  We investigate optimized quantum state preparation for quantum metrology
applications in noisy environments. Using the QFI-Opt package, we simulate a
low-depth variational quantum circuit (VQC) composed of a sequence of global
rotations and entangling operations applied to a chain of qubits that are
subject to dephasing noise. The parameters controlling the VQC are numerically
optimized to maximize the quantum Fisher information, which characterizes the
ultimate metrological sensitivity of a quantum state with respect to a global
rotation. We find that regardless of the details of the entangling operation
implemented in the VQC, the optimal quantum states can be broadly classified
into a trio of qualitative regimes--cat-like, squeezed-like, and product
states--associated with different dephasing rates. Our findings are relevant
for designing optimal state-preparation strategies for next-generation quantum
sensors exploiting entanglement, such as time and frequency standards and
magnetometers, aimed at achieving state-of-the-art performance in the presence
of noise and decoherence.
