Affiliation:
1. Computer Sciences, University of Wisconsin-Madison, Madison, United States
Abstract
Recently, Mančinska and Roberson proved that two graphs
G
and
\(G^{\prime }\)
are
quantum isomorphic
if and only if they admit the same number of homomorphisms from all
planar
graphs. We extend this result to planar #CSP with any pair of sets
\(\mathcal {F}\)
and
\(\mathcal {F}^{\prime }\)
of real-valued, arbitrary-arity constraint functions. Graph homomorphism is the special case, in which each of
\(\mathcal {F}\)
and
\(\mathcal {F}^{\prime }\)
contains a single symmetric 0-1-valued binary constraint function. Our treatment uses the framework of planar Holant problems. To prove that quantum isomorphic constraint function sets give the same value on any planar #CSP instance, we apply a novel form of
holographic transformation
of Valiant, using the quantum permutation matrix
\(\mathcal {U}\)
defining the quantum isomorphism. Due to the noncommutativity of
\(\mathcal {U}\)
’s entries, it turns out that this form of holographic transformation is only applicable to planar Holant. To prove the converse, we introduce the quantum automorphism group
\(Qut(\mathcal {F})\)
of a set
\(\mathcal {F}\)
of constraint functions/tensors and characterize the intertwiners of
\(Qut(\mathcal {F})\)
as the signature matrices of planar
\(\text{Holant}{\mathcal {F}\,|\,\mathcal {EQ})\)
quantum gadgets. Then, we define a new notion of (projective) connectivity for constraint functions and reduce their arity while preserving their inclusion in the original intertwiner space. Finally, to address the challenges posed by generalizing from 0-1 valued to real-valued constraint functions, we adapt a technique of Lovász in the classical setting for isomorphisms of real-weighted graphs to the setting of quantum isomorphisms.
Publisher
Association for Computing Machinery (ACM)