Computational complexity of quantum optics In "Requirement for quantum computation", Bartlett and Sanders summarize some of the known results for continuous variable quantum computation in the following table:  



MY question is three-fold:


Nine years later, can the last cell be filled in?
If a column is added with the title "Universal for BQP", how would the rest of the column look?
Can Aaronson and Arkhipov's 95 page masterpiece be summarized into a new row?
With respect to your third question, Aaronson and Arkhipov (A&amp;A for brevity) use a construction of linear optical quantum computing very closely related to the KLM construction. In particular, they consider the case of $n$ identical non-interacting photons in a space of $\text{poly}(n) \ge m \ge n$ modes, starting in the initial state
$$
\left|1_n\right&gt;=\left|1,\dots,1,\ 0,\dots,0\right&gt;\quad (n\text{ 1s}).
$$
In addition, A&amp;A allow beamsplitters and phaseshifters, which are enough to generate all $m\times m$ unitary operators on the space of modes (importantly, though, not on the full state space of the system). Measurement is performed by counting the number of photons in each mode, producing a tuple $(s_1, s_2, \dots, s_m)$ of occupation numbers such that $\sum_i s_i = n$ and $s_i \ge 0$ for each $i$. (Most of these definitions can be found in pages 18-20 of A&amp;A.)

Thus, in the language of the table, the A&amp;A BosonSampling model would likely best be described as "$n$ photons, linear optics and photon counting." While the classical efficiency of sampling from this model is, strictly speaking, unknown, the ability to classically sample from the A&amp;A model would imply a collapse of the polynomial hierarchy. Since any collapse of PH is generally considered extremely unlikely, it's not at all a stretch to say that BosonSampling is very probably not efficiently and classically simulatable.

As for BQP-universality of the A&amp;A model, while linear optics of non-interacting bosons alone is not known to be universal for BQP, the addition of post-selected measurement is enough to obtain full BQP universality, via the celebrated KLM theorem. The acceptance probability of the postselection in the KLM construction scales as $1/16^\Gamma$, where $\Gamma$ is the number of controlled-Z gates that appear in a given circuit. Whether that is enough to conclude that the postselected linear optics model of BQP is efficient or not is thus a matter of what one defines to be efficient, but it is universal.

Aaronson explores the postselected linear optics case more in his followup paper on the #P-hardness of the permanent. This result was earlier proved by Valiant, but Aaronson presents a novel proof based on the KLM theorem. As a side note, I find that this paper makes a very nice introduction to many of the concepts that A&amp;A use in their BosonSampling masterpiece.