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Following the recent approach of using order domains to construct Grobnerbases from general projective varieties, we examine the parity andtime-reversal arguments relating de Witt and Lyman's assertion that all pathweights associated with homotopy in dimensions d <= 2 form a faithfulrepresentation of the fundamental group of a quantum system. We then show howthe most general polynomial ring obtained for a fermionic quantum system doesnot, in fact, admit a faithful representation, and so give a generalprescription for calcluating Grobner bases for finite temperature many-bodyquantum system and show that their complexity class is BQP.

Grobner Bases for Finite-temperature Quantum Computing and their Complexity