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With quantum computers of significant size now on the horizon, we shouldunderstand how to best exploit their initially limited abilities. To this end,we aim to identify a practical problem that is beyond the reach of currentclassical computers, but that requires the fewest resources for a quantumcomputer. We consider quantum simulation of spin systems, which could beapplied to understand condensed matter phenomena. We synthesize explicitcircuits for three leading quantum simulation algorithms, employing diversetechniques to tighten error bounds and optimize circuit implementations.Quantum signal processing appears to be preferred among algorithms withrigorous performance guarantees, whereas higher-order product formulas prevailif empirical error estimates suffice. Our circuits are orders of magnitudesmaller than those for the simplest classically-infeasible instances offactoring and quantum chemistry.

Toward the first quantum simulation with quantum speedup