Man\v{c}inska and Roberson~[FOCS'20] showed that two graphs are quantum isomorphic if and only if they are homomorphism indistinguishable over the class of planar graphs. Atserias et al.~[JCTB'19] proved that quantum isomorphism is undecidable in general. The NPA hierarchy gives a sequence of semidefinite programming relaxations of quantum isomorphism. Recently, Roberson and Seppelt~[ICALP'23] obtained a homomorphism indistinguishability characterization of the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism. We prove a quantum analogue of this result by showing that each level of the NPA hierarchy of SDP relaxations for quantum isomorphism of graphs is equivalent to homomorphism indistinguishability over an appropriate class of planar graphs. By combining the convergence of the NPA hierarchy with the fact that the union of these graph classes is the set of all planar graphs, we are able to give a new proof of the result of Man\v{c}inska and Roberson~[FOCS'20] that avoids the use of the theory of quantum groups. This homomorphism indistinguishability characterization also allows us to give a randomized polynomial-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.

翻译：Mančinska 与 Roberson [FOCS'20] 证明了两个图是量子同构的，当且仅当它们在所有平面图类上是同态不可区分的。Atserias 等人 [JCTB'19] 证明了量子同构问题在一般情况下是不可判定的。NPA 层级为量子同构问题提供了一系列半定规划松弛。最近，Roberson 与 Seppelt [ICALP'23] 获得了图同构的 Lasserre 半定规划松弛层级每一层可行性的同态不可区分性刻画。我们证明了这一结果的量子类比，即表明针对图量子同构的 NPA 半定规划松弛层级的每一层，都等价于在某个适当的平面图类上的同态不可区分性。通过结合 NPA 层级的收敛性以及这些图类的并集是所有平面图这一事实，我们能够给出 Mančinska 与 Roberson [FOCS'20] 结果的一个新证明，该证明避免了使用量子群理论。这一同态不可区分性刻画还使我们能够给出一个随机多项式时间算法，用于判定针对量子同构的 NPA 半定规划松弛层级中任意固定层的精确可行性。