What can the D-Wave quantum computer do? The media are reporting the commercially sold 128-bit quantum computer from D-Wave

http://news.google.com/news?ned=us&hl=us&q=d-wave+quantum&cf=all&scoring=n

which of course sounds amazing. The gadget is described as something capable of doing quantum annealing

http://en.wikipedia.org/wiki/Quantum_annealing

which looks less convincing. I want to ask you what classes of problems the D-Wave computer can actually solve or perform. It can't run Shor's algorithm on 128 qubits, can it?
 A: It mentions: Finding a global minimum where the function of jumping from one minimum to another is handled by quantum tunneling.
I have the feeling that in order to just get an idea of what it can practically do one could look at the mentioned example of spin glasses. In other words, the spin coupling physics close to the actual hardware implementation itself 
http://en.wikipedia.org/wiki/Spin_glass.
Relevant may be the work of Giorgio Parisi (Yes, the one of the Altarelli–Parisi parton evolution equations) and his co-workers Mezard and Virasoro.
See the text of the Boltzmann 1992 Medal:

Parisi's deepest contribution concerns the solution of the
  Sherrington-Kirkpatrick mean field
  model for spin glasses. After the
  crisis caused by the unacceptable
  properties of the simple solutions,
  which used the "replication trick",
  Parisi proposed his replica symmetry
  breaking solution, which seems to be
  exact, although much more complex than
  anticipated. Later, Parisi and
  co-workers Mezard and Virasoro
  clarified greatly the physical meaning
  of the mysterious mathematics involved
  in this scheme, in terms of the
  probability distribution of overlaps
  and the ultrametric structure of the
  configuration space. This achievement
  forms one of the most important
  breakthroughs in the history of
  disordered systems. This discovery
  opened the doors to vast areas of
  application. e.g., in optimization
  problems and in neural network
  theories.

http://en.wikipedia.org/wiki/Giorgio_Parisi#Awards
A: The DWave machine stirred up quite an amount on controversy in the community when it was first announces. The machine basically attempts to solve an NP-complete optimization problem (MAX-2SAT) by encoding it as a ground state of a Hamiltonian, and tries to reach this ground state by moving adiabatically to it from the ground state of an efficiently coolable Hamiltonian.
In general, the adiabatic algorithm is not known to be able to find ground states efficiently as the proximity of low lying levels to the ground state means that the transition between Hamiltonians has to be performed slowly, and the speed at which this can occur is governed by the gap between the ground state and the lowest excited levels. It is commonly believed within the community, but not proved, that no quantum algorithm can efficiently solve NP-complete problems.
In general the ground state of a Hamiltonian can be used to encode a wider variety of problems than NP (know QMA-complete problems), and so decision to focus on NP optimization problems has led to restrictions which prevent the device from being used for general purpose quantum computing (even if noise was not an issue). Thus you can't run Shor's algorithm on the device. Further, you -can- factor any number that you could fit on a 128 qubit device by classical means. The general number field sieve puts 128 bits within reach of modern personal computers.
Noise is a real issue with DWave's device, and although there have been a number of technical papers from them playing down the issue and trying to demonstrate quantum effects, the coherence times for the individual qubits are much shorter than the time scale for the algorithm. Thus the common view within the community seems to be that this is basically an expensive classical special purpose computer.
There is an interesting subtlety as regards noise: If you add noise to the adiabatic algorithm, it degrades gracefully into one of the best classical algorithms for the same problem. Thus you can obtain the same result either way, and the only difference is in the assymptotics for large systems (which are obviously not observables). Thus even if they produce a valid answer for every problem you throw at such a device, this is not enough information to determine if you are actually performing quantum computation.
Let me add that the adiabatic model can encode universal quantum computation, however the limitations of DWave's implementation means that specific machine cannot.
