Binary Constraint System Games and Locally Commutative Reductions
Zhengfeng Ji
TL;DR
The paper develops binary constraint system (BCS) games as a unifying framework for studying quantum satisfiability in non-local games. It shows how classical NP-hard reductions can be lifted to the quantum setting using commutativity gadgets and establishes BCS encodings for quantum coloring and Kochen-Specker sets, among others. A Clifford-algebra–based parity BCS game demonstrates explicit entanglement lower bounds, while non-commutative algebra and Gröbner basis techniques underpin the gadget constructions. Overall, the work connects non-local games, operator algebra, and constraint-satisfaction problems to map quantum feasibility and entanglement requirements in structured settings.
Abstract
A binary constraint system game is a two-player one-round non-local game defined by a system of Boolean constraints. The game has a perfect quantum strategy if and only if the constraint system has a quantum satisfying assignment [R. Cleve and R. Mittal, arXiv:1209.2729]. We show that several concepts including the quantum chromatic number and the Kochen-Specker sets that arose from different contexts fit naturally in the binary constraint system framework. The structure and complexity of the quantum satisfiability problems for these constraint systems are investigated. Combined with a new construct called the commutativity gadget for each problem, several classic NP-hardness reductions are lifted to their corresponding quantum versions. We also provide a simple parity constraint game that requires $Ω(\sqrt{n})$ EPR pairs in perfect strategies where $n$ is the number of variables in the constraint system.