Provable and Verifiable Quantum Advantage in Sample Complexity

TL;DR

This paper demonstrates a quantum algorithm that achieves a provable advantage over classical methods in the task of complement sampling, requiring only a single quantum sample compared to classical methods needing many, especially when the subset size is half the universe.

Contribution

The authors introduce a simple quantum algorithm for complement sampling that achieves exponential separation in sample complexity over classical algorithms, with implications for NISQ devices and cryptography.

Findings

Quantum algorithm succeeds with probability 1 when K=N/2

Classical algorithms require order N samples for success

Supports quantum advantage in sample complexity with NISQ feasibility

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.