Grobner Bases for Finite-temperature Quantum Computing and their Complexity

Following the recent approach of using order domains to construct Grobner bases from general projective varieties, we examine the parity and time-reversal arguments relating de Witt and Lyman's assertion that all path weights associated with homotopy in dimensions d <= 2 form a faithful representation of the fundamental group of a quantum system. We then show how the most general polynomial ring obtained for a fermionic quantum system does not, in fact, admit a faithful representation, and so give a general prescription for calcluating Grobner bases for finite temperature many-body quantum system and show that their complexity class is BQP.