Variational Probe and Measurement Optimization for Structured Phase Estimation

We present a proof-of-principle study of variational quantum sensing for estimating a structured linear function of local phase parameters, in which each qubit in a spin-1/2 array accumulates a phase phi_i = alpha_i theta with known weights alpha and a global parameter theta. In a hardware-motivated regime of shallow circuits and shallow decoding measurements, we optimize the probe state with respect to the classical Fisher information (CFI) using the Covariance Matrix Adaptation Evolution Strategy. The variational ansatz is built from dipolar-interacting gates and global rotations on a polygon-centered qubit layout. To assess whether the standard Ramsey readout extracts all available information, we introduce a shallow global decoder and optimize it independently with the encoder frozen. For uniform (alpha_i = 1/N) and weighted-central (alpha_c = 1, alpha_i = 0.5) encodings with N = 2-8 qubits and depths L = 1-3, the optimized probes approach the respective entanglement-enhanced precision bounds, which reduce to the Heisenberg limit only for uniform encoding. The decoder provides systematic but modest CFI gains. For uniform encoding, these gains are smallest at the deepest circuits, confirming that fixed Ramsey readout is near-optimal for well-converged probes. For weighted encoding, a persistent component remains, reflecting the broken permutation symmetry of the generator under unequal weights. At large N, the weighted-encoding CFI also exhibits non-monotonic growth with system size, revealing an expressivity limit of the polygon-symmetric ansatz under asymmetric encoding.