I love Hacker News because it exposes me to a great deal of things like this. I intend to do as deep a dive I can muster into the provided lecture notes. But boy howdy, are certain topics I encounter here humbling.
If I were fully honest with myself I would avoid a certain subset of the content on hn. Sometimes I think it only fuels my impostor syndrome.
I spend plenty of my time dealing with my own imposter syndrome, often because there are many people in my own field who know far far more about it than I do. Peter Shor, the OP, for instance. But its fine. You do what enjoy and best at, and find satisfaction in doing it. That is the way to live.
QC and general relativity are daunting topics that don't lend themselves well to short-form, self-paced content. These are topics that need foundational perquisites and subject matter experts to catechize. It would be like hopping out of a helicopter mid K2 without any training or acclimatization and expecting to summit. Start smaller and work your way up one step or idea at a time. :)
I think in my particular case the concepts around quantum ____ are so abstract I'm unable to find anything to liken it to. It certainly doesn't help I lack any substantial background in mathematics.
Hopefully one day I'll get there. Or maybe I'm already there and not there at the same time! Who knows!?
part 1 here is essentially the idea of logic circuits. you can write these down in terms of large vectors of 0s and 1s, and operations transforming them as large matrices -- which is useful because it takes familiar ideas (to a CS student) and casts it using mathematical formalisms that are later required for weird quantum stuff.
this could be unfamiliar if you haven't learned about matrix/vector multiplication. but ultimately it's just a different way of talking about combinations of AND/OR/NOT on truth values.
part 2 just says, go from vectors of 0s and 1s to probabilities between 0 and 1. the "L1 norm" (aka Manhattan or taxicab distance) in this situation just means that all the values together have to sum up to 1, the way probabilities do. this includes part 1 as a special case, and extends it naturally.
part 3 is indeed where the weird quantum stuff comes in. no easy explanation, you do have to get an idea of how complex numbers work. but again, the familiar logic circuits of part 1 are still a special case. the "L2 norm" for real numbers is just the regular straight-line distance -- I can't provide a good intuition for complex numbers though.
I love, love, love learning but I feel like I have some… granite wall of impenetrability when it comes to quantum computing. I simply cannot figure it out. I’m a very adept person and a quick learner, but everything about quantum theory just leaves me feeling like a complete idiot.
Linear algebra are equations that do not have exponential independent variables. For example (a,b,c,d,e) are constants: aw + bx + cy + dz = e. You won't find an x^2, z^3, etc. Everything is of order 1.
"classical n-bit states are 2^n length binary vectors": Binary is 1's and 0's. A binary vector is a list of them where each entry is a particular dimension. 2-bit state vector will have 2^2 entries. <1, 0, 1, 1> is 2-bit state binary vector.
"gates are 2^n x 2^n binary matrices": A matrix is a table of values that map one vector to another vector, i.e. a transformation.
|a b|
|c d|
If matrix A is the 2x2 matrix above, then vec x * matrix A = vec y is a set of two linear equations:
a * x_0 + b * x_1 = y_0
c * x_0 + d * x_1 = y_1
<x_0, x_1, x_2, x_3> * |a b c d| = <y_0, y_1, y_2, y_3>
|e f g h|
|i j k l|
|m n o p|
a * x_0 + b * x_1 + c * x_2 + d * x_3 = y_0
e * x_0 + f * x_1 + g * x_2 + h * x_3 = y_1
i * x_0 + j * x_1 + k * x_2 + l * x_3 = y_2
m * x_0 + n * x_1 + o * x_2 + p * x_3 = y_3
"Introduce probabilistic classical computation: Now the states/vectors have real entries in [0,1] and obey the L1 norm (the critical feature). Similarly, the gate matrices.":
Now the entries in vectors and matrices are not just 0 and 1, but any real value between and including 0 and 1. This is important for statistics because it can represent the likelihood of something happening. 0% to 50% to 100% and everything in between.
The L1 norm turns a vector of numbers into just one number. L1 is particularly easy: just add up the absolute value of all numbers (all numbers are positive here anyways). ||vec x|| = x_0 + x_1 + x_2 + ...
"Now, argue that quantum computing just requires the same linear algebriac structure but we (1) work over the complex number field, (2) norm is L2.":
Now entries in the vectors are complex numbers. Complex numbers are a pair of real numbers that have a different multiplication transformation. Complex numbers have the form z = <z_0, z_1>. Adding complex vec e and vec g is straight forward, add the pairs of components: <e_0 + g_0, e_1 + g_1>. Multiplication of complex vec e * complex vec g is <e_0g_0 - e_1g_1, e_0g_1 + e_1g_0>. This is also just linear algebra.
Complex numbers are these 2 dimensional vectors, but being just "numbers" themselves, they have a short-hand notation: z=x + iy. "i" is not a variable and not a constant. It represents a second dimension to the number. To quickly do the matrix math, any time you have "i * i" replace it with -1 and you will have done the matrix multiplication. For example:
(1 + 2i) * (3 + 4i) = 3 + 4i + 6i + 8i^2
= 3 + 10i - 8
= -5 + 10i
L2 is another measure of vectors. L1 can be thought of as a calculation of a vector's perimeter (add up all the dimensions). L2 is a calculation of distance. L2 is adding up the squares of all dimensions and taking the square root (i.e., Pythagorean's theorem but generalized). Linear algebra, L2 norm, etc. all work out nicely for complex numbers.
"The reason I like this development is that it takes at least some of the mystery out of quantum mechanics. It is not a strange model of computation, completely divorced from classical. Just a variant of it, that happens to be the one the universe runs on."
The above are all classic topics in advanced high school and undergraduate math, and important to ~every discipline of engineering. Linear algebra, real numbers, complex numbers, statistics, measurements (L1, L2), etc. all have really nice geometric properties that can improve understanding, which I can't do justice here.
The other important part to remember is that all these ideas took hundreds of years to discover and be refined. Highly recommend diving into them though because they are a lot of fun. They should be pretty accessible from online texts and videos.
Cheers
edit: I can't believe how shitty the formatting options are on HN.
Enter 4 spaces before the text to write it monospaced
[0, 1]
[1, 0]
┌ ┐
│ 1 4 │
│ 2 5 │
│ 3 6 │
└ ┘