  In modelling complex processes, the potential past data that influence future
expectations are immense. Models that track all this data are not only
computationally wasteful but also shed little light on what past data most
influence the future. There is thus enormous interest in dimensional
reduction-finding automated means to reduce the memory dimension of our models
while minimizing its impact on its predictive accuracy. Here we construct
dimensionally reduced quantum models by machine learning methods that can
achieve greater accuracy than provably optimal classical counterparts. We
demonstrate this advantage on present-day quantum computing hardware. Our
algorithm works directly off classical time-series data and can thus be
deployed in real-world settings. These techniques illustrate the immediate
relevance of quantum technologies to time-series analysis and offer a rare
instance where the resulting quantum advantage can be provably established.
