A Relativizing MIP for BQP

2026-04-13

AI Summary

This work addresses the long-standing open question of whether the quantum complexity class BQP is contained in the interactive proof systems IP or MIP relative to any classical oracle. For an arbitrary classical oracle \( O \), we construct an exponentially long PCP proof system for \( \text{BQP}^O \), wherein a classical verifier accesses both the proof and the oracle via a polynomial number of queries. Our protocol constitutes the first relativized MIP protocol that holds for all classical oracles, innovatively incorporating ideas from Grover–Rudolph state preparation and using relativization as a proxy for verification efficiency. The result establishes that \( \text{BQP} \subseteq \text{MIP} \) holds relative to any classical oracle, thereby opening a new avenue toward constructing non-cryptographic, classically verifiable protocols for quantum computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• BQP
• IP
• relativization
• interactive proof
• quantum complexity

Innovation

• relativization
• BQP
• MIP
• PCP
• quantum verification