QuEra's open-source packages GenericTensorNetworks.jl and Bloqade.jl were instrumental in discovering hard instances and understanding quantum performance. A super-linear quantum speed-up was found compared to a class of generic classical algorithms. Characterizing the difficulty of the optimization problem instances with a 'hardness parameter,' the team identified cases that challenged classical computers, but that were more efficiently solved with the neutral-atom quantum processor. This combination of system size, circuit depth, and outstanding quantum control culminated in a quantum leap: problem instances were found with empirically better-than-expected performance on the quantum processor versus classical heuristics. A quantum-classical hybrid algorithm had to be deployed in a closed loop, with direct, automated feedback to the quantum processor. Unlike in previous examples of quantum optimization, the large system size and circuit depth used in this work made it impossible to use classical simulations to pre-optimize the control parameters. The calculations were performed on Harvard's quantum processor of 289 qubits operating in the analog mode, with effective circuit depths up to 32. In this landmark publication, the authors not only deploy the first implementation of efficient quantum optimization on a real quantum computer, but also showcase unprecedented quantum hardware power. Previously, neutral-atom quantum processors had been proposed to efficiently encode certain hard combinatorial optimization problems. Titled 'Quantum Optimization of Maximum Independent Set using Rydberg Atom Arrays,' was published on May 5 th, 2022, in Science Magazine. The study was co-led by Mikhail Lukin, the George Vasmer Leverett Professor of Physics at Harvard and co-director of the Harvard Quantum Initiative, Markus Greiner, George Vasmer Leverett Professor of Physics, and Vladan Vuletic, Lester Wolfe Professor of Physics at MIT.

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