This is all undergrad material. There's really barely any math here; he says it pretty explicitly:

>While we will take a brief glance at topology here, I am planning to write a separate blog post fully dedicated to the maths behind topological quantum error-correction. The goal of this post is to gain a first intuitive understanding of the surface code.

All the brief mentions of homotopy and homology (the actual topology at play) are relegated to footnotes.

I taught myself most of this last year (actually including one of this guy's other posts) so that I could then teach it to a high schooler. My background was a degree in physics+math ~10 years ago and then more recently a really lame grad class in QC (that only covered up to repetition codes). I am a PhD student but not in QC.

So how many people are like me? Probably a 200-400 minted each year just in the US - how many undergrad QC classes are there taught each year? 20-40 of 10-20 students?

The real difficulty with this stuff is not the math but all the "awe gee wow" treatment people give it so that every paragraph seems like it's dripping in profundity; junk like this

> surface code is also one of the most beautiful ideas of quantum computing, and if you ask me, of all physics.

> ...

> And indeed, the surface code has deep connections to many areas of maths and physics.

You ever wonder to yourself how it's possible that so many things are "the most beautiful idea" and have "deep connections"? In reality it's really just a bunch of cute tricks and convenient definitions that make the proofs work out (basically like all of pure math).

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.

>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).

> 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.

> 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’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.

> often employing terminology from category theory and topology that is completely alien to me.

Then it might be worthwhile to learn some basic category theory. Did you know that the language of category theory was introduced to make statements about algebraic topology precise (in particular the notion of so called "natural transformations of functors")?

The thing is, the way I’ve always learned things is by doing. See a problem, and learn different angles to see that problem, different methods to approach it, play around with it some more, and keep probing it until it makes sense through intuition.

The mathematical approach with many introductory texts to this stuff do things completely backwards, from my perspective. They don’t present scenarios, problems, or anything really. They start with definitions, lemmas, theorems. If you’re lucky, they’ll give you proofs. To me, that’s like learning C by reading the standard, and only writing a Hello World program afterwards. Sure, you can get some good understanding that way, but it’s an entirely impractical learning method unless you’re already seasoned in the concepts.

To paint a more complete picture, I came across the concept of polynomial invariants in knot theory, and I managed to get my teeth into that because you can start coming up with problems, solving them, and getting some intuition. That also lead to TL diagrams, winding in (2+1)D, and I could get that down into code pretty easily and again start playing around with it. These are concrete things, and involved memorising nothing other than a handful of relationships.

I tried reading “Category Theory for the Practicing Physicist”, and it was great to begin with - but pretty quickly started just giving names for things. Names for types of categories. Names for types of relationships. Definition definition definition definition. This is through no fault of the author, it’s just how the topic is, and it’s how people who can digest that stuff understand things.

This is thing A. This is thing B. They are related through thing AB. Thing B is related to thing C through thing BC. Thing A is related to C through BCAB. Here’s a diagram. This is a theorem. Ok now on to the definition of thing D.

OK GREAT BUT WHAT DO I DO WITH IT? It’s one hell of an uphill effort for learning nothing in particular - except for words for things. That‘s why I paraphrased Fermi - “if I could remember all the names of these particles I’d have become a botanist”.

>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.

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.

I am well aware of quantum error correction, but there is quite a lot about surface codes that I don't understand.

I would estimate there are about a couple of thousand people at most who can, without any other resources, make an accurate technical summary of the essential concepts in this post.

I think if people understand quantum computing a little bit, then his earlier posts are much better

Roughly how many people on the planet would be able to understand this blog at a casual glance?

I'm going to take a few swings at it tomorrow because it's so well written, but I feel comically far away from the target audience.

This is all undergrad material. There's really barely any math here; he says it pretty explicitly:

>While we will take a brief glance at topology here, I am planning to write a separate blog post fully dedicated to the maths behind topological quantum error-correction. The goal of this post is to gain a first intuitive understanding of the surface code.

All the brief mentions of homotopy and homology (the actual topology at play) are relegated to footnotes.

I taught myself most of this last year (actually including one of this guy's other posts) so that I could then teach it to a high schooler. My background was a degree in physics+math ~10 years ago and then more recently a really lame grad class in QC (that only covered up to repetition codes). I am a PhD student but not in QC.

So how many people are like me? Probably a 200-400 minted each year just in the US - how many undergrad QC classes are there taught each year? 20-40 of 10-20 students?

The real difficulty with this stuff is not the math but all the "awe gee wow" treatment people give it so that every paragraph seems like it's dripping in profundity; junk like this

> surface code is also one of the most beautiful ideas of quantum computing, and if you ask me, of all physics.

> ...

> And indeed, the surface code has deep connections to many areas of maths and physics.

You ever wonder to yourself how it's possible that so many things are "the most beautiful idea" and have "deep connections"? In reality it's really just a bunch of cute tricks and convenient definitions that make the proofs work out (basically like all of pure math).

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.

>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).

> 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.

> 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’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.

> often employing terminology from category theory and topology that is completely alien to me.

Then it might be worthwhile to learn some basic category theory. Did you know that the language of category theory was introduced to make statements about algebraic topology precise (in particular the notion of so called "natural transformations of functors")?

The thing is, the way I’ve always learned things is by doing. See a problem, and learn different angles to see that problem, different methods to approach it, play around with it some more, and keep probing it until it makes sense through intuition.

The mathematical approach with many introductory texts to this stuff do things completely backwards, from my perspective. They don’t present scenarios, problems, or anything really. They start with definitions, lemmas, theorems. If you’re lucky, they’ll give you proofs. To me, that’s like learning C by reading the standard, and only writing a Hello World program afterwards. Sure, you can get some good understanding that way, but it’s an entirely impractical learning method unless you’re already seasoned in the concepts.

To paint a more complete picture, I came across the concept of polynomial invariants in knot theory, and I managed to get my teeth into that because you can start coming up with problems, solving them, and getting some intuition. That also lead to TL diagrams, winding in (2+1)D, and I could get that down into code pretty easily and again start playing around with it. These are concrete things, and involved memorising nothing other than a handful of relationships.

I tried reading “Category Theory for the Practicing Physicist”, and it was great to begin with - but pretty quickly started just giving names for things. Names for types of categories. Names for types of relationships. Definition definition definition definition. This is through no fault of the author, it’s just how the topic is, and it’s how people who can digest that stuff understand things.

This is thing A. This is thing B. They are related through thing AB. Thing B is related to thing C through thing BC. Thing A is related to C through BCAB. Here’s a diagram. This is a theorem. Ok now on to the definition of thing D.

OK GREAT BUT WHAT DO I DO WITH IT? It’s one hell of an uphill effort for learning nothing in particular - except for words for things. That‘s why I paraphrased Fermi - “if I could remember all the names of these particles I’d have become a botanist”.

>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

shrugdon'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’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.

I am well aware of quantum error correction, but there is quite a lot about surface codes that I don't understand.

I would estimate there are about a couple of thousand people at most who can, without any other resources, make an accurate technical summary of the essential concepts in this post.

I think if people understand quantum computing a little bit, then his earlier posts are much better

https://arthurpesah.me/blog/2022-01-25-intro-qec-1/

https://arthurpesah.me/blog/2023-01-31-stabilizer-formalism-...

> And how did they achieve such a milestone? You guessed it, by using the surface code to protect their qubits!

In all sincerity, I did not guess it.

i got it in multiple choice

I've read Scientific American since I was a young kid, but this article makes me feel supremely inadequate in my knowledge of physics and maths.