  Fault-tolerant quantum computation with depolarization error often requires
demanding error threshold and resource overhead. If the operations can maintain
high noise bias -- dominated by dephasing error with small bit-flip error -- we
can achieve hardware-efficient fault-tolerant quantum computation with a more
favorable error threshold. Distinct from two-level physical systems,
multi-level systems (such as harmonic oscillators) can achieve a desirable set
of bias-preserving quantum operations while using continuous engineered
dissipation or Hamiltonian protection to stabilize to the encoding subspace.
For example, cat codes stabilized with driven-dissipation or Kerr nonlinearity
can possess a set of biased-preserving gates while continuously correcting
bosonic dephasing error. However, cat codes are not compatible with continuous
quantum error correction against excitation loss error, because it is
challenging to continuously monitor the parity to correct photon loss errors.
In this work, we generalize the bias-preserving operations to pair-cat codes,
which can be regarded as a multimode generalization of cat codes, to be
compatible with continuous quantum error correction against both bosonic loss
and dephasing errors. Our results open the door towards hardware-efficient
robust quantum information processing with both bias-preserving operations and
continuous quantum error correction simultaneously correcting bosonic loss and
dephasing errors.
