Circle Graph States: Theory and MBQC Implications

• Circle graph states are graph states whose underlying graphs are circle graphs—intersection graphs of chords—with key applications in MBQC and structural quantum theory.
• They enable efficient simulation through local Clifford operations and Pfaffian evaluations, offering near-linear preparation costs compared to quadratic bounds in general graph states.
• Their LU=LC equivalence and relation to planar-code states facilitate algorithmic classification, emphasizing their structural rigidity and role at the boundary of MBQC universality.

Circle graph states are graph states $\lvert G\rangle$ whose underlying simple graph $G$ is a circle graph, i.e. the intersection graph of a finite family of chords of a circle. In stabilizer language they are a natural subclass of graph states; in graph theory they are a canonical vertex-minor-closed family; and in measurement-based quantum computation (MBQC) they occupy a distinctive intermediate position. Any graph-state family with unbounded entanglement width must be able to generate every circle graph state using local Clifford operations, local Pauli measurements, and classical communication, yet circle graph states themselves are not efficiently universal resources for MBQC, assuming $\mathsf{BQP}\neq \mathsf{BPP}$ (Harrison et al., 7 Oct 2025). Recent work further shows that their full local-unitary orbit does not leave the family, that bipartite circle graph states coincide with planar code states up to local Clifford equivalence, and that their entangling-gate preparation cost admits explicit near-linear upper bounds (Hahn et al., 9 Mar 2026, Davies et al., 31 Mar 2025).

1. Graph-theoretic foundation

A circle graph is the intersection graph of a finite set of chords of a circle: vertices correspond to chords, and two vertices are adjacent exactly when the corresponding chords intersect. Equivalent descriptions play a central role in the theory. A graph is a circle graph if and only if it is the alternance graph of a double-occurrence word, and circle graphs also admit an Eulerian-tour characterization via $4$-regular multigraphs: every circle graph can be obtained as the alternance graph of the vertex sequence of an Eulerian tour of a $4$-regular multigraph, and conversely such a $4$-regular multigraph and Eulerian tour can be constructed efficiently when the input graph is a circle graph (Harrison et al., 7 Oct 2025).

This graph-theoretic meaning should be distinguished from unrelated uses of similar language. In particular, the literature on circulant graphs concerns Cayley graphs on $\mathbb Z_n$, continuous-time quantum walks, and state transfer on cycles and more general circulants, rather than chord-intersection graphs (Pal, 2017).

For graph states, the circle-graph condition is imposed on the underlying graph. Thus a circle graph state is not an additional quantum structure beyond a graph state; it is a graph state with a restricted graph family. That restriction is nevertheless highly nontrivial. Circle graphs form a canonical vertex-minor-closed class, and the family is structurally rich enough to be central in rank-width and MBQC universality theory (Davies et al., 31 Mar 2025, Harrison et al., 7 Oct 2025).

2. Stabilizer formulation and local equivalence

For a graph $G$, the associated graph state is

$$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$

with stabilizer generators

$S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$

Every stabilizer state is local-Clifford equivalent to some graph state, so graph states provide a canonical representative system for stabilizer-state analysis (Harrison et al., 7 Oct 2025).

The relevant equivalence notions are local Clifford (LC) and local unitary (LU) equivalence. For graph states, LC-equivalence is encoded graph-theoretically by local complementation. If $G$0, then local complementation at $G$1 replaces the induced subgraph on $G$2 by its complement: $G$3 More generally, LU-equivalence of graph states can be described in terms of valid $G$4-local complementations, a larger family of graph operations defined on $G$5-incident independent multisets (Hahn et al., 9 Mar 2026).

The principal structural theorem is that circle graph states satisfy $G$6: every valid $G$7-local complementation on a circle graph is implemented by a sequence of ordinary local complementations, and circle graphs are closed under $G$8-local complementation (Hahn et al., 9 Mar 2026). This yields a sharp orbit statement: the only graph states that are LU-equivalent to circle graph states are circle graph states themselves. The proof is graph-theoretic rather than purely obstruction-theoretic. It reduces a general $G$9-incident multiset to a cleaned-up normal form without twins and without degree-$\mathsf{BQP}\neq \mathsf{BPP}$0 or degree-$\mathsf{BQP}\neq \mathsf{BPP}$1 support vertices, then uses a structural lemma for independent sets in circle graphs: if $\mathsf{BQP}\neq \mathsf{BPP}$2 is an independent set with no twins and with at least one vertex of degree at least $\mathsf{BQP}\neq \mathsf{BPP}$3, then there exist $\mathsf{BQP}\neq \mathsf{BPP}$4 with exactly one common neighbor in $\mathsf{BQP}\neq \mathsf{BPP}$5. That configuration contradicts the divisibility constraints required by $\mathsf{BQP}\neq \mathsf{BPP}$6-incidence, forcing the nontrivial $\mathsf{BQP}\neq \mathsf{BPP}$7-part to vanish (Hahn et al., 9 Mar 2026).

