A Relativizing MIP for BQP

Quantum Physics [Submitted on 13 Apr 2026] A Relativizing MIP for BQP Scott Aaronson, Anand Natarajan, Avishay Tal, Agi Villanyi 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. Comments: 19 pages, 4 figures Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC) Cite as: arXiv:2604.11952 [quant-ph]   (or arXiv:2604.11952v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.11952 Focus to learn more Submission history From: Anand Natarajan [view email] [v1] Mon, 13 Apr 2026 18:45:12 UTC (51 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.CC References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)