Quantum computation as geometry
It’s probably surprising if you don’t already know it, but in the standard “quantum circuit” model quantum computers are actually quite similar to ordinary computers. A quantum computation is built up out of quantum gates that perform quantum logic operations on quantum bits. All of this proceeds pretty much by analogy with classical computers, where gates perform logical operations on bits. There are some technical differences, but the broad picture is pretty similar.
Mark Dowling, Mile Gu, Andrew Doherty and I have recently developed a rather different geometric approach to quantum computation. (Here’s the link to the abstract, and here’s the link to the full text at Science. Jonathan Oppenheim has also written a nice perspective piece (sorry, I don’t have the full text).)
Our result is pretty simple: we show that finding the best (read smallest) quantum circuit to solve a particular problem is equivalent to finding the shortest paths between two points in a particular curved geometry. Intuitively, this problem is like an orienteer or hiker trying to find the shortest path between two points in a hilly landscape, although the space we are working in is harder to visualize. There’s some technical caveats to the result, but that’s the general gist.
What use is this? At the moment it’s difficult to say – unfortunately, we don’t yet have any killer applications of the result. But our result does mean that problems in quantum computation can be viewed in terms of equivalent problems in the field known as Riemannian geometry. This opens up the possibility of using some of the deep ideas of Riemannian geometry to solve problems in quantum computing. And who knows: maybe ideas from quantum computing will have a useful stimulating effect, injecting new ideas into the study of geometry!