Peter Shor's MIT Fall 2022 course lecture notes on quantum computing

Can you link some examples?

Quantum gates/circuits are everything - unlike classical computing there is no abstraction built on top. The circuits are built and configured in a classical computer and then the configuration is loaded into the quantum computer.

Why not try playing around with some quantum circuits via qiskit, you can actually run them on a real quantum computer for a few cents with Amazon braket.

This one from Dan Boneh at Stanford is a good intro to cryptography (you can see that it's old as "crypto" (in the URL) was the abbreviation for cryptography still back then...)

https://www.coursera.org/learn/crypto

https://crypto.stanford.edu/~dabo/courses/OnlineCrypto/

(Part two has been going live anytime now for ages... but it's still not there: https://www.coursera.org/learn/crypto2)

I'm curious, why L2 norm? Each qubit lives in C^2, does it not? I've only ever seen quantum computing formulated over finite dimensional Hilbert spaces (though to be fair it was only in a math context, no real physics there)

Here's a brief example paired to some of the nomenclature in the parent post:

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. 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|

Is a 2x2 matrix. If the matrix is binary then a,b,c,d can only be the values 0 or 1. A matrix is just a condensed notation of a set of linear equations. Let's make a vec x (aka input) and vec Y (aka output) where subscripts indicate the dimension (entry) of the vector (list): vec x = , and vec Y = .

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

The math is pretty easy if it is 1's and 0's everywhere. A 2-state vector and matrix will then be:

     * |a b c d| = 
                           |e f g h|
                           |i j k l|
                           |m n o p|

(apologies the matrix is probably not going to render nicely, it is a 4x4). This is just the equations:

  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

A "gate" is a chain of transformations that result in some computation. For classic computation that's just binary. Most silicon chips are made of NAND gates: NOT(X AND Y). Most other gates can be created from combinations of NAND gates. For quantum computation this will not be just binary 1 and 0 as the output, hence...

"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 = . Adding complex vec e and vec g is straight forward, add the pairs of components: . Multiplication of complex vec e * complex vec g is g_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

"i" may be some other symbol, like "j" depending on whatever math dialect is being used. Complex numbers are algebraically closed, which basically means if you make an equation out of them, the solutions will also be in the set of complex numbers. This is not the case for real numbers, e.g., the solutions to x^2 +1 = 0 are not real numbers, they are complex numbers. Complex numbers model rotations/spinning/something "periodic" as simple algebra, which is very important for trigonometry and sinusoids, making them a nice tool for a lot of disciplines.

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.