Global quantum phase estimation via hybrid quantum--classical learning

Achieving both high precision and large dynamic range remains a central challenge in quantum metrology, as improving local sensitivity typically reduces the unambiguous estimation range. Variational quantum interferometers enhance precision but are generally limited to narrow operating regimes. Here we introduce a hybrid variational quantum--classical neural network interferometer (VQ-CNNI), where a shallow quantum circuit encodes phase-dependent measurement statistics and a neural network performs nonlinear phase reconstruction. Joint optimization enables accurate and unambiguous phase estimation over [-π,π) without loss of precision. We show that this performance requires co-optimization of quantum encoding and classical decoding. Visualization of the learned representation geometry links global estimation to well-conditioned measurement statistics across the full phase range, enabling stable inversion. Odd-symmetric activations further improve robustness by promoting global consistency. These results suggest that global quantum metrology can be understood through the learnability of the quantum--classical representation, providing a practical route to programmable interferometers with both high precision and large dynamic range.