Equivalence detection of graph states

[king-p3nguin]

Van den Nest et al. (https://arxiv.org/abs/quant-ph/0405023) shows an efficient algorithm to detect local Clifford equivalence of two given graph states. It is known that all stabilizer states are equivalent under local Clifford operations to some graph state. On the other hand, Elliott et al. state in https://arxiv.org/abs/0806.2651 that all graph states that produce a certain stabilizer state can be found by successive application of the E1 and E2 equivalence rules. Equivalence checking by E1 and E2 rules are already implemented in graphix as equivalent_graph_E1() and equivalent_graph_E2(). Thus, if we can examine the process of transforming one graph state into another LC-equivalent graph state through Van den Nest et al.'s algorithm and find that it consists only of E1 and E2 rules, we know that the two graph states produce the same stabilizer state.