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Provable and Verifiable Quantum Advantage in Sample Complexity

Marcello Benedetti1,*, Harry Buhrman1,2,3,†, and Jordi Weggemans2,4,‡

  • *Contact author: marcello.benedetti@quantinuum.com
  • †Contact author: harry.buhrman@quantinuum.com
  • ‡Contact author: jrw@cwi.nl
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Phys. Rev. Lett. 136, 040601 – Published 27 January, 2026

DOI: https://doi.org/10.1103/q55v-wm7y

Abstract

Consider a fixed universe of N=2n 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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Outline

Information

Phys. Rev. Lett. 136, 040601– Published 27 January, 2026
  • Received 11 August 2025
  • Revised 20 October 2025
  • Accepted 9 December 2025
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