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Probabilistically Checking Quantum Proofs, with Interaction

Authors Baocheng Sun , Thomas Vidick

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LIPIcs.CCC.2026.4.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum complexity theory
  • quantum probabilistically checkable proofs
  • interactive oracle proofs
  • quantum locally testable codes
  • QMA

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Abstract

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.

Cite As Get BibTex

Baocheng Sun and Thomas Vidick. Probabilistically Checking Quantum Proofs, with Interaction. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 4:1-4:49, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.4

Author Details

Baocheng Sun
  • Weizmann Institute of Science, Rehovot, Israel
Thomas Vidick
  • School of Computer and Communication Sciences and Center for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne, Switzerland
  • Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel

Acknowledgements

Both authors were supported by AFOSR Grant No. FA9550-22-1-0391 and ERC Consolidator Grant VerNisQDevS (101086733).

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