  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.
