Sampling Supremacy

Abstract: We give new evidence that quantum circuits are substantially more powerful than classical circuits. We show, relative to a random oracle, that polynomial-size quantum circuits can sample distributions that subexponential-size classical circuits cannot approximate even to TV distance $1-o(1)$. Prior work of Aaronson and Arkhipov (2011) showed such a separation for the case of exact sampling (i.e.~TV distance $0$), but separations for approximate sampling were only known for uniform algorithms. A key ingredient in our proof is a new hardness amplification lemma for the classical query complexity of the Yamakawa--Zhandry (2022) search problem. We show that the probability that any family of query algorithms collectively finds $k$ distinct solutions decays exponentially in $k$.

Abstract: Recently, a quantum algorithm called Decoded Quantum Interferometry (DQI) was introduced that achieves an apparent exponential speedup for Optimal Polynomial Intersection (OPI) problem, which has previously been studied in the contexts of cryptography and error correcting codes. However, this left open the question of how many logical gates and logical qubits would be needed to solve a classically intractable instance of OPI. Here, we develop optimized implementations of DQI which greatly reduce its resource requirements. We establish that DQI for OPI is the first known candidate for verifiable quantum advantage with optimal asymptotic speedup: solving instances with classical hardness $O(2^N)$ requires only $\widetilde{O}(N)$ quantum gates, matching the theoretical lower bound. To realize this, we overcome the primary bottleneck of reversible Reed-Solomon decoding by introducing novel quantum circuits for the Extended Euclidean Algorithm (EEA) that reduce the leading-order space complexity to the theoretical minimum of $2nb$ qubits. These improvements are broadly applicable, including to Shor's algorithm for the discrete logarithm. We analyze OPI over binary extension fields $\GF(2^b)$, assess hardness against new classical attacks, and identify resilient instances. Our resource estimates show that classically intractable OPI instances (requiring $>10^{23}$ classical trials) can be solved with approximately 5.72 million Toffoli gates. This is roughly $1000$ times fewer gates than required for factoring RSA-2048 and, remarkably, is also less than the leading interactive protocol for computational proof of quantumness, positioning DQI as a compelling candidate for practical, verifiable quantum advantage.