As a young grad student, I remember going to a talk by Bennett where he explained how a Quantum Computer allows manipulation in a 2^N dimensional hilbert space, while the outputs measurements give you only N bits of information. The trick is to somehow encode the result in the final N bits.
I felt this was a much better layman explanation of what a quantum computer does than simply saying a quantum computer runs all possible paths in parallel.
> I felt this was a much better layman explanation of what a quantum computer does than simply saying a quantum computer runs all possible paths in parallel.
Relevant concerning your point:
> "The Talk"
> https://www.smbc-comics.com/comic/the-talk-3
Thanks for this! I guess i need to read up on Hilbert Space.
...and Shor's Algorithm
Don't let the terminology intimidate you. The interesting ideas in quantum computing are far more dependent upon a foundation in linear algebra rather than a foundation in mathematical analysis.
When I started out, I was under the assumption that I had to understand at least the undergraduate real analysis curriculum before I could grasp quantum algorithms. In reality, for the main QC algorithms you see discussed, you don't need to understand completeness; you can just treat a Hilbert space as a finite-dimensional vector space with a complex inner product.
For those unfamiliar with said concepts from linear algebra, there is a playlist [1] often recommended here which discusses them thoroughly.
[1] https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...