Mapping graph state orbits under local complementation

Graph states, and the entanglement they posses, are central to modern quantum computing and communications architectures. Local complementation---the graph operation that links all local-Clifford equivalent graph states---allows us to classify all stabliser states by the entanglement they posses. Here, we study the structure of the orbits generated by local complementation, mapping them up to 9 qubits and revealing a rich hidden structure. We observe correlations of these orbits' properties, for example their colourability, with known entanglement metrics, such as Schmidt measure, and find relationships between an orbit's connectivity and the entanglement of its graph states. We provide data for each of the 587 orbits up to 9 qubits as well as means to visualise them, uncovering their exquisite patterns. It is well known that graph theory and quantum entanglement have strong interplay---our exploration deepens this relationship, providing new tools with which to probe the nature of nonlocal stabiliser states.

pends on a graph-specific parameter, the rank width 14,21 . It is also known that counting single-qubit Clifford equivalent graph states is #P-complete 22 .
Recently we showed that interspersing local complementations with CZ operations can lead to a reduction in CZ complexity 23 . However, little is known about the structure of the orbits that are generated by local complementation. These orbits are themselves graphs, in which each orbit vertex represents a graph state and edges between them are induced by local complementation of different graph state vertices (see Fig. 1). Here, we refer to the object that links graphs via local comeplemtation as their 'orbit', and we refer to those component graphs as 'graph states'. These orbits, which are wildly complex, give a fresh perspective for the study of stabiliser entanglement and graph states, while providing new tools for optimising quantum protocols.
In this work, we map the space of graph states and their relationship via local complementation, generating the orbit of each of the 587 entanglement classes up to n ≤ 9 qubits. We explicitly relate properties of these orbits to properties of their component graph states, and observe correlations between orbit parameters and Schmidt measure. We also plot an illustrative selection of these orbits, and identify promising avenues for both analysis and applications. All the data generated in this study are available online 24 , together with a means to plot them.
Remarkably, graph states can be locally equivalent, despite having different constructions via nonlocal CZ gates 11,12 . Specifically, graphs are locally equivalent if and only if they can be transformed into one another by successive applications of local complementation.
Local complementation of a vertex α, LC α , applied to a graph, G(V, E), acts to complement the neighbourhood of the vertex α. That is, in the neighbourhood of α, it removes edges if they are present, and adds any edges are missing  Fig. 1: Local complementation and its orbits. Orbit edges are labelled with the vertex that undergoes local complementation. a. A guide to local complementation. The neighbourhood of qubit α is complemented to yield the output graph. b. The orbit L 3 (GHZ entanglement of four qubits). c. The orbit C 3 , where isomorphic graph states are considered equal. d. The orbit L 4 (cluster state entanglement of four qubits). This is one of three equivalent orbits, which together contain every isomorphism of the contained graph states. e. The orbit C 4 . f. The orbit C 19 . Graph state vertices are labelled descending clockwise from noon (see b). We use directed edges when drawing C i orbits as only one isomorphism of the graph states can be drawn on an orbit.
(see Fig. 1a). More formally: where Here, K N G (α) is the set of edges of the complete graph on the vertex set N G (α), the neighbourhood of α, and, ∆ is the symmetric difference. On graph states, the following local unitary implements local complementation 11,12 : Where U LC α |G = |LC α (G) . Repeated application of local complementation is guaranteed to hit every member of a entanglement class of locally equivalent graph states, given any member of that class as a starting point 11,12 . This defines graph (and therefore stabiliser) entanglement classes, each with their own orbit under local complementation. So far, these classes have been uniquely specified 17 up to n = 12, while the structure of the orbits generated by local complementation has not yet been studied.
All n-vertex graphs can be locally complemented in n different ways, generating up to n different graphs. Each of these can be locally complemented further, generating up to n−1 new graphs (local complementation is self inverse). We can repeatedly local complement graphs until we find no new ones, concluding that all graphs in the class have been found. By performing every local complementation on every graph in the class, the orbit is mapped (see Section 2.3). We will denote these orbits L i for entanglement class i, canonically indexed as in ref. 17. This orbit is itself naturally represented as a graph-its vertices are graph states and the edges that link them are local complementations on the graph state's vertices (see Fig. 1). Edges of the orbits are labelled with a vertex index indicating which local complementation links the two graph states on the orbit vertices. Since local complementation is selfinverse, these edges are undirected. Some simple examples of orbits are shown in Figs. 1b,d.

