Quantum Computing and the Hidden Subgroup Problem(daniellowengrub.com) |
Quantum Computing and the Hidden Subgroup Problem(daniellowengrub.com) |
I particularly appreciate the choice of Simon's Algorithm as a "toy example" and the quick recap of character theory; both were very helpful for a non-QC person with an interest in this stuff.
Can anyone ELI5?
Naively, you'd evaluate the functions at every point by trial & error until they much the shape of the given graph. Or use the symmetry of sin & cos to combine them constructively and destructively (peaks and valley) and to match the given shape.
FT & QFT are "shortcuts" that help to decipher the correct combination of basis functions.
The high level point is that many algorithms for which quantum speedups are possible can be reduced to the Hidden Subgroup Problem, which requires a few weeks of a group theory course to understand.
The Wikipedia phrases it backwards (as do you): "quantum computers" don't solve the hidden subgroup problem, the quantum fourier transform, which "measures" f in parallel, can be used to solve the hidden subgroup problem efficiently. The QFT is the fundamental thing, not the HSP, and it's the building block for basically any/all useful quantum algorithms.
The Hidden Subgroup Problem (HSP) is the mathematical framework that explains why quantum computers are so good at breaking encryption. Famous quantum algorithms like Shor's (for factoring numbers) and Simon's are all just different versions of solving the same underlying pattern-finding problem. The quantum "superpower" comes from using the Quantum Fourier Transform to reveal hidden mathematical patterns that classical computers can't efficiently find.
It's not hard to explain the problem, as most math undergraduates will have taken some group theory course. But even "subgroup" means nothing to most computer scientists (speaking from personal experience interacting with computer scientist academics).
Yes because we're talking about pure math here not popcorn and soda.
"Efficiency of hw arithmetic is not the bottleneck, lack of a poly time algo to factor prime numbers is"
&, the closely related
"Ability to reduce everything to HSP is not the bottleneck, lack of a scaleable implementation of qFT is"