Graph-Structured States

• Graph-structured states are multipartite quantum states defined by graphs that determine interaction patterns, stabilizer structures, or edge constraints.
• They are constructed using controlled entangling gates and generalized via symmetric Hadamard matrices to extend to qudits, continuous variables, directed, and weighted systems.
• These states enable scalable protocols in quantum computing and error correction, offering efficient preparation, verification, and networking techniques in diverse physical platforms.

Graph-structured states are multipartite quantum states for which a graph specifies either the interaction pattern, the stabilizer structure, the nullifier constraints, or a more general edge-operator construction. In the canonical qubit setting, a simple undirected graph $G=(V,E)$ determines a unique graph state $|G\rangle$ both as the common $+1$ eigenstate of commuting vertex stabilizers and as the result of applying controlled-$Z$ gates along the edges to $|+\rangle^{\otimes |V|}$ (Ionicioiu et al., 2011). This construction has been extended in several directions: to qudits via symmetric complex Hadamard matrices, to continuous-variable Gaussian states via nullifiers, to directed and weighted graphs through generalized edge operators, and to code constructions that go beyond stabilizer codes (Cui et al., 2015). The subject therefore spans a core stabilizer formalism, a family of representation theorems, and a set of operational protocols for preparation, transformation, distribution, and verification.

1. Canonical graph states

For a simple undirected graph $G=(V,E)$ on $n=|V|$ vertices, the standard $n$-qubit graph state is defined by

$$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$

with $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$ and $|G\rangle$0 (Zhao et al., 2015). Equivalently, it is the unique simultaneous $|G\rangle$1-eigenstate of the commuting stabilizer generators

$|G\rangle$2

where $|G\rangle$3 is the neighborhood of vertex $|G\rangle$4 (Corli et al., 2022). The stabilizer group generated by $|G\rangle$5 is abelian and has size $|G\rangle$6 (Corli et al., 2022).

This dual circuit–stabilizer description is central because it makes graph states simultaneously combinatorial and algebraic. In the circuit picture, the edge set controls entangling gates; in the stabilizer picture, the same graph controls the support pattern of Pauli generators. Several standard resource states arise as special graphs. Complete graphs are locally equivalent to GHZ states, and star graphs likewise give GHZ-type states under local unitaries (Vandré et al., 2024). Path, cycle, and lattice graphs produce the 1D, ring, and cluster-state families widely used in measurement-based quantum computation and error correction (Unnikrishnan et al., 2020).

A broader axiomatic formulation replaces the specific qubit-CZ choice by a map $|G\rangle$7 satisfying separability on disjoint unions, covariance under graph isomorphism, and a universal edge operator $|G\rangle$8 that implements edge addition (Ionicioiu et al., 2011). Under these axioms, the total Hilbert space is forced to have tensor-product form $|G\rangle$9, and the state of a graph is built by applying $+1$0 to every occupied edge of the empty-graph product state $+1$1 (Ionicioiu et al., 2011). Standard qubit graph states are recovered by choosing $+1$2, $+1$3, and $+1$4.

2. Generalizations beyond the qubit stabilizer setting

A direct generalization replaces the qubit Hadamard gate by a symmetric complex Hadamard matrix $+1$5 satisfying $+1$6 and $+1$7. With the two-qudit controlled-phase gate defined by

$+1$8

one sets

$+1$9

thereby obtaining generalized graph states on $Z$0 qudits (Cui et al., 2015). When $Z$1 and $Z$2 is the qubit Hadamard, this reduces to the usual graph state; when $Z$3, the $Z$4-point discrete Fourier matrix, one recovers the standard qudit graph state (Cui et al., 2015).

This Hadamard-matrix construction shifts the emphasis from abelian Pauli stabilizers to encoding circuits determined by the pair $Z$5. It also produces a natural generalization of the Pauli $Z$6 pair through the notion of an $Z$7-symmetry: a permutation–diagonal pair $Z$8 satisfying $Z$9 (Cui et al., 2015). These symmetries induce local stabilizers of the state, and in the qubit Fourier case they reduce to the familiar $|+\rangle^{\otimes |V|}$0 and $|+\rangle^{\otimes |V|}$1 stabilizers (Cui et al., 2015).

The same axiomatic framework accommodates directed and weighted graphs by relaxing the symmetry condition on the edge operator. For directed graphs, one uses generally distinct operators $|+\rangle^{\otimes |V|}$2 subject to a set of commutativity constraints; for weighted graphs, one lets the operator $|+\rangle^{\otimes |V|}$3 depend on edge parameters while preserving mutual commutativity (Ionicioiu et al., 2011). A separate, noncommutative directed-edge construction uses the two-qudit gate $|+\rangle^{\otimes |V|}$4, which requires an orientation and a vertex ordering. In that setting, complete directed graphs with hierarchical orientation generate totally antisymmetric multipartite states for odd $|+\rangle^{\otimes |V|}$5, whereas the ordinary undirected CZ construction yields full permutation symmetry if and only if the underlying graph is complete (Jesus et al., 27 Jan 2026).

