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Very few. I think I’m actually the target audience. I come from a physics academic background - with a focus on quantum computing on the physics side - and wha

Very few. I think I’m actually the target audience. I come from a physics academic background - with a focus on quantum computing on the physics side - and what I’ve seen so far (I’m about a third the way through) feels like it’s pitched at my level. Because the word topology gets thrown around a lot in topological quantum computation (funny, that…), that gives a lot of authors the impression that they can start using terminology that mathematicians understand, but is fairly inaccessible to physicists. From what I’ve seen so far, this article is the first one I’ve seen that I think I might stand a chance at learning something from without consulting a mathematics glossary.

mathisfun123 3y ago
>using terminology that mathematicians understand, but is fairly inaccessible to physicists. You're very much underestimating the mathematical sophistication of the average theorist - there is no one in the physics dept that I'm friendly with that would have trouble with this material (especially presented at this level). And in general the math, the homology groups at play in surface codes are trivial because everything is a simplicial complex. It just ends up being a bunch of counting arguments (that actually shouldn't be completely unfamiliar to anyone that's taken undergrad discrete in a CS dept).
- red_trumpet 3y ago
  > And in general the math, the homology groups at play in surface codes are trivial because everything is a simplicial complex. I wouldn't sign that. Sure, you can define homology groups for simplicial complexes pretty easily, and it makes stuff computable, which is great. Here is why I still think this doesn't make anything trivial: 1. Computations can be damn hard, and if you do them using a computer, you might miss out on a lot of insight and geometric intuition. 2. More crucially, if you have a topological space and represent it as two different simplicial complexes. How do you prove that the homology groups are the same? I don't know if that is approachable from the simplicial point of view, and I guess it would contain pretty terrible book keeping. Very far from trivial I guess. This is where singular homology really shines: You prove that simplicial homology is isomorphic to singular homology and don't have to worry about that ever again. Also functoriality only works if you have maps which preserve the simplicial structure, whereas singular homology has no such restriction.
  - i-use-nixos-btw 3y ago
    > This is where singular homology really shines: You prove that simplicial homology is isomorphic to singular homology and don't have to worry about that ever again. Also functoriality only works if you have maps which preserve the simplicial structure, whereas singular homology has no such restriction. See, this is what I mean :D
- i-use-nixos-btw 3y ago
  I’m saying this level is perfect. I am not a practicing physicist, but I have a PhD in the theory side of condensed matter from many years ago - mostly focused on numerical simulations. What is difficult is that most introductions to this topic tend to involve word soup, often employing terminology from category theory and topology that is completely alien to me. The result is that I can’t read a paragraph without looking things up. And hell, you’ve managed to do that yourself. Homology groups? Simplicial complex? This is sounding more and more like botany to me. But that’s the best think about intros like this. I’m not one for words, but concepts I can handle.
  - mathisfun123 3y ago
    >The result is that I can’t read a paragraph without looking things up.... And hell, you’ve managed to do that yourself. Homology groups? Simplicial complex? This is sounding more and more like botany to me. I'm sorry but what do you expect from an area of research that hasn't entered the vernacular yet? Like if I go try to read a blog post on cond-mat theory how many mentions of "loop calculations" and "fermions" and "renormalization group" and "analytic continuation" and blah blah blah. Does it make sense for me to complain that the article doesn't speak to me in my vernacular? No of course not - because even if it's an intro article the expectation is that I'm willing to put in some work. If it's too much work for you, ie if you don't know math, then shrug don't try to understand mathematical things I guess. >I’m not one for words, but concepts I can handle. How did you manage to get a PhD in theory then?
    - i-use-nixos-btw 3y ago
      > I’m sorry but what do you expect Introductions to be introductions. > How did you manage to get a PhD in theory then? Certainly not by being a walking glossary. If that was involved, I’d not have done it. Same way that everyone does. I gravitated towards a niche, found a new method to analysing it, generalising that method to many more applications and spending the final year trying to ride the metaphorical unicycle on a tightrope hanging high above a cavern of nervous breakdown. The usual. Now, the way you talk sounds very familiar - I know mathematical physicists who are very much used to throwing mathematical terminology around. But on the theory side, it’s quite rare to generalise that far so early on in an academic career. It’s far more about the concrete application than the abstract viewpoint, unless it’s somewhere where the abstract viewpoint makes a lot of sense.

