Data re-uploading with a single qudit
Quantum two-level systems, i.e. qubits, form the basis for most quantum
machine learning approaches that have been proposed throughout the years.
However, higher dimensional quantum systems constitute a promising alternative
and are increasingly explored in theory and practice. Here, we explore the
capabilities of multi-level quantum systems, so-called qudits, for their use in
a quantum machine learning context. We formulate classification and regression
problems with the data re-uploading approach and demonstrate that a quantum
circuit operating on a single qudit is able to successfully learn highly
non-linear decision boundaries of classification problems such as the MNIST
digit recognition problem. We demonstrate that the performance strongly depends
on the relation between the qudit states representing the labels and the
structure of labels in the training data set. Such a bias can lead to
substantial performance improvement over qubit-based circuits in cases where
the labels, the qudit states and the operators employed to encode the data are
well-aligned. Furthermore, we elucidate the influence of the choice of the
elementary operators and show that a squeezing operator is necessary to achieve
good performances. We also show that there exists a trade-off for qudit systems
between the number of circuit-generating operators in each processing layer and
the total number of layers needed to achieve a given accuracy. Finally, we
compare classification results from numerically exact simulations and their
equivalent implementation on actual IBM quantum hardware. The findings of our
work support the notion that qudit-based algorithms exhibit attractive traits
and constitute a promising route to increasing the computational capabilities
of quantum machine learning approaches.
Noah L. Wach Manuel S. Rudolph Fred Jendrzejewski Sebastian Schmitt