  A quantum instruction set is where quantum hardware and software meet. We
develop new characterization and compilation techniques for non-Clifford gates
to accurately evaluate different quantum instruction set designs. We
specifically apply them to our fluxonium processor that supports mainstream
instruction $\mathrm{iSWAP}$ by calibrating and characterizing its square root
$\mathrm{SQiSW}$. We measure a gate fidelity of up to $99.72\%$ with an average
of $99.31\%$ and realize Haar random two-qubit gates using $\mathrm{SQiSW}$
with an average fidelity of $96.38\%$. This is an average error reduction of
$41\%$ for the former and a $50\%$ reduction for the latter compared to using
$\mathrm{iSWAP}$ on the same processor. This shows designing the quantum
instruction set consisting of $\mathrm{SQiSW}$ and single-qubit gates on such
platforms leads to a performance boost at almost no cost.
