Statistical mechanics of classical and quantum computational complexity

The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (``NP-complete'') and those which are likely hard for quantum computers (``QMA-complete'') by indirect methods. That is, they identify problems of comparable worst-case difficulty without directly determining the individual hardness of any given instance. Statistical mechanical methods can be used to complement this classification by directly extracting information about particular families of instances---typically those that involve optimization---by studying random ensembles of them. These pose unusual and interesting (quantum) statistical mechanical questions and the results shed light on the difficulty of problems for large classes of algorithms as well as providing a window on the contrast between typical and worst case complexity. In these lecture notes we present an introduction to this set of ideas with older work on classical satisfiability and recent work on quantum satisfiability as primary examples. We also touch on the connection of computational hardness with the physical notion of glassiness.