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arXiv:2604.11952

A Relativizing MIP for BQP

Scott Aaronson, Anand Natarajan, Avishay Tal, Agi Villanyi

quant-phcs.CCSubmitted Apr 13, 2026• Announced Apr 15, 2026• 0 citations
v1 · Apr 13, 2026v2 · Oct 5, 2026
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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.

Comments
19 pages, 4 figures

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Complexity class containments involving interactive proof classes are famously nonrelativizing: although IP=PSPACE\mathsf{IP} = \mathsf{PSPACE}IP=PSPACE, Fortnow and Sipser showed that that there exists an oracle relative to which coNP⊈IP\mathsf{coNP} \not\subseteq \mathsf{IP}coNP⊆IP. In contrast, the question of whether the containment BQP⊆IP\mathsf{BQP} \subseteq \mathsf{IP}BQP⊆IP is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment BQP⊆MIP\mathsf{BQP} \subseteq \mathsf{MIP}BQP⊆MIP holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle OOO, a PCP\mathsf{PCP}PCP proof system for BQPO\mathsf{BQP}^{O}BQPO where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle OOO. 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 IP\mathsf{IP}IP for BQP\mathsf{BQP}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.
BQP⊆IP\mathsf{BQP} \subseteq \mathsf{IP}BQP⊆IP Does Not Relativize
We construct an oracle relative to which BQP⊈IP\mathsf{BQP} \not\subseteq \mathsf{IP}BQP⊆IP, resolving a long-standing open question in quantum complexity theory. Together with recent work due to Aaronson et al., our work also gives the first oracle separation between IP\mathsf{IP}IP and MIP\mathsf{MIP}MIP, answering a question dating back to Fortnow's thesis. Our separation is based on the Forrelation problem, where given Boolean functions fff and ggg, the goal is to determine if fff is correlated with the Fourier spectrum of ggg. While this task is solvable by a query-efficient quantum algorithm, we show that it admits no classical interactive protocol with polynomial communication and a polynomial-query verifier. Our proof is based on (i) a new structural result showing how to approximate Avg-Max circuits (which are well-known to capture the power of interactive proofs in the oracular setting) by convex functions with small first and second derivatives and (ii) a novel analysis establishing that the Forrelation distribution suggested by Aaronson and Ambainis fools such functions. Our results imply that any prover-efficient classical interactive protocol for BQP\mathsf{BQP}BQP must rely on non-relativizing techniques. This might serve as a partial explanation for the lack of progress towards doubly-efficient, unconditionally sound classical verification of quantum computation.
We initiate the study of quantum Interactive Oracle Proofs (qIOPs), a generalization of both quantum Probabilistically Checkable Proofs and quantum Interactive Proofs, as well as a quantum analogue of classical Interactive Oracle Proofs. In the model of quantum Interactive Oracle Proofs, we allow multiple rounds of quantum interaction between the quantum prover and the quantum verifier, but the verifier has limited access to quantum resources. This includes both queries to the prover's messages and the complexity of the quantum circuits applied by the verifier. The question of whether QMA admits a quantum interactive oracle proof system is a relaxation of the quantum PCP Conjecture. We show the following two main constructions of qIOPs, both of which are unconditional: - We construct a qIOP for QMA in which the verifier shares polynomially many EPR pairs with the prover at the start of the protocol and reads only a constant number of qubits from the prover's messages. - We provide a stronger construction of qIOP for QMA in which the verifier not only reads a constant number of qubits but also operates on a constant number of qubits overall, including those in their private registers. However, in this stronger setting, the communication complexity becomes exponential. This leaves open the question of whether strong qIOPs for QMA, with polynomial communication complexity, exist. As a key component of our construction, we introduce a novel single prover many-qubits test, which may be of independent interest.
In recent years, the quantum oracle model introduced by Aaronson and Kuperberg (2007) has found a lot of use in showing oracle separations between complexity classes and cryptographic primitives. It is generally assumed that proof techniques that do not relativize with respect to quantum oracles will also not relativize with respect to classical oracles. Aaronson (2009) showed that this is not the case by showing a complexity class containment that relativizes with respect to classical oracles but not with quantum oracles. However, their result only works for zero-error quantum complexity classes and they leave open the problem for bounded-error complexity classes. We show that there is a quantum oracle problem that is contained in the class QMA, but not in a class we call polyQCPH. However, with respect to classical oracles, QMA is contained in polyQCPH, because polyQCPH is equal to PSPACE with respect to classical oracles. Our result works for polyQCPH, which is a bounded-error complexity class, thus it resolves the open problem from Aaronson (2009). We also show that the same separation holds relative to a distributional oracle, which is a model introduced by Natarajan and Nirkhe (2024). We believe our findings show the need for some caution when using these non-standard oracle models, particularly when showing separations between quantum and classical resources.
