  Quantum error correction with biased-noise qubits can drastically reduce the
hardware overhead for universal and fault-tolerant quantum computation. Cat
qubits are a promising realization of biased-noise qubits as they feature an
exponential error bias inherited from their non-local encoding in the phase
space of a quantum harmonic oscillator. To confine the state of an oscillator
to the cat qubit manifold, two main approaches have been considered so far: a
Kerr-based Hamiltonian confinement with high gate performances, and a
dissipative confinement with robust protection against a broad range of noise
mechanisms. We introduce a new combined dissipative and Hamiltonian confinement
scheme based on two-photon dissipation together with a Two-Photon Exchange
(TPE) Hamiltonian. The TPE Hamiltonian is similar to Kerr nonlinearity, but
unlike the Kerr it only induces a bounded distinction between even- and
odd-photon eigenstates, a highly beneficial feature for protecting the cat
qubits with dissipative mechanisms. Using this combined confinement scheme, we
demonstrate fast and bias-preserving gates with drastically improved
performance compared to dissipative or Hamiltonian schemes. In addition, this
combined scheme can be implemented experimentally with only minor modifications
of existing dissipative cat qubit experiments.
