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this article doesn't get into what kind of future quantum computing will enable.

can anyone describe a program which cannot be written (efficiently) today, which might run on quantum computers eventually?

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Shor's algorithm is the classic (pun intended) example: http://en.wikipedia.org/wiki/Shors_algorithm

I think this is a better writeup: http://www.scottaaronson.com/blog/?p=208

Factorisation is the common example. You'll be able to factor gigantic numbers like an RSA public key pretty much instantaneously, as thanks to a qubit's super-position of states you can operate on all possible states at once. But in general it's all about parallelisation - being able to perform an operation on all states (ie numbers) at once. Due to the probabilistic/uncertain nature of measurements of a quantum system though I believe operations have to be performed a number of times in order to build statistics around the "right" answer.

While the other answer about factorization hit the nail on the head, let me say it again in more direct terms. There is a strong belief among those who pay serious attention to encryption (NSA, etc.) that quantum computing would/will render all traditional encryption (minus things like one time pads) obsolete. An actor with a sufficiently advanced quantum computer would be able to break everyone else's comms.

I understand how quantum computers can break RSA; by factoring numbers quickly. That's exploiting a known "weakness" in RSA.

I'm curious as to how quantum computers would break other encryption algorithms like AES or ElGamal. Do you have any details?

There's much more concern with asymmetric algorithms like RSA or elliptic curve crypto. I don't know the details on symmetric ciphers like AES, but nobody really seems to be worried about them.

However, it's also not necessarily true that all asymmetric cryptosystems would be broken by quantum computing. For example, the McEliece cryptosystem is gaining attention lately because it's not known to be vulnerable to any quantum attacks. If you're interested in finding out more, google for post-quantum cryptography; there's lots of interesting info out there.

Quantum algorithms halve the "strength in bits" of symmetric encryption algorithms by using the Grover's search algorithm (AFAIK):

http://en.wikipedia.org/wiki/Quantum_computer#Potential

I was repeating it as explained to me by someone whose opinion I trust, however it seems likely now that I misunderstood or perhaps it was being simplified for me. I should know better than to comment on issues like that that I don't fully understand. Thanks for correcting me.

Hehe, well, don't think of it as "correcting". I didn't (and still don't) understand it fully myself, so you should thank me for clearing things up or something instead :)

You might find this useful: http://michaelnielsen.org/blog/quantum-computing-for-everyon...

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