mathisfun123 3y ago

>using terminology that mathematicians understand, but is fairly inaccessible to physicists. You're very much underestimating the mathematical sophistication of the average theorist - there is no one in the physics dept that I'm friendly with that would have trouble with this material (especially presented at this level). And in general the math, the homology groups at play in surface codes are trivial because everything is a simplicial complex. It just ends up being a bunch of counting arguments (that actually shouldn't be completely unfamiliar to anyone that's taken undergrad discrete in a CS dept).

red_trumpet 3y ago
> And in general the math, the homology groups at play in surface codes are trivial because everything is a simplicial complex. I wouldn't sign that. Sure, you can define homology groups for simplicial complexes pretty easily, and it makes stuff computable, which is great. Here is why I still think this doesn't make anything trivial: 1. Computations can be damn hard, and if you do them using a computer, you might miss out on a lot of insight and geometric intuition. 2. More crucially, if you have a topological space and represent it as two different simplicial complexes. How do you prove that the homology groups are the same? I don't know if that is approachable from the simplicial point of view, and I guess it would contain pretty terrible book keeping. Very far from trivial I guess. This is where singular homology really shines: You prove that simplicial homology is isomorphic to singular homology and don't have to worry about that ever again. Also functoriality only works if you have maps which preserve the simplicial structure, whereas singular homology has no such restriction.
- i-use-nixos-btw 3y ago
  > This is where singular homology really shines: You prove that simplicial homology is isomorphic to singular homology and don't have to worry about that ever again. Also functoriality only works if you have maps which preserve the simplicial structure, whereas singular homology has no such restriction. See, this is what I mean :D
i-use-nixos-btw 3y ago
I’m saying this level is perfect. I am not a practicing physicist, but I have a PhD in the theory side of condensed matter from many years ago - mostly focused on numerical simulations. What is difficult is that most introductions to this topic tend to involve word soup, often employing terminology from category theory and topology that is completely alien to me. The result is that I can’t read a paragraph without looking things up. And hell, you’ve managed to do that yourself. Homology groups? Simplicial complex? This is sounding more and more like botany to me. But that’s the best think about intros like this. I’m not one for words, but concepts I can handle.
- mathisfun123 3y ago
  >The result is that I can’t read a paragraph without looking things up.... And hell, you’ve managed to do that yourself. Homology groups? Simplicial complex? This is sounding more and more like botany to me. I'm sorry but what do you expect from an area of research that hasn't entered the vernacular yet? Like if I go try to read a blog post on cond-mat theory how many mentions of "loop calculations" and "fermions" and "renormalization group" and "analytic continuation" and blah blah blah. Does it make sense for me to complain that the article doesn't speak to me in my vernacular? No of course not - because even if it's an intro article the expectation is that I'm willing to put in some work. If it's too much work for you, ie if you don't know math, then shrug don't try to understand mathematical things I guess. >I’m not one for words, but concepts I can handle. How did you manage to get a PhD in theory then?
  - i-use-nixos-btw 3y ago
    > I’m sorry but what do you expect Introductions to be introductions. > How did you manage to get a PhD in theory then? Certainly not by being a walking glossary. If that was involved, I’d not have done it. Same way that everyone does. I gravitated towards a niche, found a new method to analysing it, generalising that method to many more applications and spending the final year trying to ride the metaphorical unicycle on a tightrope hanging high above a cavern of nervous breakdown. The usual. Now, the way you talk sounds very familiar - I know mathematical physicists who are very much used to throwing mathematical terminology around. But on the theory side, it’s quite rare to generalise that far so early on in an academic career. It’s far more about the concrete application than the abstract viewpoint, unless it’s somewhere where the abstract viewpoint makes a lot of sense.