NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability

2024-07-15

arXiv.org

AI Summary

This work establishes an exact correspondence between the Navascués-Pironio-Acín (NPA) hierarchy and graph homomorphism indistinguishability, specifically for the class of planar graphs. We prove that the $k$-th level of the NPA hierarchy is equivalent to $k$-local planar graph homomorphism indistinguishability—yielding the first purely combinatorial characterization of all NPA levels, independent of quantum group theory, and endowing the SDP relaxations for quantum graph isomorphism with combinatorial semantics. Methodologically, we provide a combinatorial reformulation and novel proof of the Mančinska–Roberson theorem, circumventing operator-algebraic machinery. Our key contributions are: (1) a fully combinatorial reconstruction of the Mančinska–Roberson theorem; and (2) the first algorithm that, for any fixed $k$, decides NPA-$k$ feasibility exactly in randomized polynomial time. This work bridges quantum information theory and structural graph theory, achieving simultaneous advances in conceptual clarity and computational efficiency.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Manv{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 Manv{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.

Problem

• Characterizing quantum isomorphism via homomorphism indistinguishability over planar graphs
• Proving equivalence between NPA hierarchy levels and specific graph classes
• Developing a randomized polynomial-time algorithm for NPA hierarchy feasibility

Innovation

• NPA hierarchy SDP relaxations for quantum isomorphism
• Homomorphism indistinguishability over planar graphs
• Randomized polynomial-time algorithm for NPA feasibility