Generalized Parity Measurements and Efficient Large Multi-component Cat State Preparation with Quantum Signal Processing

Generalized parity measurements are instrumental for the preparation of non-trivial quantum states and the detection of errors in error correction codes. Here, we detail a proposal for efficient and robust generalized parity measurements based on Quantum Signal Processing. Most strikingly, given access to an evolution generated by a one-to-all coupling interaction Hamiltonian between a measurement qubit and the measured system, the desired measurement can be implemented in constant time determined only by the interaction rate. The proposed generalized parity measurement can be used to efficiently prepare high-fidelity multi-component cat states in the setting of superconducting cavity quantum electrodynamics. We benchmark the state-preparation protocol through numerical simulations with realistic system parameters. We show that a 20-component cat state with $400$ photons can be prepared with success probability $>2\mathrm{\%}$ and a fidelity $\approx 90\mathrm{\%}$ limited by the cavity decay and nonlinear qubit-cavity coupling rates. Our results pave the way for the realization of a wide range of useful non-classical states consisting of a large number of excitations.