Probabilistically Checking Quantum Proofs, with Interaction

2026-06-08

AI Summary

This work addresses the problem of efficiently verifying QMA statements while measuring only a vanishingly small number of qubits. We construct a quantum interactive oracle proof (qIOP) system in which a classical or quantum verifier, interacting with an unbounded prover, can complete verification by reading only a polylogarithmic number of qubits from the total communication transcript. Our protocol provides the first information-theoretically secure, locally verifiable, and robust interactive characterization of QMA, without relying on the unproven quantum PCP conjecture. The key technical ingredient combines quantum locally testable codes (LTCs) with classical PCPPs, leveraging the local indistinguishability of LTCs to avoid complex many-body measurements. The overall communication complexity is polynomial, completeness is exponentially close to one, and soundness enjoys a constant gap.

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 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 $1$ 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. 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.

Problem

• quantum interactive oracle proofs
• QMA
• local verification
• probabilistically checkable proofs
• quantum locally testable codes

Innovation

• quantum interactive oracle proofs
• QMA
• quantum locally testable codes
• probabilistically checkable proofs of proximity
• local verification