  In quantum error correction, information is encoded in a high-dimensional
system to protect it from the environment. A crucial step is to use natural,
low-weight operations with an ancilla to extract information about errors
without causing backaction on the encoded system. Essentially, ancilla errors
must not propagate to the encoded system and induce errors beyond those which
can be corrected. The current schemes for achieving this fault-tolerance to
ancilla errors come at the cost of increased overhead requirements. An
efficient way to extract error syndromes in a fault-tolerant manner is by using
a single ancilla with strongly biased noise channel. Typically, however,
required elementary operations can become challenging when the noise is
extremely biased. We propose to overcome this shortcoming by using a
bosonic-cat ancilla in a parametrically driven nonlinear cavity. Such a
cat-qubit experiences only bit-flip noise and is stabilized against
phase-flips. To highlight the flexibility of this approach, we illustrate the
syndrome extraction process in a variety of codes such as qubit-based toric
codes, bosonic cat- and Gottesman-Kitaev-Preskill (GKP) codes. Our results open
a path for realizing hardware-efficient, fault-tolerant error syndrome
extraction.