A quantum Rubik's cube
Local complementation orbits have an entertaining analogy with a Rubik's cube (a popular puzzle toy). Each face of a Rubik's cube is a different colour, which is itself separated into 3 × 3 = 9 individual squares. This is the cube's solved state. The toy has 6 basic moves, which rotate the different faces of the cube by 90 • . By applying these six moves in a random combination, a random state of the cube is generated. The challenge is then to return the cube to its solved state. For a mathematician, the challenge is to understand the cube's symmetry, and solve it in the general case.
Using about one billion seconds (35 years) of CPU time, the Rubik's cube Cayley graph-the orbit of the states of the cube-has been computed 25  In our analogy, the many states of the toy are our graph states, and rotating the different faces of the cube corresponds to local complementation of different graph vertices. As evidenced by the ratio of its cardinality to its diameter (∼10 18 ), the orbit of the Rubik's cube is highly dense (though each vertex only has six edges). Each of the ∼1.3 million entanglement classes of 12 qubits has its own unique orbit-each of them is another Rubik's cube (with 12 rather than 6 moves). Note there are factorially many entanglement classes as n is increased. God's number (the orbit diameter) for local complementation orbits depends on the class. Using about a week of CPU time on a standard desktop computer, we compute the diameter of local complementation is maximally 9 for 9-qubit graph states. That is, any two locally equivalent graph states are at most 9 local complementations distant from one another).

Isomorphic graph states
Graphs can be symmetric under relabelling of their vertices-such graphs are said to be isomorphic. Graph states which are isomorphic share the same variety of entanglement. This is an important feature for the implementation of protocols where qubit relabelling is non-trivial-this includes most quantum information processing and communication scenarios. Here we consider both cases. We denote orbits C i when isomorphic graphs are considered equal (unlabelled graph states), and L i otherwise (labelled graph states). C i contain on average 1 /8 as many graph states as their partner L i orbits for n < 9 qubits. This greatly reduces the computational resources needed to map and analyse them. We note that all C i are subgraph of their partner L i , formed by merging all orbit vertices corresponding to isomorphic graph states. This can be seen by observing that isomorphic graph states have isomorphic neighbuorhoods in L i .
We find there are typically more than one C i orbit (for fixed i), as most C i orbits do not contain every isomorphism of it's member graph states (e.g. Fig. 1d)-the entanglement possessed is distributed in different ways between the parties. For example, there are three equivalent orbits of L 4 (one of which is shown in Fig. 1), each containing different isomorphisms of their component graph states. Some entanglement classes have only one orbit, which contains every isomorphism of the graph states. For example, the classes which contain the 'star' and fully-connected graph states. These orbits are composed of |L i | = n + 1 graph states (vertices) and are themselves a 'star' graph (e.g. Fig. 1b).
As in L i orbits, edges of a C i orbit are undirected. However, as a guide to the eye we display directed edges for C i orbits when those edges are labelled, as this allows the reader to identify which of the displayed graph state vertices are locally complemented to reach the output graph (see Fig.  1c,e,f).

Orbit exploration
Mapping the orbit of the i th entanglement class, M i containing a graph state |G , is a graph exploration problem. Here, we use an exhaustive breadth-first exploration to traverse the entire orbit, cataloguing each graph state (vertices of the orbit) along with how local complementation links them (edges of the orbit). We start with a single graph state G, taken from ref. 17, in our catalogue, and perform each possible local complementation on it. In doing so, we discover up to n new orbit vertices and up to n new orbit edges. Then we perform every possible local complementation on those output graph states and catalogue the outputs by comparing them to graph states which we have already found. This is repeated until every local complementation has been performed on every graph state in the catalogue (and no new graph states or edges are found).
