Analogue simulation provides a way to solve hard computational problems by building physical devices that mimic those problems. The history of such devices goes back at least 2000 years to the intricate clockwork mechanisms used to make complex astronomical predictions, before the advent of all-purpose digital computers. But today there remain many important problems that are intractable, even for the fastest supercomputers. An important class of such problems relates to simulating fundamental models of quantum matter, which underpin our understanding of nanoscale processes and bulk materials. Since universal quantum computers capable of tackling such problems are still far off, an emerging paradigm is to sacrifice generality for power by constructing devices with quantum components to perform analogue quantum simulation. In this Colloquium I will give an introduction to the field of analogue simulation, and present results from a recent experiment-theory collaboration in which a bespoke nano-electronics circuit was used to solve a complex quantum many-body model. The quantum device realizes a new quantum impurity model that supports a novel "non-Fermi liquid" quantum critical point and hosts so-called 'anyons' -- exotic emergent quasiparticles that are neither fermionic nor bosonic in nature. Finally, I discuss recent progress in developing methods to measure the fractional entropy associated with such anyons.