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In 1987, Vaidman, Aharanov, and Albert put forward a puzzle called the MeanKing's Problem (MKP) that can be solved only by harnessing quantumentanglement. Prime-powered solutions to the problem have been shown to exist,but they have not yet been experimentally realized for any dimension beyondtwo. We propose a general first-of-its-kind experimental scheme for solving theMKP in prime dimensions ($D$). Our search is guided by the digital discoveryframework PyTheus, which finds highly interpretable graph-based representationsof quantum optical experimental setups; using it, we find specific solutionsand generalize to higher dimensions through human insight. As proof ofprinciple, we present a detailed investigation of our solution for the three-,five-, and seven-dimensional cases. We obtain maximum success probabilities of$72.8 \%$, $45.8\%$, and $34.8 \%$, respectively. We, therefore, posit that ourcomputer-inspired scheme yields solutions that exceed the classical probability($1/D$) twofold, demonstrating its promise for experimental implementation.

Experimental Solutions to the High-Dimensional Mean King's Problem