To map an n-qubit orbit, M i , which contains |M i | graph states requires O(n|M i |) local complementations and graph comparisons. By 'graph comparison', we mean evaluating if two graphs are equal, or calling GRAPHISOMORPHISM (depending on whether M i = L i or M i = C i respectively). Linear savings can be made by noting that local complementation is self inverse, and has no effect when applied to a vertex of   Figure 11 for a table showing a representative graph state from each class of n < 9 qubits). Here, |Q| is the number of qubits of the orbit's graph states, |e| is smallest number of edges of any graph state member of C i . The class's Schmidt measure, E s , is written a < b to compactly express lower (a) and upper (b) bounds, when an exact value is not known 11 . rwd is the class's rank width, |C i | is the size of the orbit, |E i | is the number of edges on the orbit, χ g is the minimum chromatic number of the graph states in the class, χ C i is the orbits chromatic number, χ e g is the minimum chromatic index of the graph states in the class (which corresponds to the minimum number of CZ gates required to prepare them), d i are the distances between vertices on the orbit (max(d i ) is therefore the diameter of the orbit), 'Tree' is whether the orbit is a tree (excluding self-loops), '2D' is whether the orbit is planar, 'Loop' is whether the orbit has any self-loops, 'E.' ('H.') is whether the graph has a cycle in which each edge (vertex) of the orbit is visited precisely once. We note the correlation of orbit diameter, and orbit chromatic index with the Schmidt measure and rank width of the entangled state. Properties of L i orbits may differ to their C i equivalent. Similarly, the distance matrix, D, gives the distance between two vertices: D i j is equal to the minimum number of edges that must be traversed to get from vertex i to vertex j. degree 1. We use this method to explore the L i for n ≤ 8 and C i for n ≤ 9, that is, up to graph state entanglement class i = 146 and i = 586 respectively. The largest of these orbits contains 3248 and 8836 graph states, respectively. GRAPHISO-MORPHISM is a costly routine, belonging to the complexity class NP. Exploration of C i makes heavy use of GRAPHI-SOMORPHISM, calling it up to n|C i | times. However, since |C i | |L i |, and our graph states are of modest size, exploring C i up to 9 qubits required less compuational time than exploring L i up to 8 qubits.
When exploring C i orbits, we notice that local complementing symmetric vertices of an input graph state will result in the same output graph state. For example, in Fig. 1c, local complementing any of the vertices of the fully connected graph state will result in the same output, while for the adjacent 'star' tree graph state, there are two inequivalent local complementation operations-the centre and the leaves.
The sets of vertices which result in isomorphic graphs under local complementation can be found by computing the automorphism group of each graph state-vertices that are exchanged in an automorphism result in isomorphic graphs Hence, by computing the automorphism group of each graph state as it is discovered, and only local complementing the reduced subset of graph state vertices that are not equivalent, a saving can be made. Here, onlyñ|C i | comparisons (and hence calls to GRAPHISOMORPHISM) need be made (whereñ = |E i |/|C i | is the mean number of non-symmetric vertices on the graph states of C i . In practice, the AUTO-MORPHISMGROUP is computed in order to solve GRAPHI- SOMORPHISM 26 . Hence a linear speedup is achieved. By examining our set of computed orbits, we find this technique reduces the number of calls to GRAPHISOMORPHISM by at least half for n ≤ 9.

Results
We compute a variety of graph properties of C i orbits of 3-7 qubits and display them in Table 1. For example we display the Schmidt measure, E S , which is known to be a useful entanglement monotone for graph states 11,27 . We also compute the graph state's rank width, rwd(G) 14,28 . We also compute parameters of the orbit such as the chromatic number, chromatic index, the mean and maximum distances (diameter), and the size of the orbits automorphism group.
As per the canonical indexing of graph state entanglement classes, we list the minimum degree of each orbit: the smallest number of edges of any of the orbit's graph states. This corresponds to the minimum number of CZ gates required to generated that entanglement class. We also provide the graph state's minimum chromatic index (minimal edge colouring number), which corresponds to the minimum number of time steps required to generate a state in that entanglement class 17 (assuming CZs can be performed between each qubit arbitrarily).
