A Relativizing MIP for BQP

• URL: http://arxiv.org/abs/2604.11952v1
• Date: Mon, 13 Apr 2026 18:45:12 GMT
• Title: A Relativizing MIP for BQP
• Abstract summary: We show that the containment $mathsfBQP subseteq mathsfMIP$ holds with respect to any classical oracle.<n>We propose relativization as a proxy for prover efficiency, and hope that progress towards an $mathsfIP$ for $mathsfBQP$ in the oracle world will lead to a non-cryptographic interactive protocol.
• Score: 2.060642030400714
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Complexity class containments involving interactive proof classes are famously nonrelativizing: although $\mathsf{IP} = \mathsf{PSPACE}$, Fortnow and Sipser showed that that there exists an oracle relative to which $\mathsf{coNP} \not\subseteq \mathsf{IP}$. In contrast, the question of whether the containment $\mathsf{BQP} \subseteq \mathsf{IP}$ is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment $\mathsf{BQP} \subseteq \mathsf{MIP}$ holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle $O$, a $\mathsf{PCP}$ proof system for $\mathsf{BQP}^{O}$ where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle $O$. Our construction is inspired by the state synthesis algorithm of Grover and Rudolph, and serves as a complement to the "exponential PCP" constructed by Aharonov, Arad, and Vidick, which achieves similar parameters but which is based on different ideas and does not relativize. We propose relativization as a proxy for prover efficiency, and hope that progress towards an $\mathsf{IP}$ for $\mathsf{BQP}$ in the oracle world will lead to a non-cryptographic interactive protocol for proving any quantum computation to a classical skeptic in the unrelativized world, which is a longstanding open problem in quantum complexity theory.

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