  Quantum support vector machines have the potential to achieve a quantum
speedup for solving certain machine learning problems. The key challenge for
doing so is finding good quantum kernels for a given data set -- a task called
kernel alignment. In this paper we study this problem using the Pegasos
algorithm, which is an algorithm that uses stochastic gradient descent to solve
the support vector machine optimization problem. We extend Pegasos to the
quantum case and and demonstrate its effectiveness for kernel alignment. Unlike
previous work which performs kernel alignment by training a QSVM within an
outer optimization loop, we show that using Pegasos it is possible to
simultaneously train the support vector machine and align the kernel. Our
experiments show that this approach is capable of aligning quantum feature maps
with high accuracy, and outperforms existing quantum kernel alignment
techniques. Specifically, we demonstrate that Pegasos is particularly effective
for non-stationary data, which is an important challenge in real-world
applications.

Quantum Physics;Machine Learning