Continuous-variable graph states provide another extension. For $|+\rangle^{\otimes |V|}$6 modes with quadratures $|+\rangle^{\otimes |V|}$7 and real symmetric adjacency matrix $|+\rangle^{\otimes |V|}$8, the ideal state is the joint zero-eigenstate of the nullifiers

$|+\rangle^{\otimes |V|}$9

In practice one prepares Gaussian states with small nullifier variances $G=(V,E)$0 (Cooper et al., 2022). This nullifier formalism places graph-structured states within the broader theory of Gaussian cluster states and networked bosonic entanglement.

3. Entanglement, symmetry, and local equivalence

Generalized Hadamard-based graph states display a strong local entanglement property: for any connected $G=(V,E)$1 and any symmetric Hadamard $G=(V,E)$2, each single-qudit reduced density matrix of $G=(V,E)$3 is maximally mixed, $G=(V,E)$4 (Cui et al., 2015). A key ingredient is the star-graph case, which is locally unitary equivalent to the $G=(V,E)$5-qudit GHZ state

$G=(V,E)$6

The same work shows that if $G=(V,E)$7, then $G=(V,E)$8 factorizes, up to qudit reordering, as $G=(V,E)$9 (Cui et al., 2015).

Questions of local equivalence have been central to the theory. For standard graph states, local Clifford equivalence is completely captured by local complementation of the underlying graph (Claudet, 27 Nov 2025). A more subtle issue is local unitary equivalence. The conjecture that LU-equivalent graph states are always LC-equivalent is false in general; a 27-vertex counterexample exists (Claudet, 27 Nov 2025). Nevertheless, the structure is far from arbitrary. A hierarchy of $n=|V|$0-local complementations has been introduced that fully captures LU-equivalence, yields a quasi-polynomial algorithm for deciding LU-equivalence, and implies that LU=LC for graph states on at most 19 qubits (Claudet, 27 Nov 2025). Independently, a marginal-based analysis increases the bound to 10 qubits in the setting where qubit permutations are allowed (Vandré et al., 2024).

Certain graph families are especially rigid. Circle graphs are closed under $n=|V|$1-local complementation, so the only graph states LU-equivalent to circle graph states are circle graph states themselves; on this family, LU-equivalence coincides with LC-equivalence (Hahn et al., 9 Mar 2026). For bipartite circle graph states there is a one-to-one correspondence with planar code states, and this correspondence underlies the statement that measurement-based quantum computation on all circle graph states is efficiently classically simulable (Hahn et al., 9 Mar 2026).

Symmetry statements also admit graph-theoretic characterizations. In the standard CZ-based construction, a graph state is invariant under all particle permutations if and only if the graph is complete (Jesus et al., 27 Jan 2026). In the directed $n=|V|$2-based construction, complete directed graphs with appropriate orientations generate fully antisymmetric states for odd $n=|V|$3 (Jesus et al., 27 Jan 2026). These results tie exchange symmetry directly to completeness and orientation, rather than to an external symmetrization postulate.

4. Structural representations and invariants

Graph states admit compressed algebraic descriptions that exploit graph topology. One reduced representation selects a subset of “hub” vertices $n=|V|$4 such that every edge appears in the neighborhood of at least one hub, and writes the full state as

$n=|V|$5

with $n=|V|$6 (Corli et al., 2022). For an even ring one can choose $n=|V|$7; for a star, $n=|V|$8, so the description collapses from $n=|V|$9 generators to just one (Corli et al., 2022). The algebraic expansion cost then grows like $n$0 rather than $n$1 (Corli et al., 2022).

Generalized graph states also admit a PEPS interpretation. Each edge carries the bond state

$n$2

and each vertex is projected by a map determined by the local copy of $n$3, reproducing the amplitude formula of $n$4 (Cui et al., 2015). This places the Hadamard-based construction in direct contact with tensor-network methods.

Topological stabilizer states can likewise be recast graph-theoretically. The toric code graph is composed of only two kinds of subgraphs, star graphs and half graphs (Liao et al., 2021). In this representation, topological order is identified with the existence of multiple star graphs, exposing a relation between repetition-code structure and the toric code (Liao et al., 2021). The same graph structure yields a log-depth circuit for state preparation under geometrically non-local gates and can be reduced to constant depth using ancillae and measurements at the cost of increased circuit width (Liao et al., 2021).

