"Established applications for quantum computers do exist. The best known is Peter Shor's 1994 theoretical demonstration that a quantum computer can solve the hard problem of finding the prime factors of large numbers exponentially faster than all classical schemes. Prime factorization is at the heart of breaking the universally used RSA-based cryptography, so Shor's factorization scheme immediately attracted the attention of national governments everywhere, leading to considerable quantum-computing research funding.
The only problem? Actually making a quantum computer that could do it. That depends on implementing an idea pioneered by Shor and others called quantum-error correction, a process to compensate for the fact that quantum states disappear quickly because of environmental noise (a phenomenon called "decoherence"). In 1994, scientists thought that such error correction would be easy because physics allows it. But in practice, it is extremely difficult."