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arXiv · 2502.08721

Provable and Verifiable Quantum Advantage in Sample Complexity

Marcello Benedetti·Harry Buhrman·Jordi Weggemans
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Abstract

Consider a fixed universe of $N=2^n$ elements and the uniform distribution over elements of some subset of size $K$. Given samples from this distribution, the task of complement sampling is to provide a sample from the complementary subset. We give a simple quantum algorithm that uses only a single quantum sample -- a single copy of the uniform superposition over elements of the subset. When $K=N/2$, we show that the quantum algorithm succeeds with probability $1$, whereas any classical algorithm that succeeds with bounded probability of error requires a number of samples of the order of $N$. This shows that in a sample-to-sample setting, quantum computation can achieve the largest possible separation over classical computation. We show that the same bound can be lifted to prove average-case hardness, paving the way for demonstrations on noisy intermediate-scale quantum (NISQ) computers. It follows that under the assumption of the existence of one-way functions, complement sampling gives provable, verifiable and NISQable quantum advantage in a sample complexity setting.

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Marcello Benedetti, Harry Buhrman, Jordi Weggemans. 2025-02-12. Provable and Verifiable Quantum Advantage in Sample Complexity. https://doi.org/10.1103/q55v-wm7y

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