States and Probability distributions that the 5-qubits IBM computer can produce Update (January 2018) A new very interesting paper with various experiments on the IBM machine is Five Experimental Tests on the 5-Qubit IBM Quantum Computer by Diego García-Martín, Germán Sierra.

Update (July 2017) Extended question: Of course, the question extends to the newly built IBM 16-qubits quantum computer, and the 50-qubits quantum computers that IBM, Google and various other players expect to demonstrate in the near future. A natural conjecture is that all these computers can only approximate "noise-stable" pure states.  namely pure states that are stable under noise operator with exponential decay of high order Pauli terms, (see e.g. this paper by Montanaro and Osborne).    

Original question:

IBM has recently built a 5-qubits quantum computers based on superconducting qubits. It is even possible to make experiments over the cloud. 

The space of pure states for 5 qubits is the unit sphere in the 32-dimensional space over the complex number. When you measure the state of the computer you get a probability distribution on strings of 5 bits. 

Give a "map" of the probability distributions and of the actual states that the IBM quantum computer can reach.

Of course, the boundary between what can be reached and what is beyond reach is of special interest. And also the case of 3 and 4 qubits.

My question is motivated by this comment over Lipton abd Reagan's blog by Alexander Vlasov. 

I would expect that distributions that can actually be reached are characterized by "noise stability" which is equivalent to "low degree Fourier-type" expansion. The relevant "noise" here is independent depolarizing noise. The relevant "Fourier expansion" might be the one described by Ashley Montanaro and Tobias J. Osborne in the paper quantum Boolean function. 

Vlasov referred to the paper by Daniel Alsina and  José Ignacio Latorre Experimental test of Mermin inequalities on a 5-qubit quantum computer. A sentence from the abstract reads "The experimental results obtained using the quantum computer show violation of all Mermin inequalitites, with a clear degradation of the results in the 5 qubit case." 

Update: Andrew Morgan suggests in the comment below that the proposal for the threshold might be too naive. A specific question inspired by Andrew's comment is:

Can we well approximate with the IBM machine the state

$$\frac 14 |11111\rangle+\frac 14 |11100\rangle+ \frac 14 |11010\rangle+\frac 14 |11001\rangle+
\frac 14 |10110\rangle+\frac 14 |10101\rangle+ \frac 14 |01110\rangle+\frac 14 |01101\rangle+
\frac 14 |10011\rangle+\frac 14 |01011\rangle+ \frac 14 |00111\rangle+\frac 14 |10000\rangle+
\frac 14 |01000\rangle+\frac 14 |00100\rangle+ \frac 14 |00010\rangle+\frac 14 |00001\rangle.$$
(Superposition of the computational basis elements with an odd numbers of 1's)

(Later update: In view of Peter Shor's comment below I suppose we need to look more carefully for "noise-sensitive" states that we would like to explore.) 

Also of interest are the states $$\frac {1}{\sqrt {5}} |10000\rangle +
\frac {1}{\sqrt {5}} |01000\rangle +\frac {1}{\sqrt {5}} |00100\rangle + \frac {1}{\sqrt {5}} |00010\rangle +\frac {1}{\sqrt {5}} |00001\rangle $$ and the superposition of all basis elements with two 1's. (If somebody reading this question actually tried to program the IBM 5-qubit computers I'd love to hear about it.)
I just ran the first state you suggest (i.e. the GHZ state with negative phase in the Hadamard basis). Basically what I did was to write a circuit which creates that state, apply one of 5 stabilizer generators (XIXII,IXXII,IIXXI,IIXIX or -ZZZZZ) and then inverted the creation process. The circuit is shown below for the -ZZZZZ stabilizer.



There seems to be problem with the calibration or something, since I am seeing a bit flip occurring on the first qubit with roughly probability 2/3 that should not happen. The expected result should be 00000 in all cases, however 10000 seems to be twice as likely in almost all cases (prob ~0.1 vs ~0.2 over 1024 runs for each stabilizer generator).

Below are the probabilities of obtaining 00000 for each stabilizer generator over 1024 runs each:
    XIXII:0.10840
    IXXII:0.15625
    IIXXI:0.11621
    IIXIX:0.10645
    ZZZZZ:0.09375

And the probabilities of obtaining 10000 for each stabilizer generator:
    XIXII:0.21777
    IXXII:0.21875
    IIXXI:0.17871
    IIXIX:0.14648
    ZZZZZ:0.19726