A Relativizing MIP for BQP

arXiv:2604.11952v2 Announce Type: replace 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.

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.

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2604.11952 [quant-ph]

(or arXiv:2604.11952v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2604.11952 Focus to learn more arXiv-issued DOI via DataCite

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A Relativizing MIP for BQP

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