Grobner Bases for Finite-temperature Quantum Computing and their Complexity

TL;DR

This paper explores the application of Grobner bases to finite-temperature quantum computing, demonstrating their computational complexity and limitations in representing certain quantum systems.

Contribution

It introduces a method for calculating Grobner bases in fermionic quantum systems and establishes their complexity class as BQP.

Findings

Grobner bases can be constructed for certain quantum systems.

The complexity of these bases is within BQP.

Some quantum systems do not admit faithful polynomial ring representations.

Abstract

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.