Universal Compiling and (No-)Free-Lunch Theorems for Continuous-Variable Quantum Learning

Recent progress in quantum machine learning has shown that the number of input-output state pairs needed to learn an unknown unitary can be exponentially smaller when entanglement is used as a resource. Here, we study quantum machine learning in the context of continuous variable (CV) systems and demonstrate that similar entanglement-enhanced resource reductions are possible. In particular, we prove a range of No-Free-Lunch theorems, demonstrating a linear resource reduction for learning Gaussian unitaries using entangled coherent-Fock states and an exponential resource reduction for learning arbitrary unitaries using Two-Mode-Squeezed states. We use these results to motivate several, closely related, short depth CV algorithms for the quantum compilation of a target unitary $U$ with a parameterized circuit $V(\theta)$. We analyse the trainability of our proposed cost functions and numerically demonstrate our algorithms for learning arbitrary Gaussian operations and Kerr non-linearities.