  Copulas are mathematical tools for modeling joint probability distributions.
Since copulas enable one to conveniently treat the marginal distribution of
each variable and the interdependencies among variables separately, in the past
60 years they have become an essential analysis tool on classical computers in
various fields ranging from quantitative finance and civil engineering to
signal processing and medicine. The recent finding that copulas can be
expressed as maximally entangled quantum states has revealed a promising
approach to practical quantum advantages: performing tasks faster, requiring
less memory, or, as we show, yielding better predictions. Studying the
scalability of this quantum approach as both the precision and the number of
modeled variables increase is crucial for its adoption in real-world
applications. In this paper, we successfully apply a Quantum Circuit Born
Machine (QCBM) based approach to modeling 3- and 4-variable copulas on trapped
ion quantum computers. We study the training of QCBMs with different levels of
precision and circuit design on a simulator and a state-of-the-art trapped ion
quantum computer. We observe decreased training efficacy due to the increased
complexity in parameter optimization as the models scale up. To address this
challenge, we introduce an annealing-inspired strategy that dramatically
improves the training results. In our end-to-end tests, various configurations
of the quantum models make a comparable or better prediction in risk
aggregation tasks than the standard classical models. Our detailed study of the
copula paradigm using quantum computing opens opportunities for its deployment
in various industries.
