Variational quantum algorithms (VQAs) prevail to solve practical problems such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. For variational quantum machine learning, a variational algorithm with model interpretability built into the algorithm is yet to be exploited. In this paper, we construct a quantum regression algorithm and identify the direct relation of variational parameters to learned regression coefficients, while employing a circuit that directly encodes the data in quantum amplitudes reflecting the structure of the classical data table. The algorithm is particularly suitable for well-connected qubits. With compressed encoding and digital-analog gate operation, the run time complexity is logarithmically more advantageous than that for digital 2-local gate native hardware with the number of data entries encoded, a decent improvement in noisy intermediate-scale quantum computers and a minor improvement for large-scale quantum computing Our suggested method of compressed binary encoding offers a remarkable reduction in the number of physical qubits needed when compared to the traditional one-hot-encoding technique with the same input data. The algorithm inherently performs linear regression but can also be used easily for nonlinear regression by building nonlinear features into the training data. In terms of measured cost function which distinguishes a good model from a poor one for model training, it will be effective only when the number of features is much less than the number of records for the encoded data structure to be observable. To echo this finding and mitigate hardware noise in practice, the ensemble model training from the quantum regression model learning with important feature selection from regularization is incorporated and illustrated numerically.

翻译：变分量子算法（VQAs）在解决组合优化、量子化学模拟、量子机器学习及含噪量子计算机上的量子纠错等实际问题中占据主导地位。针对变分量子机器学习领域，目前尚缺乏一种将模型可解释性内建于算法中的变分方法。本文构建了一种量子回归算法，在采用直接编码数据振幅（反映经典数据表结构）的电路的同时，确定了变分参数与学习到的回归系数之间的直接关系。该算法特别适用于高连通性量子比特。通过压缩编码与数模门操作，其运行时间复杂度相较于基于数字双局域门原生硬件（需编码数据条目数量）具有对数级优势——在含噪中等规模量子计算机上呈现显著提升，在大规模量子计算中则表现为小幅改进。与采用相同输入数据的传统独热编码技术相比，本文提出的压缩二进制编码方法可大幅减少所需物理量子比特数量。该算法本质实现线性回归，但通过将非线性特征纳入训练数据，亦可轻松扩展用于非线性回归。就衡量模型训练优劣的代价函数而言，仅当特征数量远小于（使编码数据结构可观测的）记录数量时，该函数方能有效区分模型优劣。针对这一发现及实际硬件噪声问题，本文结合正则化重要特征选择，从量子回归模型学习中融入集成模型训练，并通过数值算例予以验证。