We find correlations between orbit parameters and compute their Pearson correlation coefficients, −1 < r(x, y) < 1, for orbit parameters x, y. For example, the graph state Schmidt measure, E S correlates with orbit diameter with r(max(d i ), E S ) = {0.77±0.02, 0.93±0.02}, for C i and L i orbits respectively. Here, if the Schmidt measure is not known, we take the average value of the bounds, which are rarely loose. Orbit chromatic number and Schmidt measure have correlation coefficients of r(χ C i , E S ) = {0.67 ± 0.02, 0.70 ± 0.45}, while orbit chromatic index and Schmidt measure correlate with r(χ e C i , E S ) = {0.81 ± 0.04, 0.82 ± 0.06} for n ≤ 8 and n ≤ 7 respectively. Meanwhile, orbit chromatic number does not correlate with minimum graph state chromatic index, r(χ C i , χ e g ) = {0.032 ± 0.04, −0.09 ± 0.09}. We note that Schmidt measure correlates with rank width and minimum edge count with r(E S , rwd) = 0.62 ± 0.03 and r(E S , |e|) = 0.78 ± 0.02, but not with graph state chromatic index r(E S , χ e g ) = −0.17 ± 0.0. Interestingly, Schmidt rank (and therefore orbit chromatic index), strongly correlates with minimum edge count (the total number of CZs required to prepare an entanglement class) but not with graph state chromatic index (the number of CZ time steps required to prepare an entanglement class). Lattice graph states, which contain states universal for quantum computation , have constant CZ time steps preparation complexity, however their rank width must grow faster than logarithmically 21 .
Some properties of a entanglement class' graph states can be deduced from properties of their orbit. For example, class no. 40 has no self-loops. This implies that none of its member graph states have a vertex of degree 1. This is the only orbit with this property up to n = 7, but 9% of L i orbits up to n = 9 have it.
We also observe that local complementations commute when the neighbourhoods of the two indices are disjoint. This creates a cycle in the graph state's orbit. Hence it can be deduced that orbits which are trees only contain graph states in which all vertices whose local complementation results in different states share parts of their neighbourhoods. We note that only GreenbergerHorneZeilinger (GHZ) entanglement gives rise to L i orbits that are trees (for n ≤ 8), and these contain only two graph states. Meanwhile there is one threevertex orbit and three four-vertex C i orbits which are trees. These are connected in a line with one and two self-loops respectively (see Fig. 1e).
Interestingly, some C i orbits are isomorphic (see Figs. 1c,f and 3). A simple example is that of GHZ entanglement of n qubits. This class always contains only the n-qubit 'star' and fully connected graph states. Hence the C i orbits of GHZ entanglement are always the two qubit connected graphanalogous states give rise to analogous orbits across qubit numbers (see Fig. 3b). In contrast, no L i orbits are isomorphic for n ≤ 8, The proportion of all graphs which are asymmetric tends towards zero as the number of vertices tends towards infinity 29 , (∼50% of unlabelled 9-vertex graphs) However,the majority of the orbits we compute are symmetric (75% of 9qubit orbits have a non-empty automorphism group), including orbits containing thousands of graph states. The relationship between an orbit's symmetry and the graph state symmetries of it's component graph states left for future work.
Many of the computed parameters, such as Schmidt measure, rank width and automorphism group are exponentially difficult problems. The rank width, while exponential in nature, can be computed exactly 30 , while the Schmidt measure requires a nonconvex, nonlinear optimisation, and so is more challenging. We rely on previously computed 17   the Schmidt measure, while computing the rank width using the software 'SAGE'. Though our graph states are small, there is an exponential number of entanglement classes as qubit number is increased. Further, many of the graph metrics discussed, such as colourability belong to complexity class NP. As such, they become challenging to compute on dense orbits with thousands of vertices. For this reason we computed the chromatic index only for n ≤ 8 and n ≤ 7 for C i and L i orbits respectively. All graph colouring computations were performed with the software 'IGraph/M'. Due to their connectivity and scale, the majority of orbits we explored are far too complex to view directly, as we did in Fig. 1. We can instead represent them with matrices. Fig. 2 shows the adjacency matrices and distance matrices of class L 10 . We order the matrix by isomorphism, edge count, and then lexicographically by their lexicographically sorted edgelists. Further, we demarcate regions of the plot which correspond to graph states that have the same number of edges and that are isomorphic to one another for C i and L i respectively. In both cases, the adjacency matrices show structure related to these regions.
