[extr]

As someone with more of a math than a physics background I really enjoyed this, but I guess I'm surprised by how, for lack of a better word, basic, it seemed? Maybe that serves as a compliment to the author, in the past I have been snowed by all these vocab words when it turns out it's mostly just linear algebra. A lot of what I enjoyed here were the sidebars explaining that concept X was really just rebranded Y, detail Z is not really important for intuition, etc...

A random question along those lines: why represent states as 2d complex vectors instead of quarterions? Aren't they the same thing? As soon as I read that I spent the rest of the article wondering if everything it would make even more sense cast that way.

[Strilanc]

> why represent states as 2d complex vectors instead of quarterions? Aren't they the same thing?

Only in the sense that they are both require four real coefficients. The quaternions have a particular multiplicative structure that just doesn't apply to quantum states, so it doesn't make sense to use them.

That being said, the space of single-qubit operations is very much analogous to rotations in 3d and so is well described by quaternions. In fact, the Pauli matrices times i (iX,iY,iZ) are isomorphic to the quaternions (i,j,k). For example, iX * iY * iZ = -I.

[foxes]

I think if you want to study quantum computing, you do not need as much of a physics background as you think (perhaps for physically implementing a qc, but that is more experimental).

If you have had more exposure to maths and computer science, it will be easier for you than someone with a "pure" physics background.

As for quarternions, yes they are isomorphic, but generally for useful applications, people consider quantum computers with n qubits. So your state is an element of C^(2^n). Apart from the measurement step, you can idealise any quantum computation as a unitary transformation, so an element of the unitary group U(2^n), acting on this complex vector.

An element of U(2^n) is representable as a 2^n x 2^n matrix U, with complex entries, st U.U^{\dagger} = I. Here dagger represents conjugate transpose, and I is the 2^n x 2^n identity matrix. Sometimes people add the extra constraint, det(U) = 1, then this gives you the special unitary group SU(2^n).

[ahelwer]

This is the great secret of quantum computing! People assume it must be difficult because their exposure to quantum mechanics has been a hundred incomprehensible pop science articles written by people who have no idea what they're talking about. In fact quantum computing requires only extremely basic knowledge of linear algebra.

[chilukrn]

ok, extremely basic is a bit oversimplifying it. When you start reading quantum algorithms, you will inevitably come across Shor's factorization algorithm, which requires (quantum) phase estimation: https://en.wikipedia.org/wiki/Quantum_phase_estimation_algor... which requires quantum Fourier transform and some good deal of math. This is when you don't go into the physical implementations. If you want to look at that aspect, things may become a bit more complex.

This is not to discourage anyone, but underselling it as requiring elementary linear algebra is not very helpful (the pop-sci articles have already been overselling it as "magical"/"mind-blowing" etc.).

[ahelwer]

There are algorithms which involve advanced mathematics on classical computers, too. You don't have to understand them to understand how classical computers work. I've never bothered to learn the general number field sieve, and similarly I've never bothered to learn Shor's.

I say if you understand gates as unitary matrix multiplication, representing multiple qbits with the tensor product, entanglement, and projective measurement, you basically understand quantum computing. Throw in an algorithm or two to convince yourself of the benefits.

[chilukrn]

True, but: 1. Any serious quantum computing course/book will have Shor's algorithm in the first few chapters (in fact there are not a lot of quantum algorithms which have clear advantage over classical ones). One can teach quite a bit of useful classical algos (sort, binary search, tree/graph-based) without going into mathematics like FFT or jpeg coding.

2. Again valid, but IMHO measurements (and PoVMs) can lead to deep rabbit holes, and I found myself digging in much deeper.

Probably I should read easier expositions to see how effectively they teach. (I come from a EE+physics background, so I do gravitate to math-heavy rigorous explanations)