  We introduce a genetic algorithm that designs quantum optics experiments for
engineering quantum states with specific properties. Our algorithm is powerful
and flexible, and can easily be modified to find methods of engineering states
for a range of applications. Here we focus on quantum metrology. First, we
consider the noise-free case, and use the algorithm to find quantum states with
a large quantum Fisher information (QFI). We find methods, which only involve
experimental elements that are available with current or near-future
technology, for engineering quantum states with up to a 100-fold improvement
over the best classical state, and a 20-fold improvement over the optimal
Gaussian state. Such states are a superposition of the vacuum with a large
number of photons (around $80$), and can hence be seen as
Schr\"odinger-cat-like states. We then apply the two most dominant noise
sources in our setting -- photon loss and imperfect heralding -- and use the
algorithm to find quantum states that still improve over the optimal Gaussian
state with realistic levels of noise. This will open up experimental and
technological work in using exotic non-Gaussian states for quantum-enhanced
phase measurements. Finally, we use the Bayesian mean square error to look
beyond the regime of validity of the QFI, finding quantum states with precision
enhancements over the alternatives even when the experiment operates in the
regime of limited data.
