  Population leakage outside the qubit subspace presents a particularly harmful
source of error that cannot be handled by standard error correction methods.
Using a trapped $^{171}$Yb$+$ ion, we demonstrate an optical pumping scheme to
suppress leakage errors in atomic hyperfine qubits. The selection rules and
narrow linewidth of a quadrupole transition are used to selectively pump
population out of leakage states and back into the qubit subspace. Each pumping
cycle reduces the leakage population by a factor of $\sim3$, allowing for an
exponential suppression in the number of cycles. We use interleaved randomized
benchmarking on the qubit subspace to show that this pumping procedure has
negligible side-effects on un-leaked qubits, bounding the induced qubit memory
error by $\leq2.0(8)\times10^{-5}$ per cycle, and qubit population decay to
$\leq1.4(3)\times10^{-7}$ per cycle. These results clear a major obstacle for
implementations of quantum error correction and error mitigation protocols.
