Computational Complexity of Quantum Satisfiability

TL;DR

This paper analyzes the computational complexity of quantum satisfiability problems, revealing NP-completeness in two dimensions and BSS-completeness in higher dimensions, and explores their connections to classical and noncommutative polynomial feasibility.

Contribution

It introduces quantum satisfiability problems, classifies their complexity across dimensions, and links them to real computation and polynomial feasibility issues.

Findings

Quantum satisfiability is NP-complete in 2D.

Higher-dimensional cases are BSS-complete.

Strong satisfiability relates to noncommutative polynomial feasibility.

Abstract

Quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables. It generalizes, and on 1-dimensional Hilbert space coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic terms. It turns out that in dimension two both are also NP-complete. For higher-dimensional spaces R^d and C^d with d>2 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale model of real computation. This provides a unified view on both Turing and real BSS complexity theory; and extends the still relatively scarce family of NP_R-complete problems with one perhaps closest in spirit to the classical Cook-Levin Theorem. Our investigations on the dimensions a term is weakly/strongly satisfiable in lead to satisfiability problems in indefinite finite and finally in infinite dimension. Here, strong satisfiability turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations