  Quantum computing (QC) seems to show potential for application in machine
learning (ML). In particular quantum kernel methods (QKM) exhibit promising
properties for use in supervised ML tasks. However, a major disadvantage of
kernel methods is their unfavorable quadratic scaling with the number of
training samples. Together with the limits imposed by currently available
quantum hardware (NISQ devices) with their low qubit coherence times, small
number of qubits, and high error rates, the use of QC in ML at an industrially
relevant scale is currently impossible. As a small step in improving the
potential applications of QKMs, we introduce QUACK, a quantum kernel algorithm
whose time complexity scales linear with the number of samples during training,
and independent of the number of training samples in the inference stage. In
the training process, only the kernel entries for the samples and the centers
of the classes are calculated, i.e. the maximum shape of the kernel for n
samples and c classes is (n, c). During training, the parameters of the quantum
kernel and the positions of the centroids are optimized iteratively. In the
inference stage, for every new sample the circuit is only evaluated for every
centroid, i.e. c times. We show that the QUACK algorithm nevertheless provides
satisfactory results and can perform at a similar level as classical kernel
methods with quadratic scaling during training. In addition, our (simulated)
algorithm is able to handle high-dimensional datasets such as MNIST with 784
features without any dimensionality reduction.
