  Photonic quantum metrology harnesses quantum states of light, such as NOON or
Twin-Fock states, to measure unknown parameters beyond classical precision
limits. Current protocols suffer from two severe limitations that preclude
their scalability: the exponential decrease in fidelities (or probabilities)
when generating states with large photon numbers due to gate errors, and the
increased sensitivity of such states to noise. Here, we develop a deterministic
protocol combining quantum optical non-linearities and variational quantum
algorithms that provides a substantial improvement on both fronts. First, we
show how the variational protocol can generate metrologically-relevant states
with a small number of operations which does not significantly depend on
photon-number, resulting in exponential improvements in fidelities when gate
errors are considered. Second, we show that such states offer a better
robustness to noise compared to other states in the literature. Since our
protocol harnesses interactions already appearing in state-of-the-art setups,
such as cavity QED, we expect that it will lead to more scalable photonic
quantum metrology in the near future.
