Why can we detect single photons, but not single gravitons? Everybody can buy a single photon counter. Why are there no single graviton counters?
Obviously, graviton sources are rare. But why are graviton detectors so hard to make? Is it just because the typical gravitons (say, from black hole mergers) have such low frequency, and thus very low energy? 
In other words, is it a noise issue, in the sense that any graviton detector has intrinsic difficulties to distinguish a signal from noise, because the signal has such a low level?
Or is there still another reason?
 A: The best reference on this topic that I'm aware of is Rothman and Boughn, "Can Gravitons Be Detected?," http://arxiv.org/abs/gr-qc/0601043 . They argue that the cross-section for a graviton to interact with pretty much any target is on the order of the square of the Planck length. This is not obvious, and they refer to a lot of previous authors who came up with other estimates, and claim that with hindsight those estimates were wrong. So basically we can't detect single gravitons because the Planck length is very small.
A: Look at the difference in coupling constants between electromagnetism and gravity:


The coupling constants are what define the ballpark of the probability calculations in quantum mechanics. Supposing that gravitons exist ( it is an effective, not decisive, quantization of gravity that is used) the probability of their interacting with the atoms and molecules of a detector is smaller than for photons at least by $10^{35}$ times.
Since Avogadro's number( molecules in a mole of matter) is of order $10^{23}$ you can imagine how big a detector would have to be in order for a graviton to interact with one of its molecules. Then there is the probability of the interaction being non trivial in energy so some photons are produced in order to see that an interaction happened.
A: Photons interact readily with individual charged particles and have a relatively high energy compared with the energy required to trigger a chain of events that leads to a count in a detector.
Gravitons don't interact with individual particles directly, they have the effect of warping spacetime. The effect an individual graviton is expected to have is incredibly tiny compared with the impact of a photon on an atom, so a vastly more-sensitive detector would be required. 
A: It is very important to differentiate between virtual gravitons (in QFT calculations of interactions) and real gravitons (GW quanta). Now there is no consensus on this site whether gravitons are the quanta of GWs or not.
https://physics.stackexchange.com/a/215180/132371
There are two ways we can detect gravitons, either build detectors as you suggest, or create them at the LHC.
Now detecting single gravitons is very unlikely, because gravitons interact with matter very very weakly. 
$$\alpha_\mathrm{G} = \frac{G m_\mathrm{e}^2}{\hbar c} = \left( \frac{m_\mathrm{e}}{m_\mathrm{P}} \right)^2 \approx 1.7518 \times 10^{-45} $$
The gravitational coupling constant characterizes the gravitatonal attraction between elementary particles. 
$$\alpha_\mathrm{G}$$ is 42 orders of magnitude smaller then $$\alpha$$ 
Now if you use the energies at LHC, you will see that it is similarly orders of magnitude less likely to produce a single graviton at the LHC then a single photon.
https://en.wikipedia.org/wiki/Gravitational_coupling_constant
Thus, the correct answer to your question is that in the foreseeable future we will not be able to build detectors that can detect single gravitons nor will we be able to produce them at the LHC.
You are correct, it is the cross section.
Gravitons have a much lower cross section then photons to interact with matter.
https://arxiv.org/pdf/gr-qc/0607045.pdf
