  Quantum metrology offers superior measurement precision compared to classical
counterparts. Traditionally, achieving this enhancement involves employing
multi-particle entangled quantum states, necessitating state preparation and
detection schemes that are practically challenging. In this article, we present
a versatile and on-demand protocol for deterministic parameter estimation that
leverages two state-transfer operations on a single bosonic mode. Specifically,
we demonstrate this protocol in the context of phase and amplitude estimation
using the superposition of coherent states in the bosonic circuit quantum
electrodynamics (cQED) platform. With low photon numbers of up to 1.76, we
achieve quantum-enhanced precision approaching the Heisenberg scaling, reaching
a metrological gain of 7.5(6) dB and 9.3(5) dB respectively for phase and
amplitude estimation. We show that the gain or sensitivity range can be further
enhanced on the fly by tailoring the input states based on specific system
constraints. Our protocol can be adapted to further optimize the desired
figures of merit using tailored quantum states or operations, not only for
bosonic cQED hardware but also readily extensible to other continuous-variable
platforms. The realisation of this versatile and effective scheme affords a
promising path towards practical quantum-enhanced sensing.
