# Is there a geometrical picture for adiabatic quantum computation?

In adiabatic quantum computation (AQC), one encodes the solution to an optimization problem in the ground state of a [problem] Hamiltonian $H_p$. To get to this ground state, you start in an easily coolable initial (ground) state with Hamiltonian $H_i$ and "anneal" (adiabatically perturb) towards $H_p$, i.e.

$$H(s) = s H_i + (1-s) H_p$$

where $s \in [0,1]$. Details about AQC: http://arxiv.org/abs/quant-ph/0001106v1

The interesting thing about this problem is to try to understand the gap between the ground state eigenvalue and the first excited state, as this determines the problem complexity. One interesting thing to do would be to try and say something about the behavior of certain types of Hamiltonians. One can analyze the energy spectrum of small qubit cases by simulation to understand the problem complexity, but this becomes infeasible very quickly.

What I'd like to know is if there is a geometrical or topological way of looking at how certain Hamiltonians behave. Someone mentioned that the above form could be looked at as a homotopy (if the scalar functions were generalized to operators), but I'm not well-versed in higher level mathematics so I'm not sure what this implies or what I could do with it.

It might help to mention that the Hamiltonians are usually Ising spin-glass Hamiltonians (at least, that is what $H_p$ is). I'm not well-read on advanced statistical mechanics literature either, so this may be another avenue.

I wondered if anyone could provide some explanation on this, or at least provide some interesting references, keywords, etc.

• Two relevant references (that are, admittedly, still heavy on the math): arxiv.org/abs/0905.2376 and isi.edu/sites/default/files/users/jns/… – hadsed Oct 28 '13 at 3:41
• the hamiltonian is not specific to adiabatic computing of course, its a general qm/computing concept. so are you ok with more general refs on geometry in qm computing in general (which seems to be a subarea)? found two refs that seem close... it might be helpful to discriminate this more carefully from quantum geometry... – vzn Nov 11 '13 at 22:09
• Any explanation that will give some more intuition on thinking about (time-dependent) Hamiltonians geometrically is welcomed. – hadsed Nov 12 '13 at 0:03
• Another paper inspired by differential geometric control theory: arxiv.org/abs/0905.2376 – hadsed Nov 15 '13 at 0:02