Voyager is at ~130 AU. Alpha Centauri is at ~275,000 AU. With the 1/r^2 decrease in signal, that means the signal from earth will be smaller by a factor of ~3 x 10^6. Now, supposing we have ~1000 of these smartphone probes, this means that their combined signal will be weaker than the Voyager signal by a factor of ~3000. We might imagine that they are somewhat more optimized to send a stronger signal than an ordinary smartphone. Let's say they have a 30 W radio instead of 3 W. This means that they will have a weaker signal by a factor of ~300. This is a lot, but if they manage to build a detector that has a larger effective diameter than the Deep Space Network by a factor of ~15. Given the budget of the project, and the fact that they have 20 years of time to prepare after the launch, this could be feasible.
Looking at the project's website, though, it looks like they are taking the approach of using a laser onboard the probe instead of using a traditional radio transmitter [1]. This would increase the efficiency of the transmission by several orders of magnitude. It might even be possible to detect the signal from a single probe with this approach.
[1]: http://breakthroughinitiatives.org/index.php?controller=Foru...
https://en.wikipedia.org/wiki/Interstellar_medium#Radiowave_...
Not to mention there is also additional cosmic radiation outside the heliosphere. Something the size of a chip is going to be extremely vulnerable. Slower rad-hard electronics are going to be the order of the day, but even with thousands of these things, it's going to take a toll.
How do you keep it cool with a 100GW laser pointed at it over 2 minutes?
I wonder if it would be easier to string ~300 of these at ~1000 AU intervals, and use them for a relay? Or perhaps ~3000 at ~100 AU intervals.
And I wonder if that would be construed as mining a common hyperspace lane by the galactic council... ;-)
http://link.springer.com/content/pdf/10.1007%2F978-3-642-274...
Depending on what maneuvering profile these things will have, it might be desirable to have a "semi-phased" design that can be steered to a limited degree without physically moving the spacecraft.
And, it also goes without saying that this antenna probably needs to be flexible or at least articulated in order to deploy.
NASA has previously done some really interesting antenna designs with genetic algorithms. You just need to figure out what the goodness function here should be.
https://ti.arc.nasa.gov/m/pub-archive/1244h/1244%20(Hornby)....
Similar to how the retroreflectors on the moon allow you to bounce a laser off of them and send it directly back to you.