The Structure of Circle Graph States

Abstract

Circle graph states are a structurally important family of graph states. The family's entanglement is a priori high enough to allow for universal measurement-based quantum computation (MBQC); however, MBQC on circle graph states is actually efficiently classically simulable. In this work, we paint a detailed picture of the local equivalence of circle graph states. First, we consider the class of all graph states that are local unitary (LU)-equivalent to circle graph states. In graph-theoretical terms, this LU-equivalence class is the set of all graphs reachable from the family of circle graphs by applying $r$-local complementations. We prove that the only graph states that are LU-equivalent to circle graph states are circle graph states themselves: circle graphs are closed under $r$-local complementation. Second, we show that bipartite circle graph states, i.e., 2-colorable circle graph states, are in one-to-one correspondence with planar code states, on which MBQC is known to be efficiently classically simulable. Leveraging this correspondence, we present alternative, simple proofs that (1) if a planar code state is LU-equivalent to a stabilizer state, they are in fact local Clifford (LC)-equivalent to it and that (2) MBQC on all circle graph states is efficiently classically simulable. Third and finally, we demonstrate that the problem of counting the number of graph states LU-equivalent to a given graph state is $\#\mathsf{P}$-hard.

Figures (8)

• Figure 1: Circle graphs $L_5$ and $L_5 \star c$ and their chord representations.
• Figure 2: Circle graphs and bipartite circle graphs can both be characterized by excluded minors: A simple graph is a circle graph if and only if it does not have a vertex-minor isomorphic to any of the three graphs depicted in the violet box bouchetCircleGraphObstructions1994.A simple graph is a bipartite circle graph if and only if it does not have a pivot-minor isomorphic to any of the four graphs depicted in the green box. Note that all obstructions that characterize all circle graphs (bipartite or not) via excluded pivot-minors can be found in geelen2009circle.
• Figure 3: For an exemplary $9$-vertex circle graph $C$ with a choice of independent set $K=\{1,2,3,4\}$, we show how to construct the corresponding tree $T$ and multigraph $H$. Violet arrow: From the chord representation of $C$, we remove all chords not corresponding to $K=\{1,2,3,4\}$. This yields a simpler diagram of four nonintersecting chords that divide the circle into five regions. In each of the five circle regions, we then draw a vertex and connect those pairs of vertices that correspond to pairs of neighboring regions by a tree edge. We label each of the four tree edges with the name of the chord that separates the neighboring circle regions. Green arrow: Adding all vertices $e\in V(G) \setminus K$ (that lie outside the independent set $K$) as additional edges to this tree $T$, creates fundamental cycles $F_e$ in a new multigraph $H$. As an example, the fundamental cycles $\textcolor{orange}{F_5}$ and $\textcolor{brightgreen}{F_6}$ are highlighted. Note that in the chord diagram of $C$, $e$ intersects a different chord $v$ if and only if $v$ is an edge of the fundamental cycle $F_e$ of $H$. In our example, $F_5$, containing edge $2$, corresponds to the $5$ and the $2$ chords intersecting the chord diagram of $C$ (and $F_6$, containing edges $1, 2$, corresponds to $6$ intersecting both $1, 2$ in the chord diagram of $C$).
• Figure 4: Illustration of Theorem \ref{['thm:planar_code_states_are_bipartite_circle_graph_states']}. The gray box shows an example of a planar code state $\ket{\psi}$, where the corresponding planar multigraph $P$ is the $(4\times 4)$-grid graph without any multideges. The green box then shows how an arbitrary selection of a spanning tree $T$ of $P$ allows for the construction of a corresponding bipartite circle graph by applying a Hadamard gate to each qubit outside of the spanning tree.
• Figure 5: Every circle graph $C$ is a vertex-minor of some bipartite circle graph $B$. We show how to construct such a bipartite $B$ for the example where $C$ is a 5-cycle. 1. From a chord diagram of $C$, we read off (clockwise or counterclockwise) a double occurrence word ($aebacbdced$) and draw a 4-regular multigraph with vertices $a,b,c,d,e$ and edges for all adjacent letters in this word (first and last letter are considered adjacent). 2. This 4-regular multigraph generally has a nonzero crossing number, but we can planarize it. Here, the crossing number is one and thus adding a single vertex $f$ creates a planar 4-regular multigraph. We then 2-color the faces of this graph in green and white. 3. Choose one face color (w.l.o.g. green). Draw a new graph with vertices for each green face and edges for each point where the green faces touch. 4. Select a spanning tree and draw a contour around it. 5. Rotate the edges of the spanning tree so that they connect to the contour. 6. Transform the contour into a circle, pulling the edges that lie outside the spanning tree into the circle. The result is a chord diagram of a bipartite circle graph.

Theorems & Definitions (43)

• Definition 1: Graph state
• Definition 2: Clifford group
• Definition 3: Local Clifford group
• Definition 4: LC-equivalence
• Definition 5: LU-equivalence
• Definition 6: $r$-Incidence claudet2024local
• Definition 7: $r$-Local Complementation claudet2024local
• Definition 8: Vertex-minor
• Proposition 1: see Proposition 5 in Cautres2024
• Definition 9: Pivot-minor