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

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?)

• "It's all the time said that arbitrary gates can only be estimated but not necessarily be accurate." it would help if you could add where you saw this stated, to better understand the context of the sentence. Are you asking why should one consider a continuous set of gates instead of just a finite, universal set of gates? – glS Feb 25 at 10:05
• I am not sure. But in the book of Chuang and Nielsen they say that since the set of quantum gates is continuous then using a universal discrete set can only approximate it. I don't understand it. – bilanush Feb 25 at 10:10
• In chapter 4 they wrote it – bilanush Feb 25 at 10:20

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.