  Entangling gates between qubits are a crucial component for performing
algorithms in quantum computers. However, any quantum algorithm must ultimately
operate on error-protected logical qubits encoded in high-dimensional systems.
Typically, logical qubits are encoded in multiple two-level systems, but
entangling gates operating on such qubits are highly complex and have not yet
been demonstrated. Here, we realize a controlled NOT (CNOT) gate between two
multiphoton qubits in two microwave cavities. In this approach, we encode a
qubit in the high-dimensional space of a single cavity mode, rather than in
multiple two-level systems. We couple two such encoded qubits together through
a transmon, which is driven by an RF pump to apply the gate within 190 ns. This
is two orders of magnitude shorter than the decoherence time of the transmon,
enabling a high-fidelity gate operation. These results are an important step
towards universal algorithms on error-corrected logical qubits.
