The 27-qubit Counterexample to the LU-LC Conjecture is Minimal

Abstract

It was once conjectured that two graph states are local unitary (LU) equivalent if and only if they are local Clifford (LC) equivalent. This so-called LU-LC conjecture was disproved in 2007, as a pair of 27-qubit graph states that are LU-equivalent, but not LC-equivalent, was discovered. We prove that this counterexample to the LU-LC conjecture is minimal. In other words, for graph states on up to 26 qubits, the notions of LU-equivalence and LC-equivalence coincide. This result is obtained by studying the structure of 2-local complementation, a special case of the recently introduced r-local complementation, and a generalization of the well-known local complementation. We make use of a connection with triorthogonal codes and Reed-Muller codes.

Figures (4)

• Figure 1: A 27-qubit counterexample to the LU-LC conjecture Ji07Tsimakuridze17, that is, a pair of graph states that are LU-equivalent but not LC-equivalent. The graphs have 6 bottom vertices. There is one top vertex of degree 5 per set of 5 bottom vertices, and one top vertex of degree 4 per set of 4 bottom vertices; leading to $\binom{6}{5} + \binom{6}{4} = 21$ top 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 top qubits and $Z(\pi/4)$ on the bottom qubits maps one graph state to the other. Proving that these two graph states are not LC-equivalent is more involved, a proof can be found in Tsimakuridze17.
• Figure 2: A 28-qubit counterexample to the LU-LC conjecture Tsimakuridze17, that is, a pair of graph states that are LU-equivalent but not LC-equivalent. The graphs have 7 bottom vertices. The top vertices are all of degree 5, and there is one top vertex per set of 5 bottom vertices, leading to $\binom{7}{5} = 21$ top 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 top qubits and $Z(\pi/4)$ on the bottom qubits maps one graph state to the other. Proving that these two graph states are not LC-equivalent is more involved, a proof can be found in Tsimakuridze17, and a proof in the formalism of $r$-local complementation can be found in claudet2024local. The 27-qubit counterexample to the LU-LC conjecture (see Figure \ref{['fig:ce27']}) can be recovered by removing one bottom vertex.
• Figure 3: One of the 16-vertex bipartite graphs corresponding to the unique unital triorthogonal subspace in $\mathbb F^{16}_2$, from which the original magic state distillation code by Bravyi and Kitaev Bravyi2005 and the first code of the Bravyi-Haah family Bravyi2012 can be derived nezami2022. We name the bottom vertices 1, 2, 3, 4 and 5, and we name the top vertices after their neighborhood. One of the top vertices is adjacent to every bottom vertex: $\{1,2,3,4,5\}$. The $\binom{5}{3} = 10$ other top vertices are all possible vertices of degree 3: $\{1,2,3\}$, $\{1,2,4\}$, $\{1,2,5\}$, $\{1,3,4\}$, $\{1,3,5\}$, $\{1,4,5\}$, $\{2,3,4\}$, $\{2,3,5\}$, $\{2,4,5\}$ and $\{3,4,5\}$. The set of top vertices is 2-incident. A 2-local complementation on the top vertices leaves the graph invariant, i.e. does not create or remove any edge. Thus, no counterexample to the LU-LC conjecture can be derived from this graph.
• Figure 4: The leftmost graph is one of the 24-vertex bipartite graphs corresponding to the unique unital triorthogonal subspace in $\mathbb F^{24}_2$, from which the second code of the Bravyi-Haah family Bravyi2012 can be derived nezami2022. We name the bottom vertices 1, 2, 3, 4, 5, 6 and 7, and we name the top vertices after their neighborhood. One of the top vertices is adjacent to every bottom vertex: $\{1,2,3,4,5,6,7\}$. 3 top vertices are of degree 5: $\{1,2,3,4,5\}$, $\{1,2,3,4,6\}$ and $\{1,2,3,4,7\}$. The $\binom{3}{2}\times\binom{4}{1} + \binom{3}{3} = 3 \times 4 +1 = 11$ other top vertices are of degree 3, they are all possible vertices of degree 3 with 2 or more neighbors among the vertices 4, 5 and 6: $\{1,5,6\}$, $\{2,5,6\}$, $\{3,5,6\}$, $\{4,5,6\}$, $\{1,5,7\}$, $\{2,5,7\}$, $\{3,5,7\}$, $\{4,5,7\}$, $\{1,6,7\}$, $\{2,6,7\}$, $\{3,6,7\}$, $\{4,6,7\}$ and $\{5,6,7\}$. The set of top vertices is 2-incident. A 2-local complementation on the top vertices maps the leftmost graph to the rightmost graph by creating 3 edges: one between 5 and 6, one between 5 and 7, and one between 6 and 7. This transformation can also be obtained with a single local complementation on the vertex $\{5,6,7\}$. Thus, no counterexample to the LU-LC conjecture can be derived from these graphs.

Theorems & Definitions (27)

• Theorem 1
• Corollary 1
• Definition 1: claudet2024local
• Definition 2: claudet2024local
• Example 1
• Proposition 1: claudet2024local
• Proposition 2: claudet2025deciding
• Lemma 1
• Example 2
• Lemma 2