On Blockwise Symmetric Matchgate Signatures and Higher Domain #CSP
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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. the rank of the matrix form of the blockwise symmetric standard signatures, 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.
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