The National Security Agency is advising US agencies and businesses to prepare for a time in the not-too-distant future when the cryptography protecting virtually all e-mail, medical and financial records, and online transactions is rendered obsolete by quantum computing.
Quantum computers have capabilities that can lay to ruin all of the public-key cryptographic systems currently in use. These capabilities, which aren't known to be present in the classical computers of today, include the ability to almost instantly find the prime factors of extremely large numbers, using a method called Shor's algorithm. Quantum computing is also believed to be capable of tackling other mathematical problems classical computers can't solve quickly, including computing discrete logarithm mod primes and discrete logs over elliptic curves.
The difficulty of factoring and computing discrete log primes and elliptic curve discrete logs play an essential role in cryptographers' confidence in RSA, elliptic curve cryptography, and other public-key crypto systems. When implemented correctly, most scientists and cryptographers believe that the crypto can't be defeated with today's computers before the end of the universe.