Perfect Zero Knowledge for Quantum Multiprover Interactive Proofs

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 PZK-MIP * , 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 * = PZK-MIP * . This result provides a quantum analogue of the celebrated result of Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP = PZK-MIP (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. IntroductionMultiprover interactive proofs (MIPs) are a model of computation where a probabilistic polynomial time verifier interacts with several all-powerful -but non-communicating -provers to check the validity of a statement (for example, whether a quantified boolean formula is satisfiable). If the statement is true, then there is a strategy for the provers to convince the verifier of this fact. Otherwise, for all prover strategies, the verifier rejects with high probability. This gives rise to the complexity class MIP, which is the set of all languages that can be decided by MIPs. This model of computation was first introduced by Ben-Or, Goldwasser, Kilian and Wigderson [6]. A foundational result in complexity theory due to Babai, Fortnow, and Lund shows that multiprover interactive proofs are surprisingly powerful: MIP is actually equal to the class of problems solvable in non-deterministic exponential time, i.e., MIP = NEXP [2].Research in quantum complexity theory has led to the study of quantum MIPs. In one of the most commonly considered models, the verifier interacts with provers that are quantumly entangled. Even though the provers still cannot communicate with each other, they can utilize correlations arising from local measurements on entangled quantum states. Such correlations cannot be explained classically, and the study of the counter-intuitive nature of these correlations dates back to the famous 1935 paper of Einstein, Podolsky and Rosen [18] and the seminal work of Bell in 1964 [4]. Over the past twenty years, MIPs with entangled provers have provided a fruitful computational lens through which the power of s...

Classically Verifiable NIZK for QMA with Preprocessing

“…We apply a similar idea to the NIZK of [BG20] to make the verification classical. The NIZK of [BG20] is based on the fact that a QMA language can be reduced to the 5-local Hamiltonian problem with a locally simulatable history state [GSY19]. That is, an instance x corresponds to an N -qubit Hamiltonian H x of the form…”

“…Recently, Broadbent and Grilo [BG20] showed that any QMA problem can be reduced to a 5-local Hamiltonian problem with local simulatability. (See also [GSY19].) Moreover, it is easy to see that we can make the Hamiltonian H x be of the form…”

“…has exponentially small energy. Due to the local simulatability, there is an efficient deterministic algorithm that outputs the classical description of a state that is close to the reduced density matrix of the history state on at most five qubits [BG20,GSY19]. If every quantum state |ψ makes V ′ reject with probability at least ǫ, then the groundenergy of H is at least Ω( ǫ T 3 ).…”

Practical Relativistic Zero-Knowledge for NP

“…Although certain algebraic zero-knowledge multi-prover interactive proofs for NP and NEXP using explicitly no commitments at all have been presented before in [7], [8] (sound against local provers) and [9], [10] (sound against entangled provers), in the local cases making these protocols entanglement sound is absolutely non-trivial, whereas in the entangled case the multi-round structure and the amount of communication in each round makes implementing the protocol completely impractical as well. (To their defense, the protocols were not designed to be practical).…”

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