Efficient trainability of linear optical modules in quantum optical neural networks

The existence of "barren plateau landscapes" for generic discrete variable quantum neural networks, which obstructs efficient gradient-based optimization of cost functions defined by global measurements, would be surprising in the case of generic linear optical modules in quantum optical neural networks due to the tunability of the intensity of continuous variable states and the relevant unitary group having exponentially smaller dimension. We demonstrate that coherent light in $m$ modes can be generically compiled efficiently if the total intensity scales sublinearly with $m$, and extend this result to cost functions based on homodyne, heterodyne, or photon detection measurement statistics, and to noisy cost functions in the presence of attenuation. We further demonstrate efficient trainability of $m$ mode linear optical quantum circuits for variational mean field energy estimation of positive quadratic Hamiltonians for input states that do not have energy exponentially vanishing with $m$.