  Non-equilibrium quantum many-body systems, which are difficult to study via
classical computation, have attracted wide interest. Quantum simulation can
provide insights into these problems. Here, using a programmable quantum
simulator with 16 all-to-all connected superconducting qubits, we investigate
the dynamical phase transition in the Lipkin-Meshkov-Glick model with a
quenched transverse field. Clear signatures of the dynamical phase transition,
merging different concepts of dynamical criticality, are observed by measuring
the non-equilibrium order parameter, nonlocal correlations, and the Loschmidt
echo. Moreover, near the dynamical critical point, we obtain the optimal spin
squeezing of $-7.0\pm 0.8$ decibels, showing multipartite entanglement useful
for measurements with precision five-fold beyond the standard quantum limit.
Based on the capability of entangling qubits simultaneously and the accurate
single-shot readout of multi-qubit states, this superconducting quantum
simulator can be used to study other problems in non-equilibrium quantum
many-body systems.
