NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability

TL;DR

This paper establishes a new characterization of quantum isomorphism using the NPA hierarchy and homomorphism counts from planar graphs, providing both theoretical insights and a polynomial-time decision algorithm.

Contribution

It links the NPA hierarchy levels to homomorphism counts from planar graphs, offering a quantum isomorphism characterization without quantum groups and a polynomial-time decision method.

Findings

Feasibility of NPA hierarchy levels equals homomorphism counts from planar graphs.

New proof of quantum isomorphism characterization avoiding quantum groups.

Polynomial-time algorithm for deciding fixed-level NPA feasibility.

Abstract

Man\v{c}inska 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 Man\v{c}inska 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.