The spying agency hopes to harness the special qualities of quantum computers to speed up its code-cracking efforts.
The NSA is believed to have spent about $80m (£49m) on the project but it has yet to produce a working machine.
If the NSA managed to develop a working quantum computer it would be put to work breaking encryption systems used online and by foreign governments to keep official messages secure, suggest the documents excerpted in the Post.
The quantum computer is being developed under a research programme called Penetrating Hard Targets and is believed to be conducted out of a lab in Maryland.
Many research groups around the world are pursuing the goal of creating a working quantum computer but those developed so far have not been able to run the algorithms required to break contemporary encryption systems.
Current computers attempt to crack encryption via many different means but they are limited to generating possible keys to unscramble data one at a time. Using big computers can speed this up but the huge numbers used as keys to lock away data limits the usefulness of this approach.
By contrast, quantum computers exploit properties of matter that, under certain conditions, mean the machine can carry out lots and lots of calculations simultaneously. This makes it practical to try all the possible keys protecting a particular message or stream of data.
The hard part of creating a working quantum computer is keeping enough of its constituent computational elements, called qubits, stable so they can interact and be put to useful work.
The NSA is not believed to have made significant breakthroughs in its work that would put it ahead of research efforts elsewhere in the US and Europe. However, the documents passed to the Post by Edward Snowden suggest the agency’s researchers are having some success developing the basic building blocks for the machine.