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. Due to the equality of complexity classes MIP*=RE, general succinctly-presented entangled CSPs are RE-complete. In this work, we show that a wide range of NP-complete CSPs become RE-complete in this setting, including all boolean CSPs, such as 3SAT, as well as 3-colouring. This also implies that these CSP languages remain undecidable even when not succinctly presented.
To show this, we work in the weighted algebra framework introduced by Mastel and Slofstra, where synchronous strategies for a nonlocal game are represented by tracial states on an algebra. Along the way, we improve the subdivision technique in order to be able to separate constraints in the CSP while preserving constant soundness, construct commutativity gadgets for all boolean CSPs, and show a variety of relations between the different ways of presenting CSPs as games.
This work will appear at FOCS 2025 (arXiv)
I am currently working on applying the techniques of this paper to hardness of approximation and gap amplification in entangled 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. Prior work of Grilo, Slofstra, and Yuen [FOCS '19] further 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, and 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 paper, 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, and in particular allows us to use a variant of a construction due to Dwork, Feige, Kilian, Naor, and Safra [Crypto '92] which gives a classical MIP protocol for 3SAT with perfect zero knowledge. To show quantum soundness of this classical construction, we develop a toolkit for analyzing quantum soundness of reductions between BCS games, which we expect to be useful more broadly. 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.
Kieran Mastel and William Slofstra. 2024. Two Prover Perfect Zero Knowledge for MIP*. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024). Association for Computing Machinery, New York, NY, USA, 991–1002. https://doi.org/10.1145/3618260.3649702 (arXiv)
I am currently working on a follow up to this work that improves the parameters of the PZK protocol.
The n-qubit Pauli group and its normalizer the n-qubit Clifford group have applications in quantum error correction and device characterization. Recent applications have made use of the representation theory of the Clifford group. We apply the tools of (the coincidentally named) Clifford theory to examine the representation theory of the Clifford group using the much simpler representation theory of the Pauli group. We find an unexpected correspondence between irreducible characters of the n-qubit Clifford group and those of the (n+1)-qubit Clifford group. (arXiv)