Mančinska and Roberson [FOCS'20] showed that two graphs are quantum isomorphic if and only if they admit the same number of homomorphisms from any planar graph. Atserias et al. [JCTB'19] proved that quantum isomorphism is undecidable in general, which motivates the study of its relaxations. In the classical setting, Roberson and Seppelt [ICALP'23] characterized the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism in terms of equality of homomorphism counts from an appropriate graph class. The NPA hierarchy, a noncommutative generalization of the Lasserre hierarchy, provides a sequence of semidefinite programming relaxations for quantum isomorphism. In the quantum setting, we show that the feasibility of each level of the NPA hierarchy for quantum isomorphism is equivalent to equality of homomorphism counts from an appropriate class of planar graphs. Combining this characterization with the convergence of the NPA hierarchy, and noting that the union of these classes is the set of all planar graphs, we obtain a new proof of the result of Mančinska and Roberson [FOCS'20] that avoids the use of quantum groups. Moreover, this homomorphism indistinguishability characterization also yields 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，两个图是量子同构的，当且仅当它们从任意平面图出发的同态计数均相等。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，这推动了对该问题松弛形式的研究。在经典情形中，Roberson与Seppelt [ICALP'23] 通过特定图类的同态计数相等性，刻画了图同构的半定规划松弛——Lasserre层次结构每一层可行性的充要条件。NPA层次结构作为Lasserre层次结构的非交换推广，为量子同构问题提供了一系列半定规划松弛。在量子情形下，我们证明了NPA层次结构每一层对于量子同构的可行性，等价于从一类特定平面图出发的同态计数相等性。将这一刻画与NPA层次结构的收敛性相结合，并注意到这些图类的并集即为所有平面图，我们得到了Mančinska与Roberson [FOCS'20] 结果的一个新证明，该证明避免了量子群的使用。此外，这一同态不可区分性刻画还产生了一个随机多项式时间算法，可用于判定量子同构的SDP松弛——NPA层次结构中任一固定层的精确可行性。