Compiling Any $\mathsf{MIP}^{*}$ into a (Succinct) Classical Interactive Argument
Andrew Huang, Yael Tauman Kalai
TL;DR
The paper addresses the problem of classically verifying quantum computations by converting any $\mathsf{MIP}^{*}$ protocol into a succinct, classical $\mathsf{QIA}$ whose post-quantum soundness rests on the sub-exponential hardness of $\mathsf{LWE}$. The core approach combines a two-step reduction: first, show that a language with a semi-malicious $\mathsf{QIP}$ protocol lies in $\mathsf{QMATIME}(t_P)$; second, construct a succinct classical $\mathsf{QIA}$ for such languages by leveraging real-valued witnesses, Morimae–Fitzsimons measurement-based reductions, and $\mathsf{LWE}$-based semi-succinct commitments, followed by protocol compression to full succinctness. The result yields a $T$-secure classical $\mathsf{QIA}$ with poly$(\lambda)$ rounds and poly$(\lambda)$-sized messages, where the verifier runs in $\text{poly}(\lambda) + \widetilde{O}(|x|)$ time and the prover runs in poly$(t_P)$. The framework extends to $\mathsf{QIP}$ with semi-malicious soundness and provides open questions on witness restrictions, public verifiability, and extensions to $\mathsf{MIP}^{*}$-games. This work advances practical classical verification of quantum computations by connecting interactive proofs, commitment schemes, and LWE-based cryptography in a robust compiler.
Abstract
We present a generic compiler that converts any $\mathsf{MIP}^{*}$ protocol into a succinct interactive argument where the communication and the verifier are classical, and where post-quantum soundness relies on the post-quantum sub-exponential hardness of the Learning with Errors ($\mathsf{LWE}$) problem. Prior to this work, such a compiler for $\mathsf{MIP}^{*}$ was given by Kalai, Lombardi, Vaikuntanathan and Yang (STOC 2022), but the post-quantum soundness of this compiler is still under investigation. More generally, our compiler can be applied to any $\mathsf{QIP}$ protocol which is sound only against semi-malicious provers that follow the prescribed protocol, but with possibly malicious initial state. Our compiler consists of two steps. We first show that if a language $\mathcal{L}$ has a $\mathsf{QIP}$ with semi-malicious soundness, where the prover runs in time $T$, then $\mathcal{L} \in \mathsf{QMATIME}(T)$. Then we construct a succinct classical argument for any such language, where the communication complexity grows polylogarithmically with $T$, under the post-quantum sub-exponential hardness of $\mathsf{LWE}$. Note: After this work was finished, an independent and concurrent work (Baroni et al. 2025) resolved the question of quantum soundness of the KLVY compiler.