  Protected qubits such as the 0-$\pi$ qubit, and bosonic qubits including cat
qubits and GKP qubits offer advantages for fault-tolerance. Some of these
protected qubits (e.g., 0-$\pi$ qubit and Kerr cat qubit) are stabilized by
Hamiltonians which have (near-)degenerate ground state manifolds with large
energy-gaps to the excited state manifolds. Without dissipative stabilization
mechanisms the performance of such energy-gap-protected qubits can be limited
by leakage to excited states. Here, we propose a scheme for dissipatively
stabilizing an energy-gap-protected qubit using colored (i.e.,
frequency-selective) dissipation without inducing errors in the ground state
manifold. Concretely we apply our colored dissipation technique to Kerr cat
qubits and propose colored Kerr cat qubits which are protected by an engineered
colored single-photon loss. When applied to the Kerr cat qubits our scheme
significantly suppresses leakage-induced bit-flip errors (which we show are a
limiting error mechanism) while only using linear interactions. Beyond the
benefits to the Kerr cat qubit we also show that our frequency-selective loss
technique can be applied to a broader class of protected qubits.
