Verifiable quantum advantage based on polynomials with planted structures

2026-09-30

AI Summary

This study addresses the challenge that existing quantum advantage protocols are difficult to verify efficiently using only classical outputs. To overcome this limitation, this work proposes a "simulated secret" mechanism that constructs a verifiable quantum advantage scheme utilizing Instantaneous Quantum Polynomial (IQP) circuits with planted independent spaces. By integrating low-rank stabilizer decompositions, cross-entropy testing, and fine-grained complexity theory, the authors establish a polynomial gap between verification time and the computational cost for classical adversaries. Theoretically, this approach achieves an exponential time separation while requiring only approximately 100 logical qubits for practical implementation. Furthermore, the proposed framework inherently supports certified randomness generation. Overall, this research introduces a novel paradigm for practical and efficiently verifiable quantum advantage demonstrations.

Abstract

A central question in the theory of quantum advantage is whether there are quantum advantage protocols with similar resource requirements as random circuit sampling that are also verifiable just from the classical outputs of the quantum computation. Here, we develop the idea of simulation secrets for verifiable advantage. A verifier can use a simulation secret to evaluate a cross-entropy test faster than it would take a classical adversary to pass the test. We instantiate this idea using IQP circuits described by cubic polynomials with planted independent spaces. These correspond to the largest independent set in the orbit of a polynomial under the general linear group and yield a low-rank stabilizer decomposition of the corresponding state. We conjecture that large independent spaces are invisible to a computationally bounded adversary, and therefore they cannot exploit them to pass the protocol. A second conjecture regards the fine-grained complexity of producing samples that pass the cross-entropy test for uniformly random polynomials. Under these conjectures, our scheme results in a polynomial gap between the verification time and the time a classical adversary would need to pass the protocol---both are exponential. It has a potential application to generating classically certifiable randomness, since the output distributions have high min-entropy. We estimate that the planted polynomial scheme is implementable using 100 logical qubits at logical error rates around $10^{-6}$.

Problem

• quantum advantage
• verifiable quantum advantage
• random circuit sampling
• cross-entropy test

Innovation

• verifiable quantum advantage
• simulation secrets
• IQP circuits
• planted polynomials
• cross-entropy test