Secure Two-Party Quantum Computation Over Classical Channels

TL;DR

This paper explores the limitations and possibilities of secure two-party quantum computation over classical channels, introducing new security notions, assumptions, and a zero-knowledge proof compiler for classical verifiers.

Contribution

It demonstrates the impossibility of black-box secure quantum protocols over classical channels and proposes alternative approaches, including a zero-knowledge proof compiler for quantum knowledge.

Findings

Impossibility of black-box secure quantum protocols with classical channels.

A protocol satisfying one-sided simulation security using learning with errors.

Conversion of Mahadev's protocol into a zero-knowledge proof for classical verifiers.

Abstract

Secure two-party computation considers the problem of two parties computing a joint function of their private inputs without revealing anything beyond the output. In this work, we consider the setting where the two parties (a classical Alice and a quantum Bob) can communicate only via a classical channel. Our first result shows that it is in general impossible to realize a two-party quantum functionality with black-box simulation in the case of malicious quantum adversaries. In particular, we show that the existence of a secure quantum computing protocol that relies only on classical channels would contradict the quantum no-cloning argument. We circumvent this impossibility following three different approaches. The first is by considering a weaker security notion called one-sided simulation security. This notion protects the input of one party (the quantum Bob) in the standard simulation-based sense and protects the privacy of the other party's input (the classical Alice). We show how to realize a protocol that satisfies this notion relying on the learning with errors assumption. The second way to circumvent the impossibility result, while at the same time providing standard simulation-based security also against a malicious Bob, is by assuming that the quantum input has an efficient classical representation. Finally, we focus our attention on the class of zero-knowledge functionalities and provide a compiler that takes as input a classical proof of quantum knowledge (PoQK) protocol for a QMA relation R and outputs a zero-knowledge PoQK for R that can be verified by classical parties. The direct implication of our result is that Mahadev's protocol for classical verification of quantum computations (FOCS'18) can be turned into a zero-knowledge proof of quantum knowledge with classical verifiers. To the best of our knowledge, we are the first to instantiate such a primitive.