Quantum Computing and the Hidden Subgroup Problem | Hacker News Reader

Quantum Computing and the Hidden Subgroup Problem

daniellowengrub.com

40 pointslowdanie12 comments

Fantastic writeup! The exposition is extremely clear.

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.

Thanks!

I’m afraid I couldn’t follow. Too long since I used most of those terms or symbols.

Can anyone ELI5?

Claude summed it up like this:

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.

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.

That's very pedantic, like saying "computers don't solve integer addition, AND/OR/NOT/XOR gates solve it, those are the fundamental thing".

> That's very pedantic

Yes because we're talking about pure math here not popcorn and soda.

My attempt at less offensive analogies:

"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"

Very late, but I meant it would be hard to summarize the details of the blog post (e.g. QFT implementation) even to the typical math undergraduate.

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).

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.

Can quantum computers solve hard problems that aren’t HSPs?

Reply on news.ycombinator.com