Energy-dependent barren plateau in bosonic variational quantum circuits

Bosonic variational quantum circuits (VQCs) are crucial for information processing in microwave cavities, trapped ions, and optical systems, widely applicable in quantum communication, sensing and error correction. The trainability of such VQCs is less understood, hindered by the lack of theoretical tools such as t-design due to the infinite dimension of the continuous-variable systems involved. We overcome this difficulty to reveal an energy-dependent barren plateau in such VQCs. The variance of the gradient decays as 1/E^Mν, exponential in the number of modes M but polynomial in the (per-mode) circuit energy E. The exponent ν = 1 for shallow circuits and ν = 2 for deep circuits. We prove these results for state preparation of general Gaussian states and number states. We also provide numerical evidence demonstrating that the results extend to general state preparation tasks. As circuit energy is a controllable parameter, we provide a strategy to mitigate the barren plateau in bosonic continuous-variable VQCs.