Abstract

The search for an optimal solution is analogous to looking for the lowest point in a mountainous terrain with many valleys, trenches, and drops. Such a search may seem daunting in natural terrain, but imagine its complexity in high-dimensional space! This is exactly the problem to tackle when the objective function to minimise represents a real-life problem with many unknowns, parameters, and constraints. Modern supercomputers can only deal with a small subset of such problems when the dimension of the function to be minimised is small or when the underlying structure of the problem allows it to find the optimal solution quickly even for a function of large dimensionality. Even a hypothetical quantum computer, if realised, offers at best the quadratic speed-up for the “brute-force” search for the global minimum. What if instead of moving along the mountainous terrain in search of the lowest point, one fills the landscape with a magical dust that only shines at the deepest level, becoming an easily detectable marker of the solution? Our “magic dust” is created by shining a laser at stacked layers of selected atoms such as gallium, arsenic, indium, and aluminum. The electrons in these layers absorb and emit light of a specific colour. Polaritons are ten thousand times lighter than electrons and may achieve sufficient densities to form a new state of matter known as a Bose-Einstein condensate, where the quantum phases of polaritons synchronise and create a single macroscopic quantum object that can be detected through photoluminescence measurements. To create a potential landscape that corresponds to the function to be minimised and to force polaritons to condense at its lowest point we focused on a particular type of optimisation problem, but a type that is general enough so that any other hard problem can be related to it, namely minimisation of the XY model which is one of the most fundamental models of statistical mechanics. We have shown that we can create polaritons at vertices of an arbitrary graph: as polaritons condense, the quantum phases of polaritons arrange themselves in a configuration that corresponds to the absolute minimum of the objective function.