  The speed of quantum gates and measurements is a decisive factor for the
overall fidelity of quantum protocols when performed on physical qubits with
finite coherence time. Reducing the time required to distinguish qubit states
with high fidelity is therefore a critical goal in quantum information science.
The state-of-the-art readout of superconducting qubits is based on the
dispersive interaction with a readout resonator. Here, we bring this technique
to its current limit and demonstrate how the careful design of system
parameters leads to fast and high-fidelity measurements without affecting qubit
coherence. We achieve this result by increasing the dispersive interaction
strength, by choosing an optimal linewidth of the readout resonator, by
employing a Purcell filter, and by utilizing phase-sensitive parametric
amplification. In our experiment, we measure 98.25% readout fidelity in only 48
ns, when minimizing read-out time, and 99.2% in 88 ns, when maximizing the
fidelity, limited predominantly by the qubit lifetime of 7.6 us. The presented
scheme is also expected to be suitable for integration into a multiplexed
readout architecture.
