The report, which the newspaper said was based on documents leaked by former NSA contractor Edward Snowden, comes amid continuing controversy over the spy agency's program to collect the phone records Internet communications of private citizens.
Such a computer, which would be able to perform several calculations at once instead of in a single stream, could take years to develop, the newspaper said. In addition to being able to break through the cloaks meant to protect private data, such a computer would have implications for such fields as medicine, the newspaper reported.
The research is part of a $79.7 million research program called "Penetrating Hard Targets," the newspaper said. Other, non-governmental researchers are also trying to develop quantum computers, and it is not clear whether the NSA program lags the private efforts or is ahead of them.
His disclosures have sparked a debate over how much leeway to give the U.S. government in gathering information to protect Americans from terrorism, and have prompted numerous lawsuits.
Last week, a federal judge ruled that the NSA's collection of phone call records is lawful, while another judge earlier in December questioned the program's constitutionality. The issue is now more likely to move before the U.S. Supreme Court.
On Thursday, the editorial board of the New York Times said that the U.S. government should grant Snowden clemency or a plea bargain, given the public value of revelations over the National Security Agency's vast spying programs.