There is variety and scale in the 587 C i orbits and 147 L i orbits we have computed which cannot be reproduced in a single article. A curated selection of orbits is displayed in Appendix Section 1, and the full data available online.

Discussion
It is likely that future quantum information processors will have restricted two-qubit gate topology, due to the qubit's physical locations and proximity. Since single-qubit operations are commonly faster or higher-fidelity than two-qubit gates, local complementation may be used to improve a device's speed or fidelity 23 . For a prescriptive method, the relationship between orbits by nonlocal CZ gates must be known. A complete map of this type would describe how all n-qubit graph states are related to one another, and provide a look up table for optimal transformations between them. From here, the addition of vertex deletion would give a complete map of graph states under LC+LPM+CC operations (the vertex minor problem). A doubly-exponential problem, computation of these maps appears to be infeasible for even modest n. For small graphs, however, such a map may be enlightening-the exploration is left for future work.
Knowledge of the orbits of local complementation may also enable in quantum secret sharing and quantum networks 7,31 . A graph state may be distributed between separated parties, each of whom can perform local operations and communicate with their neighbours (according to the graph state structure). This allows different quantum protocols to be implemented using a resource which has already been distributed spatially. If the parties only have knowledge of their own neighbourhood, and each party performs local complementation at random, the shared state can be scrambled. Numerically, we find the stationary distributions generated by random walks on the orbits appear to tend towards uniform as orbit size increases, implying this 'scrambling' is effective. This could be formalised further by investigating mixing rates.
Local complementation allows the entanglement of a resource state to be utilised differently [31][32][33] . That is, a resource state can be transformed into any other state from its entanglement class, and used according to its shape. Though this is equivalent to changing the protocol measurement bases, considering locally equivalent graph states as a new state preserves the standard language of measurement-based protocols (measurement in the X-Y plane and Z directions). Generally, local complementation has merit in applications where qubits are in inequivalent spatial locations-it illustrates the many functions of a given entanglement.
In some quantum computer architectures, such as those for linear optical quantum computing 34 , percolated resource states are generated probabilisticly. These states have a randomly generated structure, and hence some are more powerful than others, for example they may have more favourable connectivity for pathfinding 35 or loss tolerance 36,37 , which may be optimised by local complementation. Though the entanglement class of any useful resource state will be too large to compute directly, it may be possible to develop heuristics for using local complementation to optimise local regions of the (locally connected) resource. These heuristics may be explored and verified with the algorithm of ref. 19.
Our library of orbits, available online 24 , comprises 35 MB compressed. All computations thus far were performed using a single core of an Intel i5 CPU in Mathematica. Exploration up to n = 12, where representative graph states of each orbit are known, is feasible if a compiled language and parallelism are employed. Extending the database further is a significant computational challenge, as, though an exact scaling is not known, the number of graph state entanglement classes grows super-exponentially for n ≤ 12 qubits Our exploration opens some novel lines of questioning. For example, what is the relationship between graph state properties, such as Schmidt measure, and orbit properties, such as colourability? Is it possible to completely map LC+LPM+CC operations up to 12 qubits? What new applications are possible utilising knowledge of LC orbits?
Stabiliser state entanglement is-and will continue to be-at the core of quantum information protocols. The resource we provide gives a new handle to investigate the rich relationship between graph theory, stabiliser state entanglement, and applications of quantum information. Regions in the plot correspond to graph states which are isomorphic. By separating regions corresponding to non-isomorphic graphs, we see that the orbit is composed of just two non-isomorphic graph states. These have 60 and 72 isomorphisms respectively.