Toward a topological CNOT between two Kerr-cat qubits: part 1/2

Schrödinger cat states, superpositions of coherent states in an oscillator, can encode a noise-biased qubit that is naturally protected against one Pauli error channel. Such a protected "cat qubit" has the ability to significantly reduce the overhead associated with quantum error correction in, for instance, a surface-code-style architecture. This overhead reduction relies on the ability to perform any gate in a manner that preserves the noise bias. Unlike pure two-level systems, exchanging coherent states in one oscillator conditioned on the second oscillator's state generates a noise-biased CNOT. Such an exchange-based topological gate does not depend on the path or the speed, but only presence or absence of exchange. This exchange can also be understood as correlated motion of 4 coherent states in a 4D phase space.
In the first part of this talk, we review the Kerr-cat qubit and introduce the Hamiltonian of the gate and its construction