Flying-cat parity checks for quantum error correction

Long range, multiqubit 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mo>|</a:mo><a:mi>α</a:mi><a:mo>〉</a:mo></a:mrow></a:math> of the electromagnetic field. We consider “flying-cat” parity checks based on an entangling operation that is quantum nondemolition for Schrödinger's cat states <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mrow><b:mo>|</b:mo><b:mi>α</b:mi><b:mo>〉</b:mo></b:mrow><b:mo>±</b:mo><b:mrow><b:mo>|</b:mo><b:mrow><b:mo>−</b:mo><b:mi>α</b:mi></b:mrow><b:mo>〉</b:mo></b:mrow></b:mrow></b:math>. This operation encodes parity information in the phase of maximally distinguishable coherent states <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mo>|</c:mo><c:mrow><c:mo>±</c:mo><c:mi>α</c:mi></c:mrow><c:mo>〉</c:mo></c:mrow></c:math>, 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 anticorrelated 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. Published by the American Physical Society 2024