We propose a quantum algorithm for `extremal learning', which is the process of finding the input to a hidden function that extremizes the function output, without having direct access to the hidden function, given only partial input-output (training) data. The algorithm, called quantum extremal learning (QEL), consists of a parametric quantum circuit that is variationally trained to model data input-output relationships and where a trainable quantum feature map, that encodes the input data, is analytically differentiated in order to find the coordinate that extremizes the model. This enables the combination of established quantum machine learning modelling with established quantum optimization, on a single circuit/quantum computer. We have tested our algorithm on a range of classical datasets based on either discrete or continuous input variables, both of which are compatible with the algorithm. In case of discrete variables, we test our algorithm on synthetic problems formulated based on Max-Cut problem generators and also considering higher order correlations in the input-output relationships. In case of the continuous variables, we test our algorithm on synthetic datasets in 1D and simple ordinary differential functions. We find that the algorithm is able to successfully find the extremal value of such problems, even when the training dataset is sparse or a small fraction of the input configuration space. We additionally show how the algorithm can be used for much more general cases of higher dimensionality, complex differential equations, and with full flexibility in the choice of both modeling and optimization ansatz. We envision that due to its general framework and simple construction, the QEL algorithm will be able to solve a wide variety of applications in different fields, opening up areas of further research.
翻译:我们为“极端学习”提出一个量子算法,这是一个为“极端学习”寻找输出功能的隐藏函数的输入过程,该功能将扩展功能输出,而没有直接访问隐藏函数,只提供部分输入输出(培训)数据。算法称为量子极端学习(QEL),它包含一个参数量子电路,该算法经过不同培训,可以模拟数据输入-输出关系,并且可以对输入数据数据数据进行编码的可训练量子特征地图进行分析性差异化,以便找到将模型扩展的坐标。这样,就可以将已经建立的量子机器学习模型与已经建立的量子优化结合起来,同时使用单一电路/量子计算机。我们根据离散输入变量或连续输入变量来测试我们的典型数据集范围。对于离散变量来说,我们测试我们根据 Max-Cut 问题模型生成的合成问题的算法,同时考虑输入-输出关系中的更高顺序关联性关系。对于连续变量来说,我们测试我们在1D的合成数据设置中,甚至在普通的变法函数中测试我们的测算法,对于普通变数的变数是成功的变法。我们发现一个小的变法,对于总的变法的变法的变法的变法的变法的变法,我们是如何的变法的变法。