Quantum circuit transformation (QCT) is required when executing a quantum program in a real quantum processing unit (QPU). In this case an additional exponential enhancement of the quantum Fisher information can in practice be observed with the number of atoms $N$ in the cavity, despite existing works suggesting a requirement of $N$-body coupling terms. We explicitly show its advantage in the archetypal setting of the Dicke model and explore a quantum gas coupled to a single-mode cavity field as a potential platform. We demonstrate that the Cramer-Rao bound can be saturated in our protocol through the standard homodyne detection scheme. This provides an exponential speed-up in the growth of the quantum Fisher information over existing critical quantum metrology protocols observing power law behaviour. In the thermodynamic limit these systems can exhibit an exponential divergence of the quantum Fisher information in time, whose origin is the exponential growth of the number of correlated photons on an arbitrarily fast time scale determined by the coupling strength. We present a quantum metrology protocol which relies on quenching a light-matter system exhibiting a superradiant quantum phase transition beyond its critical point.
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