Dimensionality Reduction

Author: Pablo Giuliani

As physicists, we are usually trying to find the important degrees of freedom of the system we want to study. If we consider too few, our theories become too simple and can’t describe what we see. If we include too many, then our calculations become impossible to perform (perhaps even on lightning fast quantum computers!). Finding the right degrees of freedom to consider has usually been more of an art than a science, and it is a feature that many successful theories have shared over the centuries. Dimensionality reduction techniques - a series of novel approaches mostly developed in the last decades thanks to the increase in classical computational power - have transformed many areas of science by allowing us to select the degrees of freedom that are ‘just right’.

In this chapter we will explore one such dimensionality reduction technique called the Reduced Basis Method, which comes under the umbrella of Reduced Order Models. This reduction will enable us to tackle a simple and a not-so-simple problem on a quantum computer using just a small number of qubits. We start first with the simple one: the Quantum Harmonic Oscillator.