Separating Non-Interactive Classical Verification of Quantum Computation from Falsifiable Assumptions

TL;DR

This paper proves that non-interactive classical verification of quantum computation cannot be based on falsifiable assumptions, establishing a fundamental limitation in cryptographic protocols for quantum verification.

Contribution

It provides a strong negative result showing the impossibility of non-interactive classical verification under falsifiable assumptions, assuming a QMA-QCMA gap problem exists.

Findings

No quantum black-box reduction from non-interactive verification to falsifiable assumptions.

Separation holds under the existence of a QMA-QCMA gap problem.

Construction relative to a quantum unitary oracle supports the assumption.

Abstract

Mahadev [SIAM J. Comput. 2022] introduced the first protocol for classical verification of quantum computation based on the Learning-with-Errors (LWE) assumption, achieving a 4-message interactive scheme. This breakthrough naturally raised the question of whether fewer messages are possible in the plain model. Despite its importance, this question has remained unresolved. In this work, we prove that there is no quantum black-box reduction of non-interactive classical verification of quantum computation of $\textsf{QMA}$ to any falsifiable assumption. Here, "non-interactive" means that after an instance-independent setup, the protocol consists of a single message. This constitutes a strong negative result given that falsifiable assumptions cover almost all standard assumptions used in cryptography, including LWE. Our separation holds under the existence of a $\textsf{QMA} \text{-} \textsf{QCMA}$ gap problem. Essentially, these problems require a slightly stronger assumption than $\textsf{QMA}\neq \textsf{QCMA}$. To support the existence of such problems, we present a construction relative to a quantum unitary oracle.