What's the point of quantum gates being 'continuous'?

Question

Besides the 'continuous' which I don't fully understand the term. It's all the time said that arbitrary gates can only be estimated but not necessarily be accurate. I don't understand the claim. So what if it's continuous? So do I have, uncountable many points on the Bloch sphere and I could certainly get from one to any other point using just two rotation gates.

So my question is basically, simply why aren't two rotation matrices enough not just to approximate but even to accurately calculate any arbitrary transformation so they could be described as a 'universal set' (If you need control you can add CNOT but that's it!). I mean, all you gotta do is playing around with the rotations so you switch a vector state on the surface sphere from one to another. That's enough, why not?

(I know there you may be required to use CNOT for some control qubits, but we usually define the universal set to be a bigger one, why? And why we just approximate and don't get exact or arbitrarily close to the answer?)

Answer 1

It's not clear what the exact question is, but here is how I interpret it

States can be represented on the Bloch sphere, and unitary gates correspond to rotations in the Bloch sphere. Two rotations are enough to move from any point of a sphere to another point. So why is it not enough to consider two fixed gates instead of the whole continuous set of gates?

First of all, while it is true that "two rotations" with appropriate angles are enough to send a point to any other point on the sphere, even in this case you are still using a continuous set of rotation matrices to do it. Given a pair of initial and final points on the sphere, you can find two specific rotation matrices that send one into the other. Given an initial state, you will then need a continuum of rotation matrices to send that point to all other possible points on the sphere, as every output point will correspond to different rotation angles (even though the rotation axes might be fixed).

So more precisely, given an initial state, you need a (continuous) family of rotations of the form $\{R_{\boldsymbol v}(\alpha)R_{\boldsymbol w}(\beta)\}_{\alpha,\beta\in\mathbb R}$ to describe all possibilities.

This said, you don't want to describe how a specific initial state evolves, but rather how an operation acts on each possible input. In other words, a gate is characterised by the way it transforms the whole Bloch sphere, not just one point on it. As it turns out, in this case you need three consecutive rotations to describe arbitrary rotations.