A no free lunch theorem for untrained quantum circuits in machine learning
Steven Herbert
TL;DR
This work addresses whether untrained quantum circuits can provide a theoretical advantage in classical machine learning by framing them as resources and proving a no free lunch theorem: when the resource is Haar-random (distinct and $2^{n_+}$-distinct), no circuit outperforms any other for the same set of computational effects on average. It formalizes a resource-theoretic structure, defines permutation-based equivalence classes and costs, and shows that the space of achievable effects is independent of the specific untrained circuit, with Haar randomness yielding the result almost surely. The authors extend the NFL result to non-empty inputs and discuss practical implications, conceding that empirical validation is essential to substantiate any claimed gains. The findings suggest that untrained quantum circuits cannot be assumed to offer a priori performance benefits, emphasizing the need for problem-matching and large-scale empirical demonstrations to justify their use in ML tasks.
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.