A crude explanation might be like this - if you look at graphs of sin and cos, you'd instantly recognize their symmetries, but what if you're given the graph of a linear combination of them, and asked to decipher the coefficients?
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.
I think it would be hard to explain the details to a math undergraduate.
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.
No it wouldn't? "Given f that hides a subgroup, and an oracle for f, determine the subgroup".
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.
FilosofumRex|9 months ago
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.
tylerhou|9 months ago
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.
almostgotcaught|9 months ago
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.
cut3|9 months ago
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.