  Long range, multi-qubit parity checks have applications in both quantum error
correction and measurement-based entanglement generation. Such parity checks
could be performed using qubit-state-dependent phase shifts on propagating
pulses of light described by coherent states $\vert\alpha\rangle$ of the
electromagnetic field. We consider "flying-cat" parity checks based on an
entangling operation that is quantum non-demolition (QND) for Schr\"odinger's
cat states $\vert\alpha\rangle\pm \vert-\alpha\rangle$. This operation encodes
parity information in the phase of maximally distinguishable coherent states
$\vert\pm \alpha\rangle$, which can be read out using a phase-sensitive
measurement of the electromagnetic field. In contrast to many implementations,
where single-qubit errors and measurement errors can be treated as independent,
photon loss during flying-cat parity checks introduces errors on physical
qubits at a rate that is anti-correlated with the probability for measurement
errors. We analyze this trade-off for three-qubit parity checks, which are a
requirement for universal fault-tolerant quantum computing with the subsystem
surface code. We further show how a six-qubit entangled "tetrahedron" state can
be prepared using these three-qubit parity checks. The tetrahedron state can be
used as a resource for controlled quantum teleportation of a two-qubit state,
or as a source of shared randomness with potential applications in three-party
quantum key distribution. Finally, we provide conditions for performing
high-quality flying-cat parity checks in a state-of-the-art circuit QED
architecture, accounting for qubit decoherence, internal cavity losses, and
finite-duration pulses, in addition to transmission losses.
