Complexity of Quantum Logic Satisfiability in Fixed Dimension: Realification and Rank-One Encodings

Quantum propositional logic, introduced by Birkhoff and von Neumann, formalizes reasoning about experimental propositions in quantum mechanics through closed subspaces of Hilbert spaces. We provide a comprehensive complexity analysis of the satisfiability problem for quantum logic formulas over the connectives conjunction, disjunction, and negation. Distinguishing between weak satisfiability (evaluation to a nonzero subspace) and strong satisfiability (evaluation to the full space), we establish that both notions coincide with Boolean satisfiability for dimension one, are classically NP complete for dimension two, and become complete for the Blum Shub Smale class NP_R when dimension d >= 3 is fixed.

Our principal contribution is twofold. First, we present two alternative constructive reductions from quantum logic satisfiability to the Existential Theory of the Reals (ETR): a realification based encoding that explicitly handles the complex to real translation through commutant characterizations, and a rank one column encoding with orthogonal splitting gadgets. Both reductions yield polynomial size systems of quadratic equations for fixed dimension, proving membership in the complexity class existsR and hence in PSPACE via Cannys algorithm. Second, we provide detailed size bounds distinguishing weak and strong satisfiability: weak satisfiability encodes with O(n D^2 + N_v D) scalar variables where n is the number of atoms, D the real dimension, and N_v the number of disjunctions, while strong satisfiability requires O(n D^2 + N_v D^2) variables due to basis vector witnesses. When dimension is part of the input, strong satisfiability becomes polynomial time equivalent to feasibility of noncommutative integer polynomial equations. These results position quantum logic satisfiability precisely between classical NP and existsR, highlighting dimension as a fundamental complexity parameter.