A further consequence is algorithmic and classificatory. For circle graph states, the LU orbit of graph-state representatives coincides with the LC orbit, so equivalence checking reduces to the standard local-complementation framework on graphs. The same work also states that counting the number of graph states LU-equivalent to a given graph state is P-hard, even when restricted to circle graph states; it explicitly distinguishes this from a $\mathsf{BQP}\neq \mathsf{BPP}$8-hardness statement (Hahn et al., 9 Mar 2026).

3. Preparation complexity and $\mathsf{BQP}\neq \mathsf{BPP}$9-distance

The preparation problem for circle graph states is naturally formulated in terms of the $4$0-distance. Using the gate set $4$1, where only $4$2 gates count toward cost, one defines $4$3 as the minimum number of $4$4 gates needed to prepare $4$5. Equivalently, in graph language, one may toggle single edges while allowing arbitrary local complementations for free between toggles (Davies et al., 31 Mar 2025).

For arbitrary $4$6-vertex graphs, the worst-case $4$7-distance is $4$8, and there exist graphs with $4$9. Circle graphs admit much stronger bounds. If $4$0 is an $4$1-vertex circle graph with clique number at most $4$2, then

$4$3

Since $4$4, this gives $4$5 for all circle graphs, and $4$6 when $4$7 is bounded (Davies et al., 31 Mar 2025).

The proof uses an interval-overlap representation of circle graphs. A key gadget shows that for a circle graph with an isolated helper vertex $4$8, if $4$9 is the edge set between disjoint sets $4$0 and $4$1, then one can toggle exactly the edges in $4$2 inside any ambient graph with $4$3 isolated at cost at most $4$4. This is achieved by sweeping interval endpoints left to right and updating the neighborhood of $4$5 so that it tracks the currently active intervals. A divide-and-conquer argument on a proper coloring then yields

$4$6

for an $4$7-vertex circle graph with chromatic number at most $4$8. The clique-number bound follows from the theorem that every circle graph with clique number at most $4$9 is $\mathbb Z_n$0-colourable (Davies et al., 31 Mar 2025).

These results place circle graph states in a favorable preparation regime: the entangling-gate overhead is near-linear in the number of qubits, in strong contrast with the almost quadratic worst-case behavior for unrestricted graph states (Davies et al., 31 Mar 2025).

4. Vertex-minors, rank width, and the planar-code correspondence

Circle graph states are central in MBQC resource theory because entanglement width and vertex-minor structure meet precisely on this family. For graph states, entanglement width equals rank width, and a family of graphs has unbounded rank width if and only if it contains all circle graphs as vertex minors. Together with the fact that reachability under local Clifford operations, local Pauli measurements, and classical communication is equivalent to vertex-minor containment, this implies that a graph-state family has unbounded entanglement width if and only if it can produce every circle graph state under those operations (Harrison et al., 7 Oct 2025).

This centrality does not mean that circle graph states are low-complexity only because of bounded width. On the contrary, recent work proves that there are infinitely many $\mathbb Z_n$1-vertex circle graphs with rank-width $\mathbb Z_n$2, using $\mathbb Z_n$3 comparability grids as examples (Hahn et al., 9 Mar 2026). A common misconception is therefore excluded: large rank width, or even the unbounded entanglement width required by universality theorems, is not sufficient for MBQC universality.

The bipartite subclass is especially rigid. Every planar code state is LC-equivalent to a bipartite circle graph state, and conversely every bipartite circle graph state is LC-equivalent to a planar code state (Hahn et al., 9 Mar 2026). In the planar-code-to-graph-state direction, one starts from a planar multigraph $\mathbb Z_n$4 with qubits on edges, chooses a spanning tree $\mathbb Z_n$5, and applies Hadamards on the non-tree edges. The resulting graph state is bipartite, with one color class given by $\mathbb Z_n$6 and the other by $\mathbb Z_n$7; adjacency is determined by membership in the corresponding fundamental cycles. The underlying graph is the fundamental graph of the planar multigraph, and de Fraysseix’s theorem identifies bipartite circle graphs exactly with such fundamental graphs (Hahn et al., 9 Mar 2026).

This correspondence extends beyond the bipartite case via vertex-minors. Every $\mathbb Z_n$8-vertex circle graph is a vertex-minor of some $\mathbb Z_n$9-vertex bipartite circle graph (Hahn et al., 9 Mar 2026). A plausible implication is that many structural and algorithmic questions for general circle graph states can be reduced to the bipartite, and therefore planar-code, setting with polynomial overhead.

5. MBQC simulability and non-universality

The decisive computational result is that circle graph states are not efficiently universal resources for MBQC, assuming $G$0 (Harrison et al., 7 Oct 2025). The proof is not based on low entanglement but on a hidden fermionic structure.

Starting from a circle graph $G$1, one uses its Eulerian-tour representation by a $G$2-regular multigraph $G$3. The $G$4 half-edges of $G$5 are treated as Majorana modes grouped into $G$6 blocks of four, and the perfect matching induced by the multigraph defines a pure fermionic Gaussian matching state $G$7. Under Kitaev’s four-Majorana-per-qubit encoding, the circle graph state $G$8 appears exactly as the $G$9-gauge sector of $$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$0; more precisely, the matching state decomposes as a uniform superposition over gauge sectors with $$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$1, and $$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$2 (Harrison et al., 7 Oct 2025).

This correspondence turns MBQC probabilities into Pfaffian evaluations. For a product state $$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$3,

$$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$4

and for partial adaptive measurement probabilities on $$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$5 measured qubits,

$$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$6

Since Pfaffians are computable in polynomial time, arbitrary adaptive single-qubit measurement patterns on circle graph states are efficiently classically simulable (Harrison et al., 7 Oct 2025).

A second proof route is available through the planar-code correspondence. MBQC on planar code states is efficiently classically simulable, bipartite circle graph states are exactly planar code states up to LC-equivalence, and every circle graph is a vertex-minor of a polynomially larger bipartite circle graph. This yields an alternative simulation argument for all circle graph states (Hahn et al., 9 Mar 2026).

The combination of these results gives circle graph states a sharply defined status. They are unavoidable witnesses of unbounded entanglement width, yet they remain efficiently classically simulable. This suggests that the family marks a boundary between structural necessity for universality and computational sufficiency for universality.

6. Representation theory, complements, and structural characterizations

The graph theory of circle graphs is unusually rich, and several results bear directly on circle graph states through the graph-state dictionary.

The partial representation extension problem asks whether a circle representation with some pre-drawn chords can be extended to the whole graph. For circle graphs this problem is solvable in $$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$7 time. The proof develops a structural description of all circle representations using maximal split decomposition: for a nontrivial maximal split, every representation decomposes into paired blocks corresponding to equivalence classes, while trivial maximal splits reduce to arranging component representations around an articulation-like vertex (Chaplick et al., 2013). This provides a recursive grammar for circle representations and, indirectly, a way to reason about constrained realizations of circle graph states.

Circle graphs are not generally closed under complement, but an important special closure does hold: if a bipartite graph $$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$8 is the complement of a circle graph, then $$\lvert G\rangle=\left(\prod_{\{u,v\}\in E(G)} CZ_{uv}\right)\lvert +\rangle^{\otimes |V(G)|},$$9 is itself a circle graph (Esperet et al., 2019). The proof is geometric. In a chord model of $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$0, the bipartition of $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$1 becomes two families of pairwise intersecting chords; a bisecting-line lemma produces a line meeting every chord, and reversing the endpoint order on one side of that line toggles all intersections, thereby converting a model of $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$2 into a model of $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$3 (Esperet et al., 2019).

A deeper algebraic characterization comes from multimatroid theory. Let $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$4 be the isotropic matroid of a graph $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$5, with ground set partitioned into triples $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$6, and let $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$7 be the isotropic $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$8-matroid obtained by restricting the rank function to subtransversals. Then $S_v=X_v\prod_{w\in \mathcal N_G(v)} Z_w.$9 is a circle graph if and only if $G$00 is representable over every field, equivalently over some field of characteristic different from $G$01 (Brijder et al., 2017). The full isotropic matroid is generally not regular, but regularity is recovered at the $G$02-matroid level. This characterization connects circle graphs with isotropic systems, local equivalence, and vertex-minor obstructions.

At the infinite-graph level, the ambient graph $G$03 of all chords of $G$04 has automorphism group exactly $G$05, and the countable subgraph $G$06 induced by rational chords is strongly universal for countable circle graphs: every countable circle graph embeds into it as an induced subgraph (Georgakopoulos, 13 Jan 2025). The same graph is invariant under local complementation, a rare property otherwise known in this context only for $G$07 and the Rado graph (Georgakopoulos, 13 Jan 2025). For graph-state theory this is structurally suggestive, because local complementation is the graph operation corresponding to local Clifford transformations.

Taken together, these results show that circle graph states are not merely a convenient subclass of graph states. They sit at the intersection of chord-diagram geometry, vertex-minor theory, isotropic-system algebra, and MBQC resource theory. Their significance lies precisely in that combination: they are structurally rigid enough to admit strong classification and simulation theorems, yet rich enough to encode the canonical obstruction family for unbounded entanglement width.