Reading group: on the average number of real roots of a random algebraic equation, M. Kac

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.

Reading group: Quantum algorithm for the Laughlin wave function, arXiv:0902.4797

In this post I’d like to experiment by sharing my thoughts on a recent paper as I read through it critically. I’m thinking of trying to emulate something like a reading group presentation.

While this isn’t original research, I’m sure that reading papers certainly does form an integral part of the workflow of any researcher: critically reading papers allows you to learn new ideas and techniques and, crucially, by asking difficult questions while reading a paper you often discover new research directions that, otherwise, would never occur to you. This has often happened to me. Indeed, the main inspiration for several of my papers has come from the critical evaluation of (and, sometimes, the resulting frustrations from reading) a recent paper.

Due mostly to time constraints I don’t really read that many papers these days (I tend to skim quite a few, but actually read only a couple a year). Nevertheless, I hope to do something like this post on a semi-regular/sporadic basis.

I’m going to be rather polite and not actually make any direct criticisms. I don’t see the point of being totally negative anyways: if there is a criticism it probably means there is some aspect of the paper that could be explored further. This, to my mind, equals a new research project. So, let’s be positive and consider every quibble — if there are any — as an opportunity.

The paper I’d like to discuss today is the latest from José-Ignacio Latorre and coworkers. This intriguing paper discusses how quantum computers could help prepare physically interesting quantum states.

As will be evident, I’m not an expert in the area of this paper and it is entirely possible that I’ll make several wrong statements. Any question I raise here is probably due to a misunderstanding on my behalf. Finally, in the true spirit of a reading group discussion, if you think you can answer any of my questions, or clarify the description anywhere then please do not hesitate to comment!