $Z$8 and $Z$9, and biadjacency matrix $1$00; qubits indexed by $1$01 are dual qubits, while qubits indexed by $1$02 are primal qubits. In this framework, foliated codes appear as a special subclass obtained by embedding a length-$1$03 CSS complex into a special length-$1$04 complex representing time evolution. However, the framework also contains constructions that cannot be realized as foliations of any code. The central example is a new self-dual fault-tolerant cluster state whose underlying graph state has degree $1$05; that work emphasizes that such a construction necessarily cannot be realized as the foliation of any code. The broader design problem is then recast as a search over crystal structures and cellulations, with self-duality and low graph-state degree emerging as competing architectural desiderata (Newman et al., 2019).

5. Assembly primitives, probabilistic stitching, and architectural overhead

Although many implementation papers do not discuss foliated cluster states explicitly, they supply the primitives by which foliated resources can be assembled. At the encoded-graph-state level, edge local complementation can be implemented in a restricted graph-state setting by applying Hadamard gates to the two edge qubits, and in the five-qubit-code construction a single physical $1$06 operation together with local operations is sufficient to create a logical $1$07 operation between two logical qubits. This is indirectly relevant to foliation because it provides a compact primitive for generating encoded graph-state layers and encoded entangling links (Joo et al., 2011). In photonic constructions, generalized type II fusion gives a systematic classification of how two cluster fragments can be stitched together: some successful branches yield cluster states up to local corrections, some yield weighted graph states, and some yield only product states. However, the success probability for obtaining a maximally entangled state is bounded by $1$08, the bound is saturated only under a specific matrix condition on the optical unitary, and the only states obtainable with $1$09 success are product states. This suggests that generalized fusion enlarges the class of usable successful branches but does not remove the probabilistic bottleneck for genuine cluster-compatible inter-layer links (Rimock et al., 2024).

Resource generation can also be analyzed at the architectural level. In just-in-time cluster growth with probabilistic entangling operations, the required qubit coherence time satisfies

$1$10

with geometry-dependent penalties that worsen as additional transverse links are required. That analysis is not specific to foliation, but it is directly relevant to layered cluster-state construction because foliated architectures also consume one front of the resource while future graph fragments are still being assembled (Waddington et al., 2013). Qubus-based layer-by-layer generation of $1$11 cluster states gives concrete operation-count formulas, including $1$12 when previously created links are not destroyed and $1$13 when one allows a more complicated scheme that can destroy links. The related “Lego brick” construction reduces the asymptotic operation count to $1$14 for brick length $1$15. These results concern $1$16-dimensional clusters rather than foliated ones, but they are directly suggestive for repeated-slice generation because they formalize modular assembly, boundary sharing, and bus-reuse scheduling (Brown et al., 2011, Horsman et al., 2010).

At larger scale, several architectures realize or approximate foliated resources by stochastic growth. One percolation-based architecture for atomic memories generates a random graph state with missing edges on a chosen lattice and uses single-qubit-measurement renormalization to obtain a regular universal cluster state with constant overhead once the bond probability exceeds the percolation threshold; for a square lattice the threshold is $1$17, and the time to percolation is

$1$18

A complementary photonic analysis studies a streaming $1$19 cluster with one time-like direction and two spatial directions, showing that a finite physical depth $1$20 of order $1$21–$1$22 layers can suffice for long-range online pathfinding in the idealized lossless model. Most directly, a neutral-atom protocol constructs an RHG cluster state by first generating star-graph resource states via counterfactual carving and then connecting them by heralded cavity-mediated C-phase gates; with cooperativity $1$23, star size $1$24, and non-carving infidelity $1$25, the simulations place both Pauli noise and loss-like edge failures about a factor of ten below the cited RHG thresholds. Although these works are not framed primarily in the language of foliation, they are directly relevant as physical routes to assembling the same $1$26-dimensional entanglement scaffold that a foliated-code description uses (Choi et al., 2017, Morley-Short et al., 2017, Stewart et al., 26 Jul 2025).

6. Chain-complex, symmetry, and bulk–boundary formulations

A recent chain-complex reformulation makes the topology of foliated cluster states explicit. A CSS code is written as the chain complex

$1$27

where $1$28 labels code qubits, $1$29 labels $1$30-stabilizers, and $1$31 labels $1$32-stabilizers. Foliation introduces a $1$33-dimensional direction $1$34 and tensors this CSS complex with the cell complex of a $1$35-dimensional lattice, giving a new complex whose qubit-bearing groups are

$1$36

The bipartite cluster graph is then encoded by the incidence matrix of the foliated differential $1$37, and the cluster-state stabilizers take the generalized form

$1$38

for $1$39 and $1$40. In this formulation, the standard foliated cluster state is not merely a graph-state stack but the tensor-product complex of a CSS code with a $1$41-dimensional repetition-code complex (Okuda et al., 2024).

This chain-complex description is used to argue that the foliated cluster state has symmetry-protected topological order protected by generalized global symmetries supported on cycles of the foliated complex. For open boundaries in the foliation direction, the original CSS code appears on the boundary after measuring all bulk qubits except the code-qubit boundary layers in the $1$42 basis and postselecting on $1$43. Bulk cycles induce boundary logical operators, and the gauge transformations of the bulk and boundary partition functions match, which is presented as an explicit anomaly-inflow relation between the CSS code and its foliated cluster-state bulk. The same machinery extends to fractonic models, with worked examples including the $1$44-dimensional plaquette Ising model, the X-cube model, and the checkerboard model. In this view, foliated cluster state construction is simultaneously a graph-state preparation protocol, a tensor-product operation on chain complexes, and a bulk–boundary mechanism by which a measured higher-dimensional cluster reproduces the original code on its boundary (Okuda et al., 2024).