The complexity of constraint system games with entanglement

Abstract

Constraint satisfaction problems (CSPs) are a natural class of decision problems where one must decide whether there is an assignment to variables that satisfies a given formula. Schaefer's dichotomy theorem and its extension to all alphabets due to Bulatov and Zhuk, shows that CSP languages are either efficiently decidable or NP-complete. It is possible to extend CSP languages to quantum assignments using the formalism of nonlocal games. The recent MIP*=RE theorem of Ji, Natarajan, Vidick, Wright, and Yuen shows that the complexity class MIP* of multiprover proof systems with entangled provers contains all recursively enumerable languages. As a consequence, general succinctly presented CSPs are RE-complete. We show that a wide range of NP-complete CSPs become RE-complete when the players are allowed entanglement, including all boolean CSPs, such as 3SAT and 3-colouring. This implies that these CSP languages remain undecidable even when not succinctly presented. Prior work of Grilo, Slofstra, and Yuen shows (via a technique called simulatable codes) that every language in MIP* has a perfect zero knowledge (PZK) MIP* protocol. The MIP*=RE theorem uses two-prover one-round proof systems. Hence, such systems are complete for MIP*. However, the construction in Grilo, Slofstra, and Yuen uses six provers, and there is no obvious way to get perfect zero knowledge with two provers via simulatable codes. This leads to a natural question: are there two-prover PZK-MIP* protocols for all of MIP*? In this work, we show that every language in MIP* has a two-prover one-round PZK-MIP* protocol, answering the question in the affirmative. For the proof, we use a new method based on a key consequence of the MIP*=RE theorem, which is that every MIP* protocol can be turned into a family of boolean constraint system (BCS) nonlocal games. This makes it possible to work with MIP* protocols as boolean constraint systems. In particular, it allows us to use a variant of a CSP due to Dwork, Feige, Kilian, Naor, and Safra that gives a classical MIP protocol for 3SAT with perfect zero knowledge. To prove our results, we develop a toolkit for analyzing the quantum soundness of reductions between constraint system (CS) games, which we expect to be useful more broadly. In this formalism, synchronous strategies for a nonlocal game correspond to tracial states on an algebra. We equip the algebra with a finitely supported weight that allows us to gauge the players' performance in the corresponding game using a weighted sum of squares. The soundness of our reductions hinges on guaranteeing that specific measurements for the players are close to commuting when their strategy performs well. To this end, we construct commutativity gadgets for all boolean CSPs and show that the commutativity gadget for graph 3-colouring due to Ji is sound against entangled provers. We define a broad class of CSPs that have simple commutativity gadgets. We show a variety of relations between the different ways of presenting CSPs as games. This toolkit also applies to commuting operator strategies, and our argument shows that every language with a commuting operator BCS protocol has a two prover PZK commuting operator protocol.