Recent works have demonstrated that quantum oracles have subtle behavior, as access to inverse, conjugate or controlled queries can exponentially change the query complexity of certain tasks. Inspired by these works, we introduce the notion of meta-complexity of quantum relativization. We ask: for any two quantum complexity classes, under which "types" of quantum oracles are they equal or separated? Different pantheons of oracles (or quantum oracle types, e.g. unitary vs state, poly- vs superpoly-dimensional, closed under inverse or not) form a partially ordered set based on their power in separating complexity classes. Moreover, two oracle pantheons A and B are separated if there exists a pair of complexity classes that are separated under an oracle from pantheon A but yet the complexity classes are equivalent under all oracles from pantheon B. We show that this meta-complexity can be nontrivial by giving a complete classification, within the family of oracle pantheons defined in this paper, of which models can separate the complexity classes PostBQP\mathsf{PostBQP}PostBQP and PreciseBQP\mathsf{PreciseBQP}PreciseBQP, the exponentially precise variant of BQP\mathsf{BQP}BQP. Both classes equal PP\mathsf{PP}PP in the unrelativized setting. Within our taxonomy, they remain equal relative to real or polynomial-dimensional unitary oracles and whenever inverse or conjugate access is supplied. In contrast, we give a separation relative to forward-only complex diagonal unitaries of superpolynomial dimension, as well as a separation relative to single-qubit state-preparation oracles. We view this as a test case for the meta-complexity of oracles which underscores the subtlety inherent to the relativization of quantum complexity classes.
Since the introduction of the complexity class QMA as a quantum-verifier analogue of NP (Kitaev, 1997), many have wondered whether quantum proofs are necessary or classical proofs suffice - that is, whether QMA = QCMA or QCMA != QMA (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC '07). This longstanding question was recently answered by works of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC '26) and Bostanci, Huang, and Vaikuntanathan (FOCS '26), which showed that quantum proofs are more powerful than classical ones in the classical-oracle setting. However, it remains unclear what exactly makes quantum proofs more powerful than classical ones. In the information-theoretic setting, a family of quantum states is not classicalizable if and only if it is unclonable. Indeed, these recent works also explicitly highlight the unclonability of their quantum proofs as a key mechanism behind their separations, and their arguments crucially rely on this property. This raises the question of whether unclonability is necessary for quantum proofs to be more powerful than classical ones. In this work, we show that even clonable quantum proofs can be more powerful than classical ones relative to a classical oracle by constructing a classical oracle O such that QCMA^O != ClonableQMA^O. This resolves the open question of Nehoran and Zhandry (ITCS '24), who established the analogous separation relative to a quantum oracle. We also show a classical-oracle separation between BQP/clonableqpoly and BQP/poly, and give applications of our results to quantum cryptography.
Whether some problems require quantum proofs has been a central question in quantum complexity (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC 2007). Recently, breakthrough work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), followed by a simpler separation due to Bostanci, Huang, and Vaikuntanathan (FOCS 2026), established a classical oracle separation between QMA\mathsf{QMA}QMA and QCMA\mathsf{QCMA}QCMA. However, while they are not in QCMA\mathsf{QCMA}QCMA, the problems used in both separations still lie in AM\mathsf{AM}AM: they admit a two-message public-coin proof system with a classical verifier. In this work, we ask whether some problems truly require quantum proofs. More formally, we consider the complexity class QCIP\mathsf{QCIP}QCIP, introduced by Buhrman, Le Gall, and Weggemans (2024), where an efficient quantum verifier interacts with an unbounded prover over a classical channel for an arbitrary polynomial number of rounds. We construct a classical oracle O\mathcal{O}O such that QMAO⊈QCIPO\mathsf{QMA}^{\mathcal{O}}\not\subseteq\mathsf{QCIP}^{\mathcal{O}}QMAO⊆QCIPO, thus showing that some languages indeed require quantum proofs, with no classical replacements. Since QCMA=QCIP[1]\mathsf{QCMA}=\mathsf{QCIP}[1]QCMA=QCIP[1], this strengthens the earlier QMA\mathsf{QMA}QMA--QCMA\mathsf{QCMA}QCMA separations, which now follow as a special case of our result. As a technical contribution, we extend to the complexity theory setting the techniques developed by Cakan, Goyal, and Shmueli (CRYPTO 2026) in the context of cryptography for analyzing classically interacting quantum machines. We believe this may be of independent interest.