A no free lunch theorem for untrained quantum circuits in machine learning

Abstract

This paper proves that if an untrained quantum circuit is used as a resource in a machine learning workflow, then on average no quantum circuit is better than any other that can achieve the same set of computational effects. This is the titular no free lunch theorem. The paper also proves a supporting theorem that even if the idealisations of the no free lunch theorem are omitted, the average quantum advantage remains negligible at best. These results cast serious doubt on several proposals to use untrained quantum circuits in machine learning workflows: at best such claims should be substantiated empirically, as this paper proves there is no a priori theoretical reason to suppose that introducing an untrained quantum circuit will increase performance.