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/EMν, 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.

中文翻译：

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玻色子变分量子电路中的能量依赖性贫瘠平台化


玻色子变分量子电路 （VQC） 对于微波腔、囚禁离子和光学系统的信息处理至关重要，广泛应用于量子通信、传感和纠错。这种 VQC 的可训练性不太了解，由于所涉及的连续变量系统的无限维度，缺乏 t-design 等理论工具而受到阻碍。我们克服了这一困难，在此类 VQC 中揭示了能量依赖性的贫瘠平台。梯度的方差衰减为 1/EMν，在模式 M 的数量中呈指数，但在（每个模式的）电路能量 E 中呈多项式。指数 ν = 1 用于浅电路，ν = 2 用于深电路。我们证明了这些结果，用于一般高斯态和数态的状态准备。我们还提供了数字证据，证明结果扩展到一般的状态准备任务。由于电路能量是一个可控参数，我们提供了一种缓解玻色子连续变量 VQC 中贫瘠平台的策略。