Parameterized quantum circuits as machine learning models are typically well described by their representation as a partial Fourier series of the input features, with frequencies uniquely determined by the feature map's generator Hamiltonians. Ordinarily, these data-encoding generators are chosen in advance, fixing the space of functions that can be represented. In this work we consider a generalization of quantum models to include a set of trainable parameters in the generator, leading to a trainable frequency (TF) quantum model. We numerically demonstrate how TF models can learn generators with desirable properties for solving the task at hand, including non-regularly spaced frequencies in their spectra and flexible spectral richness. Finally, we showcase the real-world effectiveness of our approach, demonstrating an improved accuracy in solving the Navier-Stokes equations using a TF model with only a single parameter added to each encoding operation. Since TF models encompass conventional fixed frequency models, they may offer a sensible default choice for variational quantum machine learning.

翻译：参数化量子电路作为机器学习模型，通常可通过其对输入特征的偏傅里叶级数表示来很好地描述，其中频率由特征映射的生成哈密顿量唯一确定。通常，这些数据编码生成器是预先选定的，从而固定了可表示的函数空间。本研究提出了一种量子模型的泛化形式，在生成器中引入一组可训练参数，由此构建了可训练频率（TF）量子模型。我们通过数值实验展示了TF模型如何学习具有理想特性的生成器，以解决特定任务，包括频谱中不规则间隔的频率以及灵活的频谱丰富性。最后，我们展示了该方法在实际应用中的有效性：在仅向每个编码操作添加单个参数的情况下，使用TF模型求解纳维-斯托克斯方程时，准确性得到了提升。由于TF模型涵盖了传统的固定频率模型，它可能成为变分量子机器学习中合理且通用的默认选择。