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arXiv:2506.19792

Collapses in quantum-classical probabilistically checkable proofs and the quantum polynomial hierarchy

Kartik Anand, Kabgyun Jeong, Junseo Lee

Published Jun 24, 2025• 0 citations
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Abstract

We investigate structural properties of quantum proof systems by establishing collapse results that uncover simplifications in their complexity landscape. We extend classical results such as the Karp-Lipton theorem to quantum polynomial hierarchy with quantum proofs and establish uniqueness preservation for quantum-classical probabilistically checkable proof systems. Our main contributions are threefold. First, we prove that restricting quantum-classical PCP systems to uniqueness does not reduce computational power: $\mathsf{UniqueQCPCP} = \mathsf{QCPCP}$ under $\mathsf{BQ}$-operator and randomized reductions, demonstrating robustness similar to the $\mathsf{UniqueQCMA} = \mathsf{QCMA}$ result. Second, we establish a non-uniform quantum analogue of the Karp-Lipton theorem, showing that if $\mathsf{QMA} \subseteq \mathsf{BQP}/\mathsf{qpoly}$, then $\mathsf{QPH} \subseteq \mathsf{QΣ}_2/\mathsf{qpoly}$, extending the classical collapse theorem to quantum complexity with quantum advice. Third, we introduce a consistent variant of the quantum polynomial hierarchy ($\mathsf{CQPH}$) with consistency constraints across interaction rounds while maintaining product-state proofs, proving its unconditional collapse $\mathsf{CQPH} = \mathsf{CQΣ}_2$. This contrasts with prior work on quantum-entangled polynomial hierarchy, showing that consistency rather than entanglement drives the collapse. These results contribute to understanding structural boundaries in quantum complexity theory and the interplay between constraint types in quantum proof systems.

DOI
10.48550/arxiv.2506.19792

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The Polynomial-Time Hierarchy (PH\mathsf{PH}PH) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to ''quantum advantage'' analyses for near-term quantum computers. Quantumly, however, despite the fact that at least \emph{four} definitions of quantum PH\mathsf{PH}PH exist, it has been challenging to prove analogues for these of even basic facts from PH\mathsf{PH}PH. This work studies three quantum-verifier based generalizations of PH\mathsf{PH}PH, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings (QCPH\mathsf{QCPH}QCPH) and quantum mixed states (QPH\mathsf{QPH}QPH) as proofs, and one of which is new to this work, utilizing quantum pure states (pureQPH\mathsf{pureQPH}pureQPH) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for QCPH\mathsf{QCPH}QCPH. Then, for our new class pureQPH\mathsf{pureQPH}pureQPH, we show one-sided error reduction for pureQPH\mathsf{pureQPH}pureQPH, as well as the first bounds relating these quantum variants of PH\mathsf{PH}PH, namely QCPH⊆pureQPH⊆EXPPP\mathsf{QCPH}\subseteq \mathsf{pureQPH} \subseteq \mathsf{EXP}^{\mathsf{PP}}QCPH⊆pureQPH⊆EXPPP.
The model of interactive oracle proofs (IOP) generalizes the notion of probabilistically checkable proof (PCP), in which a static proof is verified probabilistically by querying a small number of bits, to the interactive setting: a polynomial-time verifier interacts with an unbounded prover, but is restricted to only reading a small number of bits, in total, from the messages sent by the prover. IOPs provide a relaxed setting in which to study local probabilistic verification. They have proved instrumental in devising efficient methods for verification through subsequent compilation into non-interactive or succinct protocols. We study a quantum analogue of interactive oracle proofs (qIOP) in which the verifier and communication are both allowed to be quantum; yet the verifier is restricted to perform measurements only on a small number of qubits received from the prover. Our main result is a qIOP for any language in QMA, in which the total communication is polynomial but the verifier only reads a polylogarithmic number of qubits in total. The protocol has completeness parameter exponentially close to 111 and soundness bounded away from 111 by a constant. In the absence of a quantum PCP theorem, this provides the first information-theoretically sound local and robust characterization of QMA, albeit interactive. Our protocol combines the use of a quantum locally testable code (LTC) with classical techniques, notably probabilistically checkable proofs of proximity (PCPP). We avoid the necessity for complex multi-qubit tests employed in other settings by leveraging the local indistinguishability property of the quantum LTC.
The model of interactive oracle proofs (IOP) generalizes the notion of probabilistically checkable proof (PCP), in which a static proof is verified probabilistically by querying a small number of bits, to the interactive setting: a polynomial-time verifier interacts with an unbounded prover, but is restricted to only reading a small number of bits, in total, from the messages sent by the prover. IOPs provide a relaxed setting in which to study local probabilistic verification. They have proved instrumental in devising efficient methods for verification through subsequent compilation into non-interactive or succinct protoocls. We study a quantum analogue of interactive oracle proofs (qIOP) in which the verifier and communication are both allowed to be quantum; yet the verifier is restricted to perform measurements only on a small number of qubits received from the prover. Our main result is a qIOP for any language in QMA, in which the total communication is polynomial but the verifier only reads a polylogarithmic number of qubits in total. The protocol has completeness parameter exponentially close to 1 and soundness bounded away from 1 by a constant. In the absence of a quantum PCP theorem, this provides the first information-theoretically sound local and robust characterization of QMA, albeit interactive. Previous works in the information-theoretic setting either considered two isolated but entangled quantum provers or quantum verifiers whose effort in a single round is small but remains polynomial when aggregated across all rounds of the protocol. Our protocol combines the use of a quantum locally testable code (LTC) with classical techniques, notably probabilistically checkable proofs of proximity (PCPP). We avoid the necessity for complex multi-qubit tests employed in other settings by leveraging the local indistinguishability property of the quantum LTC.
We give a corrected proof that if PP⊆BQP/qpoly\mathsf{PP} \subseteq \mathsf{BQP/qpoly}PP⊆BQP/qpoly (probabilistic polynomial time can be efficiently simulated by quantum circuits with quantum advice), then the Counting Hierarchy collapses, as originally claimed by Aaronson (CCC'06). This recovers the related unconditional claim that PP\mathsf{PP}PP does not have circuits of any fixed-polynomial size nkn^knk even with quantum advice. Our result is based on proving that YQP∗\mathsf{YQP^*}YQP∗, an oblivious version of QMA∩coQMA\mathsf{QMA}\cap\mathsf{coQMA}QMA∩coQMA, is contained in APP\mathsf{APP}APP, a PP\mathsf{PP}PP-low subclass of PP\mathsf{PP}PP with an arbitrarily small but nonzero promise gap.
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.