Shor's algorithm is the main practical example of an algorithm that runs more quickly on a quantum computer than a classical computer - at least in theory. Shor's algorithm allows large numbers to be factored
into their component prime factors quickly.
In reality, existing quantum computers do not have nearly
enough memory to factor interesting numbers using Shor's algorithm, despite decades of research.
A new paper provides a major step
in that direction, however. While still impractical on today's quantum computers, the recent discovery
cuts the amount of memory needed to attack 256-bit elliptic-curve cryptography by a factor of 20. More interesting, however, is that the researchers chose to publish a zero-knowledge proof demonstrating that they know a quantum circuit that shows these improvements, rather than publishing the actual
knowledge of how to do it.

https://lwn.net/Articles/1066156/
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