A fundamental problem in quantum many-body physics is that of finding ground states of localHamiltonians. A number of recent works gave provably efficient machine learning (ML) algorithmsfor learning ground states.

A fundamental problem in quantum many-body physics is that of finding ground states of localHamiltonians. A number of recent works gave provably efficient machine learning (ML) algorithmsfor learning ground states. Science 2022], introduced an approach for learningproperties of the ground state of an n -qubit gapped local Hamiltonian H from only n {\mathcal{O}(1)} datapoints sampled from Hamiltonians in the same phase of matter. Nature Communications 2024], to \mathcal{O}(\log) samples when the geometry of the n -qubit system is known.In this work, we introduce two approaches that achieve a constant sample complexity, independentof system size n, for learning ground state properties. Our first algorithm consists of a simplemodification of the ML model used by Lewis et al. and applies to a property of interest known beforehand.

We study social choice mechanisms in an implicit utilitarian framework with a metric constraint, where the goal is to minimize \textit{Distortion}, the worst case social cost of an ordinal mechanism relative to underlying cardinal utilities. We consider two additional desiderata: Constant sample complexity and Squared Distortion. Constant sample complexity means that the mechanism (potentially randomized) only uses a constant number of ordinal queries regardless of the number of voters and alternatives. Squared Distortion is a measure of variance of the Distortion of a randomized mechanism. Our primary contribution is the first social choice mechanism with constant sample complexity \textit{and} constant Squared Distortion (which also implies constant Distortion). We call the mechanism Random Referee, because it uses a random agent to compare two alternatives that are the favorites of two other random agents. We prove that the use of a comparison query is necessary: no mechanism that only elicits the top-k preferred alternatives of voters (for constant k) can have Squared Distortion that is sublinear in the number of alternatives. We also prove that unlike any top-k only mechanism, the Distortion of Random Referee meaningfully improves on benign metric spaces, using the Euclidean plane as a canonical example. Finally, among top-1 only mechanisms, we introduce Random Oligarchy. The mechanism asks just 3 queries and is essentially optimal among the class of such mechanisms with respect to Distortion. In summary, we demonstrate the surprising power of constant sample complexity mechanisms generally, and just three random voters in particular, to provide some of the best known results in the implicit utilitarian framework.