Local Equivalences of Graph States

TL;DR

This work addresses the core question of when graph states share the same entanglement under local operations by introducing a graphical formalism that extends local complementation to r-local complementation (LC_r). It proves an infinite strict hierarchy between LC and LU equivalences, provides a quasi-polynomial algorithm for LU-equivalence, and shows LU=LC for graph states on up to 19 qubits and for several universal classes via MLS and generalized completions. The framework connects stabilizer theory, graph theory, and algebraic techniques (Bouchet-style equations), enabling efficient LU-equivalence tests and deeper insight into graph-state entanglement. The results have practical implications for MBQC resource design, stabilizer codes, and quantum networks, offering both theoretical completeness and scalable algorithms for LU/LC classifications. Overall, the thesis advances understanding of entanglement equivalence in graph states and provides tools to navigate the LU-LC landscape with graph-theoretic constructs.

Abstract

Graph states form a large family of quantum states that are in one-to-one correspondence with mathematical graphs. Graph states are used in many applications, such as measurement-based quantum computation, as multipartite entangled resources. It is thus crucial to understand when two such states have the same entanglement, i.e. when they can be transformed into each other using only local operations. In this case, we say that the graph states are LU-equivalent (local unitary). If the local operations are restricted to the so-called Clifford group, we say that the graph states are LC-equivalent (local Clifford). Interestingly, a simple graph rule called local complementation fully captures LC-equivalence, in the sense that two graph states are LC-equivalent if and only if the underlying graphs are related by a sequence of local complementations. While it was once conjectured that two LU-equivalent graph states are always LC-equivalent, counterexamples do exist and local complementation fails to fully capture the entanglement of graph states. We introduce in this thesis a generalization of local complementation that does fully capture LU-equivalence. Using this characterization, we prove the existence of an infinite strict hierarchy of local equivalences between LC- and LU-equivalence. This also leads to the design of a quasi-polynomial algorithm for deciding whether two graph states are LU-equivalent, and to a proof that two LU-equivalent graph states are LC-equivalent if they are defined on at most 19 qubits. Furthermore, we study graph states that are universal in the sense that any smaller graph state, defined on any small enough set of qubits, can be induced using only local operations. We provide bounds and an optimal, probabilistic construction.

Figures (26)

• Figure 1: The Bloch sphere. A single-qubit quantum state $|{\psi} \rangle$ corresponds to a point on the surface of the sphere. More precisely, a point corresponds to quantum states up to an irrelevant global phase, that is, $|{\psi} \rangle$ and $e^{i\phi}|{\psi} \rangle$ share the same point on the sphere. The axis corresponds to the Pauli gates X, Y and Z, and the endpoints corresponds to their respective eigenvectors.
• Figure 2: Each of the 24 single-qubit Clifford gates (up to global phase) and the action of their conjugation over the Pauli gates.
• Figure 3: Construction of a 3-qubit graph state. (Left) We start with a tensor product of 3 qubits in the state $|{+} \rangle$. That is, $|{G_1} \rangle = |{+++} \rangle = \frac{1}{2\sqrt 2}\left(|{000} \rangle+|{001} \rangle+|{010} \rangle+|{100} \rangle+|{011} \rangle+|{101} \rangle+|{110} \rangle+|{111} \rangle\right)$. (Middle) We create an edge between vertices 1 and 2 by applying a $CZ$ gate on the qubits 1 and 2, leading to the graph state $|{G_2} \rangle = CZ_{12} |{+++} \rangle = \frac{1}{2\sqrt 2}\left(|{000} \rangle+|{001} \rangle+|{010} \rangle+|{100} \rangle+|{011} \rangle+|{101} \rangle-|{110} \rangle-|{111} \rangle\right)$. (Right) We create an edge between vertices 2 and 3 by applying a $CZ$ gate on the qubits 2 and 3, leading to the graph state $|{G_3} \rangle = CZ_{23} CZ_{12} |{+++} \rangle = \frac{1}{2\sqrt 2}\left(|{000} \rangle+|{001} \rangle+|{010} \rangle+|{100} \rangle-|{011} \rangle+|{101} \rangle-|{110} \rangle+|{111} \rangle\right)$.
• Figure 4: A 27-vertex counter-example to the LU=LC conjecture, that is, a pair of graphs that are LU-equivalent but not LC-equivalent. The graphs have 6 bottom vertices. There is one upper vertex per set of 5 bottom vertices, and one upper vertex per set of 4 bottom vertices; leading to $\binom{6}{5} + \binom{6}{4} = 21$ upper vertices. In the leftmost graph, the bottom vertices form an independent set, while in the rightmost graph, the bottom vertices are fully connected. Applying $X(\pi/4)$ on the upper vertices and $Z(\pi/4)$ on the bottom vertices maps one graph state to the other. Proving that those two graphs are not LC-equivalent is more involved, a proof can be found in Tsimakuridze17.
• Figure 5: Example of a local complementation. Vertex 2 is related to vertices 1,3 and 4. Thus, the local complementation on vertex 2 toggles each edge between vertices 1,3 and 4. That is, the edge between vertices 3 and 4 is removed; while vertices 1 and 3 (resp. 1 and 4) share no edge, thus an edge is created. Formally, the graph $G$ is mapped to $G \star 2 = G\Delta K_{N_G(2)} = G\Delta K_{1,3,4}$.

Theorems & Definitions (227)

• Proposition
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• Proposition 1
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• Proposition 4: Euler's decomposition
• Proposition 5
• Definition 1
• Proposition 6