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Prime factorization is an essential part of modern public key cryptographic protocols, it’s most important for privacy & security in the electronic world.
As quantum computation can theoretically factorize numbers in exponentially fewer steps than classical computation. Besides, they can be used to crack any modern cryptographic protocol.
Another problem is called sorting, it can be treated very efficiently by quantum computation.
It’s possible to search databases in 〜√N queries rather than 〜N queries with using quantum computers.
