UPDATE: Graham recently contributed to this article in The Economist on Post-Quantum Cryptography.
Computers that exploit quantum mechanical properties offer the promise of (supposedly) unbreakable cryptography and other exciting applications, but they will also cause a huge, immediate problem: the day a large, practical quantum computer is developed, all existing widely-used asymmetric cryptography will be broken.
This will have serious consequences: massive amounts of secret information exchanged over the internet under public-key or hybrid cryptography could be revealed. Computers will no longer be able to ensure the updates they are downloading and installing are legitimate, since code signing will be broken. The owner of the first powerful quantum computer (which will probably be a large state organisation, who will keep it a secret) could have the power to take over almost every computer and mobile phone connected to the Internet. Even though nobody knows when this will occur, it makes sense to start preparing.