As luck would have it, I submitted two papers to the quant-ph preprint archive today. They should both show up in a day or two. Both contain what I unabashedly think is some pretty neat stuff, so I’ll describe them here.
The first (joint work with Chris Dawson, Henry Haselgrove, Andrew Hines, Duncan Mortimer, and Tobias Osborne) paper relates quantum computing to something that, on the surface, you’d think had absolutely nothing to do with quantum computing: counting solutions to sets of polynomial equations.
In particular, we show that the problem of determining the output of a quantum computation is actually equivalent to the problem of being able to estimate the number of solutions to certain sets of polynomial equations, which we explicitly construct.
This is really quite odd. After all, it is believed pretty likely that quantum computers can be used to simulate any physical system. If you believe this, then what this result tells you is that simulating nature can be boiled down to counting solutions to polynomial equations.
What the heck does the simulation of physical systems have to do with counting solutions to polynomial equations? A priori, you wouldn’t think they’d be all that related. Yet the two problems turn out to be the same, in some sense. I find this really quite remarkable.
I should say, by the way, that other people have shown some related results in the past. I won’t give an account of that history here, though, as we have a detailed discussion in the paper, and I won’t do it justice in this small space.
The second paper (joint with Denes Petz, who has been visiting us at UQ), is a short expository note. It’s a simple elementary account of what may be just about the most important result in quantum information theory – the strong subadditivity inequality for entropy, first proved by Lieb and Ruskai in 1973. Almost everything we know about entropy comes from this inequality.
In the 1980s Denes found a simple (and, I think, extremely beautiful) proof of this result. Our note gives an account of this proof that is (a) short; (b) easy-to-understand (we assume only basic linear algebra and quantum mechanics); and (c) where you can learn something interesting at pretty much every step, if you pay attention. That’s the kind of proof I love to read, myself, which is why I’m so excited about this note!