  Multipartite entangled states are significant resources for both quantum
information processing and quantum metrology. In particular, non-Gaussian
entangled states are predicted to achieve a higher sensitivity of precision
measurements than Gaussian states. On the basis of metrological sensitivity,
the conventional linear Ramsey squeezing parameter (RSP) efficiently
characterises the Gaussian entangled atomic states but fails for much wider
classes of highly sensitive non-Gaussian states. These complex non-Gaussian
entangled states can be classified by the nonlinear squeezing parameter (NLSP),
as a generalisation of the RSP with respect to nonlinear observables, and
identified via the Fisher information. However, the NLSP has never been
measured experimentally. Using a 19-qubit programmable superconducting
processor, here we report the characterisation of multiparticle entangled
states generated during its nonlinear dynamics. First, selecting 10 qubits, we
measure the RSP and the NLSP by single-shot readouts of collective spin
operators in several different directions. Then, by extracting the Fisher
information of the time-evolved state of all 19 qubits, we observe a large
metrological gain of 9.89$^{+0.28}_{-0.29}$ dB over the standard quantum limit,
indicating a high level of multiparticle entanglement for quantum-enhanced
phase sensitivity. Benefiting from high-fidelity full controls and addressable
single-shot readouts, the superconducting processor with interconnected qubits
provides an ideal platform for engineering and benchmarking non-Gaussian
entangled states that are useful for quantum-enhanced metrology.
