  Coherent storage of quantum information is crucial to many quantum
technologies. Long coherence times have been demonstrated in trapped-ion
qubits, typically using the hyperfine levels within the ground state of a
single ion. However, recent research suggests qubits encoded in metastable
states could provide architectural benefits for quantum information processing,
such as the possibility of effective dual-species operation in a single-species
system and erasure-error conversion for fault-tolerant quantum computing. Here
we demonstrate long-lived encoding of a quantum state in the metastable states
of a trapped ion. By sympathetically cooling with another ion of the same
species and constantly monitoring for erasure errors, we demonstrate a
coherence time of 136(42) seconds with a qubit encoded in the metastable
$5D_{5/2}$ state of a single $^{137}$Ba$^+$ ion. In agreement with a model
based on empirical results from dynamical-decoupling-based noise spectroscopy,
we find that dephasing of the metastable levels is the dominant source of error
once erasure errors are removed.
