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.
a very challenging/advanced/provocative question; following, a brief/sketchy/tentative answer [maybe/hopefully better than none] considering geometry in QM computing in general &amp; a few refs/leads. geometry is used in a variety of ways in QM in general, and it appears to be somewhat of an open question and challenging work-in-progress how to determine a coherent/natural "geometric picture" for QM, and there apparently are multiple ways to do it, and currently no generally-agreed, unified or standard approach. also, some directions can be highly abstract reflecting the direction of mathematical research developed largely independently of physics.

the 2-qubit state has been more extensively studied and there is more chance of creating a picture there 1st and maybe using it as a somewhat "toy" area that can be expanded later. (note that adiabatic QM computing is still based on qubits.) also there is a relatively new study of "quantum dischord" which is seen as promising by some (but also controversial) &amp; might be part of the answer as in the following ref.


the Bloch sphere is a clear geometric picture for a single qubit and has some generalization for pure states.
Geometric picture of quantum discord for two-qubit quantum states Shi, Jiang, Sun, Du
Geometry of Discrete Quantum Computing Hanson, Ortiz, Sabry, Tai
Quantum computing: The power of discord Merali/Nature