Gregory D. Kahanamoku-Meyer, Soonwon Choi, Umesh V. Vazirani et al.
We propose and analyze a novel interactive protocol for demonstrating quantum computational advantage, which is efficiently classically verifiable. Our protocol relies upon the cryptographic hardness of trapdoor claw-free functions (TCFs). Through a surprising connection to Bell's inequality, our protocol avoids the need for an adaptive hardcore bit, with essentially no increase in the quantum circuit complexity and no extra cryptographic assumptions. Crucially, this expands the set of compatible TCFs, and we propose two new constructions: one based upon the decisional Diffie-Hellman problem and the other based upon Rabin's function, $x^2 \bmod N$. We also describe two independent innovations which improve the efficiency of our protocol's implementation: (i) a scheme to discard so-called "garbage bits", thereby removing the need for reversibility in the quantum circuits, and (ii) a natural way of performing post-selection which significantly reduces the fidelity needed to demonstrate quantum advantage. These two constructions may also be of independent interest, as they may be applicable to other TCF-based quantum cryptography such as certifiable random number generation. Finally, we design several efficient circuits for $x^2 \bmod N$ and describe a blueprint for their implementation on a Rydberg-atom-based quantum computer.
Recently, quantum computing experiments have for the first time exceeded the capability of classical computers to perform certain computations -- a milestone termed "quantum computational advantage." However, verifying the output of the quantum device in these experiments required extremely large classical computations. An exciting next step for demonstrating quantum capability would be to implement tests of quantum computational advantage with efficient classical verification, such that larger system sizes can be tested and verified. One of the first proposals for an efficiently-verifiable test of quantumness consists of hiding a secret classical bitstring inside a circuit of the class IQP, in such a way that samples from the circuit's output distribution are correlated with the secret (arXiv:0809.0847). The classical hardness of this protocol has been supported by evidence that directly simulating IQP circuits is hard, but the security of the protocol against other (non-simulating) classical attacks has remained an open question. In this work we demonstrate that the protocol is not secure against classical forgery. We describe a classical algorithm that can not only convince the verifier that the (classical) prover is quantum, but can in fact can extract the secret key underlying a given protocol instance. Furthermore, we show that the key extraction algorithm is efficient in practice for problem sizes of hundreds of qubits. Finally, we provide an implementation of the algorithm, and give the secret vector underlying the "$25 challenge" posted online by the authors of the original paper.