Computational Complexity of Quantum Satisfiability

Abstract

Quantum logic generalizes, and in dimension one coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic formulas; and show both 퓝 퓟-complete in dimension two as well. For higher-dimensional spaces ℝ and ℂ with ≥ 3 fixed, on the other hand, we show the problem 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 adds (a perhaps more natural and combinatorially flavoured) one to the still sparse list of 퓝 퓟ℝ-complete problems, mostly pertaining to real algebraic geometry. Our proofs rely on (a careful examination of) works by John von Neumann as well as contributions by Hagge et.al (2005,2007,2009). We finally investigate the problem over indefinite finite dimensions and relate it to noncommutative semialgebraic geometry.

On the consistency problem for modular lattices and related structures

“…For a field F and V = {F d F | d < ℵ 0 }, if satisfiability of conjunctions of ring equations is decidable for F , then the reasoning of [17,Theorem 4.10] shows that the consistency problem for L(V) is solvable if and only if there is a recursive function δ that for every conjunction ψ of lattice equations one has the following: If ψ is of binary length n and satisfiable in L(F d F ) for some d then ψ is also satisfiable in L(F d F ) for some d ≤ δ(n). By Theorem 8, no such δ exists if F is the field of real or complex numbers.…”

“…On the other hand, in the presence of an orthocomplementation, Example 4.2(b) in [17] gives a recursively defined sequence t k (x) of terms of length O(k) in 2k + 1 variables such that t k (x) = 1 is satisfiable in L(F d F ) if d = 2 k but not for d < 2 k . We provide an analogous recursive sequence without orthocomplementation and with fixed number of variables.…”

“…Here, in contrast, deciding whether ∃x t(x) > 0 holds in some L ⊥ (V ) ("weak satisfiability") is decidable (cf. [12]) and an upper complexity bound has been derived in [17]. Proof.…”

“…This applies, in particular, to the class of all L(V ) where V is finite dimensional over a fixed field of characteristic 0. In case the V are real or complex Hilbert spaces, unsovability extends to the associated ortholattices (Section 8), thus giving a negative answer to the question raised in [16, §III.C], the decision problem for "quantum satisfiability in indefinite (yet finite) dimensions" (see also [17]).…”

Satisfiability of cross product terms is complete for real nondeterministic polytime Blum-Shub-Smale machines

“…• Several questions about systems of polynomials [CuRo92,Koir99] • Stretchability of pseudoline arrangements [Shor91] • Realizability of oriented matroids [Rich99] • Loading neural networks with real weights [Zhan92] • Several geometric properties of graphs [Scha10] • Satisfiability in Quantum Logic QSAT, starting from dimension 3 [HeZi11].…”