The potential for significant experimental achievements in quantum algorithms is the source of much enthusiasm in the emerging era of the small digital quantum computer. As such, attention is increasingly focused on practical concerns regarding spatially constant factors and gate complexity, in contrast to traditional asymptotic analyses. A predominant approach to compiling quantum algorithms into a minimum of qubits and quantum gates revolves around sophisticated mathematical optimization and the translation of classical digital circuits into reversible quantum logic. It would therefore be surprising if physical intuition was relevant to this abstract task. On the contrary, I demonstrate with some motivating, diverse but accessible examples how a rigorous understanding of the simplest physics – rotations on a sphere – leads to an unexpected and very efficient form of quantum compilation. This results in a quantum algorithm for Hamiltonian simulation with optimal query complexity in all parameters, within asymptotic and non-asymptotic bounds, and extraordinarily low overhead.

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