In this study, we apply 1D quantum convolution to address the task of time series forecasting. By encoding multiple points into the quantum circuit to predict subsequent data, each point becomes a feature, transforming the problem into a multidimensional one. Building on theoretical foundations from prior research, which demonstrated that Variational Quantum Circuits (VQCs) can be expressed as multidimensional Fourier series, we explore the capabilities of different architectures and ansatz. This analysis considers the concepts of circuit expressibility and the presence of barren plateaus. Analyzing the problem within the framework of the Fourier series enabled the design of an architecture that incorporates data reuploading, resulting in enhanced performance. Rather than a strict requirement for the number of free parameters to exceed the degrees of freedom of the Fourier series, our findings suggest that even a limited number of parameters can produce Fourier functions of higher degrees. This highlights the remarkable expressive power of quantum circuits. This observation is also significant in reducing training times. The ansatz with greater expressibility and number of non-zero Fourier coefficients consistently delivers favorable results across different scenarios, with performance metrics improving as the number of qubits increases.

翻译：本研究将一维量子卷积应用于时间序列预测任务。通过将多个数据点编码至量子电路中以预测后续数据，每个数据点转化为特征，使该问题演变为多维问题。基于先前研究的理论基础（该理论表明变分量子电路可表示为多维傅里叶级数），我们探讨了不同架构与拟设的潜在能力。此分析涉及电路可表达性及贫瘠高原现象的概念。在傅里叶级数框架下分析问题，促使我们设计了一种包含数据重上传的架构，从而提升了性能。我们的发现表明，相较于自由参数数量必须超过傅里叶级数自由度的严格约束，即便参数数量有限，亦可生成高阶傅里叶函数。这凸显了量子电路卓越的表达能力，该观察结果对缩短训练时间同样具有重要意义。具有更高可表达性及更多非零傅里叶系数的拟设，在不同场景下持续提供优异结果，且性能指标随量子比特数增加而提升。