On blockwise symmetric matchgate signatures and higher domain #CSP

Abstract

For any n ≥ 3 and q ≥ 3, we prove that the Equality function (= n ) on n variables over a domain of size q cannot be realized by matchgates under holographic transformations. This is a consequence of our theorem on the structure of blockwise symmetric matchgate signatures. This has the implication that the standard holographic algorithms based on matchgates, a methodology known to be universal for #CSP over the Boolean domain, cannot produce P-time algorithms for planar #CSP over any higher domain q ≥ 3.