Marginals provide another structural lens. For a subset $n$5, the reduced stabilizer

$n$6

has size $n$7, and the reduced state can be written

$n$8

with $n$9 (Vandré et al., 2024). The quantities $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$0, their frequency vectors $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$1, and the marginal-dimension tensors $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$2 are LU-invariants. These invariants uniquely identify the entanglement classes of every graph state up to 8 qubits, although they cease to be complete for larger systems (Vandré et al., 2024).

5. Transformations, networking, and certification

Graph states support a rich calculus of graph transformations implemented by low-depth quantum operations. As a quantum data structure for undirected graphs, they permit edge complementation by a single $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$3, neighborhood complementation by one CNOT, and local complementation by the five-gate unitary $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$4, each with $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$5 gate complexity and depth $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$6 (Zhao et al., 2015). Larger interset and intraset complementations can be carried out ancilla-assisted in time linear in the set size (Zhao et al., 2015). Equality testing, automorphism testing, vertex-comparison tests, and full adjacency-matrix readout also admit explicit quantum procedures; full readout consumes $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$7 copies, with total expected copies used $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$8 (Zhao et al., 2015).

Bell-state measurements induce another transformation theory. For graph states, a complete set of fusion rules has been derived for five inequivalent Bell-fusion success cases, each expressed as an update rule for the neighborhoods of vertices relative to the fused qubits $$|G\rangle=\Bigl(\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\Bigr)|+\rangle^{\otimes n},$$9 and $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$0 (Löbl et al., 2024). These rules provide a set-algebra description of how stabilizer graphs transform under probabilistic fusions and are intended for constructing graph codes or simulating fusion networks (Löbl et al., 2024).

In network settings, graph states can be distributed rather than prepared in situ. The Graph State Transfer protocol begins from a root node, creates a local copy of $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$1, and moves the graph-state qubits to their destination nodes by sequences of connection transfers along selected network paths (Fischer et al., 2020). The number of EPR pairs consumed is

$|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$2

with the worst-case upper bound

$|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$3

and the completion-time minimization reduces to a network-flow problem testable in polynomial time, $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$4 (Fischer et al., 2020).

Verification protocols address adversarial sources and dishonest network parties. One protocol uses only multiple copies of the graph state, local Pauli measurements, and classical communication, and is globally efficient for a large family of useful graph states including cluster states, GHZ states, and cycle graph states (Unnikrishnan et al., 2020). For general graph states, efficiency with respect to the security parameter is maintained, though there is a cost increase with the graph size (Unnikrishnan et al., 2020).

6. Physical realizations and neighboring families

Deterministic generation architectures have been proposed for photonic graph states using quantum-dot emitters coupled to a waveguide, optical fiber delay lines, passive CZ scattering blocks, and at most one optical switch (Shapourian et al., 2022). In a 3D graph state on a Raussendorf–Harrington–Goyal lattice, the reported circuit-error threshold is $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$5; the upper bound on correctable loss is $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$6 dB for direct generation and $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$7 dB for indirect construction from a simple cubic cluster state (Shapourian et al., 2022). These figures place graph-structured states directly within fault-tolerant photonic architectures.

Continuous-variable graph states have been engineered in atomic spin ensembles using global photon-mediated interactions in an optical cavity combined with local spin rotations (Cooper et al., 2022). A four-mode square graph state with adjacency matrix

$|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$8

was realized through a four-pulse sequence of local rotations, cavity squeezing, and global spinor rotations (Cooper et al., 2022). The measured nullifier variances were approximately $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$9, averaging $|G\rangle$00, and bidirectional EPR steering was reported with $|G\rangle$01 and $|G\rangle$02 (Cooper et al., 2022).

Not all graph-structured quantum states are graph states in the stabilizer sense. The $|G\rangle$03-graph states are density operators defined by

$|G\rangle$04

where $|G\rangle$05 is the adjacency matrix, $|G\rangle$06 the degree matrix, and $|G\rangle$07 (Kumar et al., 17 Dec 2025). These states are explicitly different from the standard graph states arising from stabiliser formalism (Kumar et al., 17 Dec 2025). Their positivity is characterized by the smallest eigenvalue of $|G\rangle$08 and the minimum degree, and their entanglement can be detected through PPT conditions and the $|G\rangle$09-PPT criterion expressed in graph invariants such as $|G\rangle$10, degree sums, and triangle counts in the partial-transpose graph (Kumar et al., 17 Dec 2025).

Taken together, these constructions show that “graph-structured states” names a family of theories rather than a single formalism. At one pole lies the orthodox stabilizer graph state, exactly specified by a simple graph and controlled-$|G\rangle$11 entangling gates; at the other lie qudit, continuous-variable, directed, weighted, noncommutative, and mixed-state generalizations in which the graph continues to organize locality, symmetry, and correlation, but no longer fixes a unique Pauli-stabilizer description.