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Sampling problems demonstrating beyond classical computing power with noisyintermediate-scale quantum (NISQ) devices have been experimentally realized. Inthose realizations, however, our trust that the quantum devices faithfullysolve the claimed sampling problems is usually limited to simulations ofsmaller-scale instances and is, therefore, indirect. The problem of verifiablequantum advantage aims to resolve this critical issue and provides us withgreater confidence in a claimed advantage. Instantaneous quantumpolynomial-time (IQP) sampling has been proposed to achieve beyond classicalcapabilities with a verifiable scheme based on quadratic-residue codes (QRC).Unfortunately, this verification scheme was recently broken by an attackproposed by Kahanamoku-Meyer. In this work, we revive IQP-based verifiablequantum advantage by making two major contributions. Firstly, we introduce afamily of IQP sampling protocols called the \emph{stabilizer scheme}, whichbuilds on results linking IQP circuits, the stabilizer formalism, codingtheory, and an efficient characterization of IQP circuit correlation functions.This construction extends the scope of existing IQP-based schemes whilemaintaining their simplicity and verifiability. Secondly, we introduce the\emph{Hidden Structured Code} (HSC) problem as a well-defined mathematicalchallenge that underlies the stabilizer scheme. To assess classical security,we explore a class of attacks based on secret extraction, including theKahanamoku-Meyer's attack as a special case. We provide evidence of thesecurity of the stabilizer scheme, assuming the hardness of the HSC problem. Wealso point out that the vulnerability observed in the original QRC scheme isprimarily attributed to inappropriate parameter choices, which can be naturallyrectified with proper parameter settings.

IQP Sampling and Verifiable Quantum Advantage: Stabilizer Scheme and Classical Security