This morning I had a call with IQC’s Andrew Childs and we pinned down our summer project. The goal is to investigate the analog of “Levinson’s theorem” (originally formulated for general scattering problems) for the case of scattering on graphs. Levinson’s theorem basically relates the phase shift of a scattered state to the bound states of the potential. Intuitively, you can think of it as though the number of ways for a particle to get “trapped” in the potential affects what happens to the particle when it interacts with that potential. Scattering on graphs differs from scattering off a general potential in several ways, including that finite graphs end abruptly so our “potential” cuts off sharply, whereas the usual garden-variety potential you’d meet bumping into an atom decays gradually at large distances but never quite goes to zero.
Finding a nice result here could be helpful in several ways. One, it could help us better understand the effect of weakly bound states on scattering, which might help us compute with graphs more efficiently. Two, every algorithm I’ve ever seen on graphs relies on scattering states; none exploit the bound states. If we better understood the bound states, maybe we could design some nifty new algorithms using them (for instance, we might start with an initial state bounded on the graph, scatter things off it to “compute”, then measure the resulting bound states). Three, relating bound states to their effect on scattering might also help with the “inverse problem” of finding a graph that yields specific scattering behavior, another trick that could help design algorithms.
Though we want to take a look at the literature on similar results for discrete potentials and such, we also wanted to get our hands dirty with some calculations on graphs. Our first target will be to examine the scattering and bound states for the very simple case of scattering off of a single weighted edge. I’ll take a stab at this sometime over the weekend.
Tags: physics · quantum computing · quantum information · scattering on graphs · science3 Comments
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[...] recent hacks at trying to find the scattering and bound states of semi-infinite line attached to a single weighted edge (the simplest graph imaginable) [...]
How on earth do you map QM over to graphs? This just boggles my mind.
Actually, the simplest models of electrons running around in metals (lattices) involve hopping around on graphs. An impurity in the lattice can be treated like a scattering target and voila! – you’ve got yourself a QM problem mapped on to a graph.
There’s a series of beautiful lectures on this model (the independent electron model) in Volume 3 of the Feynman lectures.