Flying-cat parity checks for quantum error correction

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.