Suppose that tomorrow is July 7, 2050 and Snowden is still stayed in Sheremetyevo airport. Further, let us suppose that quantum computation and quantum commincation are available to everyone, and Snowden's laptops are also quantumly-made. My question is: Who can or how to say something confidentially to Snowden? (Please refer to Classical Cryptographic Challenge: How to Say Something Confidentially to Snowden?)
Sorry for my boring and stolen answer (from https://crypto.stackexchange.com/a/9026/6961) :
Generate a file of cryptographically strong random data at least as long as the message to be sent. This will allow communicating the secret using the random data as a one-time-pad. I.e., produce the ciphertext by using a bit-by-bit combining function such as XOR.
Purchase a plane ticket for an international flight connecting through Sheremetyevo airport.
Burn a copy of the OTP data onto a CD-ROM. Label it "Lady Gaga" and lock it in a briefcase handcuffed to yourself.
Take the flight. During your layover, locate Snowden in the international travel area.
Whisper the secret to Snowden.
Both of you then giggle loudly something about Lady Gaga.
Drop the CD-ROM in the trash so the spooks don't have to report home empty-handed.
(Optional) exchange public keys with Snowden to secure your future communications.
Quantum computing does not affect the "one-time-pad" at all, assuming your cryptographically strong random data is properly generated (using a geiger counter or similar).
Only point 8 falls over because AFAIK there are no strong asymmetric cryptography systems assuming quantum computing yet.
However, you could still have security authenticated future communications with Snowden if you use a secret key. Quantum computing only halves symmetric crypto's security, so if you use a 256-bit cipher with a strong MAC you should be able to have secure authenticated communication.
I'd like to refer you to the other answer in the original question. Suppose your laptop and Snowden's laptop are both capable of quantum communication. Suppose there is also a quantum phone book, or a quantum DNS service. When his passport got revoked, Snowden's address is also removed from the phone book (makes sense right?).
Assuming you managed to grab a copy of the phone book before the address was removed, and Snowden still has the original laptop, you have absolutely no problem in communicating with him and the problem is resolved. Even if yours or Snowden's laptop had spyware on it, you could exchange keys (assuming an asymmetric scheme that is post-quantum secure) and use them on newly-purchased devices.
The problem is, what happens if you didn't manage to get Snowden's address? It still exists, you can still reach him on it but you don't know where to send the messages. In which case the quantum part becomes irrelevant and you are faced with the original problem.
Edit: In the original question there is a comment made by @Ali:
Perhaps Snowden can share a public key and receive encrypted data
This is really the only time you can contact Snowden, or anyone else who has dropped off the grid: When they want to be contacted.
In the same way a reporter got the original interview from him, the new video could have Snowden holding up his public key and email to the camera (in which case he'd be flooded by emails but that's a different story).
The usual concerns of doctored videos, dopplegangers, edited keys and reporters colluding with the FBI still apply.
The equivalent to rubber-hose cryptanalysis also applies in this case. You could track down the original reporter and ask him of Snowden's whereabouts but guess who did that already. You can deploy a network of spies across Russia and have them feed you information, but again, someone did that with no results.
The only way to go would be to post agents in all airfields and ports of Venezuela and wait untill Snowden gets there. In this problem, cryptography (quantum or not) doesn't seem to make a difference. Because Snowden doesn't want to be contacted.
To conclude: There is no way at this time to contact Snowden.