At the moment I am in between research projects: I am “working” on a bunch of old projects, some of which are many years old, but I haven’t had any new ideas for any of them in a long time and, hence, I haven’t made any progress whatsoever. At the same time I am thinking about beginning work on some new projects. Most notably, I want to spend some time understanding quantum systems with static and annealed disorder, and the connections between these systems and computational complexity. Unfortunately the literature on disordered quantum systems is vast, to say the least. Hence, I am putting off learning it. So now I am procrastinating by learning about a whole bunch of new ideas in the hope of learning something that will make the entry into the disordered systems literature a little smoother.

Basically I am now going to play out my learning curve through this blog.

The type of problems I eventually hope to study will be of the following form. Take some family of computational problems , and consider a random instance from this family. What is the running time, on average, of some quantum algorithm to solve the problem? At the same time I’d also like to consider families of quantum problems (eg. a family of quantum systems) and understand the running time, on average, of *either* classical or quantum algorithms to calculate approximations to, eg., expectation values of local observables, of a random instance. In both cases there is obviously some quantum system (i.e., the quantum computer in the first case, and the quantum system studied in the second case), and there is obviously some disorder present. The hope, broadly speaking, is to exploit the ubiquitous phenomenon of Anderson localisation to understand what happens in each case.

However, except in some very special cases, the problems I want to study don’t reduce in any obvious way to systems studied in the disordered quantum systems literature (at least, not so far as I can see). So I’m now casting around looking for interesting stuff which might have some relevance…

At the most abstract and high level all of the problems I want to eventually consider require that one understands the critical points of a random function (which is usually related to the running time). With a bit of luck one will be able write this expression as a polynomial. Hence it’d be nice to understand, say, the roots of random polynomials. Read the rest of this entry »