Perfect zero knowledge for quantum multiprover interactive proofs

TL;DR

This paper proves that in quantum multiprover interactive proofs, the class of problems with perfect zero knowledge protocols equals the class of all problems decidable by such protocols, extending classical results into the quantum realm.

Contribution

It establishes that every quantum multiprover interactive proof can be transformed into a perfect zero knowledge protocol, showing the equivalence of MIP* and PZKMIP* classes.

Findings

MIP* equals PZKMIP* class.

All co-recursively enumerable languages have zero knowledge MIP* protocols.

Quantum zero knowledge proofs can be efficiently constructed from any MIP* protocol.

Abstract

In this work we consider the interplay between multiprover interactive proofs, quantum entanglement, and zero knowledge proofs - notions that are central pillars of complexity theory, quantum information and cryptography. In particular, we study the relationship between the complexity class MIP$^*$, the set of languages decidable by multiprover interactive proofs with quantumly entangled provers, and the class PZKMIP$^*$, which is the set of languages decidable by MIP$^*$ protocols that furthermore possess the perfect zero knowledge property. Our main result is that the two classes are equal, i.e., MIP$^* =$ PZKMIP$^*$. This result provides a quantum analogue of the celebrated result of Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP $=$ PZKMIP (in other words, all classical multiprover interactive protocols can be made zero knowledge). We prove our result by showing that every MIP$^*$ protocol can be efficiently transformed into an equivalent zero knowledge MIP$^*$ protocol in a manner that preserves the completeness-soundness gap. Combining our transformation with previous results by Slofstra (Forum of Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we obtain the corollary that all co-recursively enumerable languages (which include undecidable problems as well as all decidable problems) have zero knowledge MIP$^*$ protocols with vanishing promise gap.