We describe an algorithm with quasi-polynomial runtime $n^{\log_2(n)+O(1)}$ for deciding local unitary (LU) equivalence of graph states. The algorithm builds on a recent graphical characterisation of LU-equivalence via generalised local complementation. By first transforming the corresponding graphs into a standard form using usual local complementations, LU-equivalence reduces to the existence of a single generalised local complementation that maps one graph to the other. We crucially demonstrate that this reduces to solving a system of quasi-polynomially many linear equations, avoiding an exponential blow-up. As a byproduct, we generalise Bouchet's algorithm for deciding local Clifford (LC) equivalence of graph states by allowing the addition of arbitrary linear constraints. We also improve existing bounds on the size of graph states that are LU- but not LC-equivalent. While the smallest known examples involve 27 qubits, and it is established that no such examples exist for up to 8 qubits, we refine this bound by proving that LU- and LC-equivalence coincide for graph states involving up to 19 qubits.

Stabilizer states form a ubiquitous family of quantum states that can be graphically represented through the graph state formalism. A fundamental property of graph states is that applying a local complementation - a well-known and extensively studied graph transformation - results in a graph that represents the same entanglement as the original. In other words, the corresponding graph states are LU-equivalent. This property served as the cornerstone for capturing non-trivial quantum properties in a simple graphical manner, in the study of quantum entanglement but also for developing protocols and models based on graph states and stabilizer states, such as measurement-based quantum computing, secret sharing, error correction, entanglement distribution... However, local complementation fails short to fully characterise entanglement: there exist pairs of graph states that are LU-equivalent but cannot be transformed one into the other using local complementations. Only few is known about the equivalence of graph states beyond local complementation. We introduce a generalisation of local complementation which graphically characterises the LU-equivalence of graph states. We use this characterisation to show the existence of a strict infinite hierarchy of equivalences of graph states. Our approach is based on minimal local sets, which are subsets of vertices that are known to cover any graph, and to be invariant under local complementation and even LU-equivalence. We use these structures to provide a type to each vertex of a graph, leading to a natural standard form in which the LU-equivalence can be exhibited and captured by means of generalised local complementation.

We study the notion of k-stabilizer universal quantum state, that is, an n-qubit quantum state, such that it is possible to induce any stabilizer state on any k qubits, by using only local operations and classical communications. These states generalize the notion of k-pairable states introduced by Bravyi et al., and can be studied from a combinatorial perspective using graph states and k-vertex-minor universal graphs. First, we demonstrate the existence of k-stabilizer universal graph states that are optimal in size with n = Θ(k²) qubits. We also provide parameters for which a random graph state on Θ(k²) qubits is k-stabilizer universal with high probability. Our second contribution consists of two explicit constructions of k-stabilizer universal graph states on n = O(k⁴) qubits. Both rely upon the incidence graph of the projective plane over a finite field 𝔽_q. This provides a major improvement over the previously known explicit construction of k-pairable graph states with n = O(2^{3k}), bringing forth a new and potentially powerful family of multipartite quantum resources.