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.

我们构建了一个量子回归算法并确定了变分参数与学习回归系数之间的直接关系，利用直接将数据编码到反映经典数据表结构的量子振幅的电路。该算法适用于连接良好的量子比特，具有对数级时间复杂度优势，并提供了与传统一热编码技术相比所需的物理比特数量显著减少的压缩二进制编码方法，同时还能进行非线性回归。通过在实践中从量子回归模型学习中进行重要特征选择的集成模型训练，我们能够减小硬件噪音的影响。