A very beautiful recent paper,
Scalable Implementation of Boson Sampling with Trapped Ions. C. Shen, Z. Zhang, and L.-M. Duan. Phys. Rev. Lett. 112 no. 5, 050504 (2014); arXiv:1310.4860
describes a proposal of an experimental realization of Boson Sampling using phonon modes in an ion trap instead of photons in a linear optical setup.
Boson Sampling, for those unfamiliar with the term, is essentially the problem of predicting the output probabilities of a large optical interferometer when some number of single photons is given to its input ports. To simulate this classically one needs to calculate a large number of permanents (determinants minus the minus signs) and this turns out to be exceptionally hard, and Scott Aaronson and Alex Arkhipov essentially proved that it is indeed a computationally 'hard' problem for a classical computer, whereas a quantum computer could do it much faster. An experimental Boson Sampling realization would show that not all physical processes can be efficiently simulated by classical computers, which is yet to be demonstrated.
Most envisioned BS implementations so far have been on linear optics and using single photon sources, with the huge disadvantage that the SPDC photon sources are non-deterministic and therefore it is hard to get more than three or four photons to coincide.
The recent paper proposes, instead, to do this on a sparse string of ions in a Paul trap. The different boson modes are the transverse phonon modes, which are localized at each ion and can therefore be initialized easily by shining a laser from the side. The ions are always coupled by their Coulomb interaction; the paper shows how to enhance this to achieve universality by adding individual phase shifts to each ion. They also show phonon-number measurements are possible using internal state measurements and clever pulse sequences.
I am struck by the beauty of this paper and the simplicity, ingenuity and generality of their method. However, they do not provide any talk of drawbacks or limitations: to read the paper, you'd think it will be performed tomorrow. This has happened before in ion traps; for example it took almost ten years between the proposal of the Cirac-Zoller gate and its implementation by the Blatt group, because of a number of limitations which are not apparent in the Cirac-Zoller paper but which quickly became apparent as people tried to implement it.
In this spirit, then, what are the possible limitations and pitfalls of Shen, Zhang and Duan's proposal? What experimental problems must still be solved before this can be experimentally realized?
