# Quantum Gates: What is the mathematical difference between the the phase shift gate and the Pauli Z rotated by pi/2?

I come from computer science, and I am trying to get into quantum computing research. I am going through Microsoft's quantum katas and I'm struggling to grasp some of the mathematical nuances associated with quantum mechanics.

I understand that the Pauli gates represent rotations of the state vector about either the $$X$$, $$Y$$, or $$Z$$ axis by $$\frac{\pi}{2}$$ radians. My main question is why in quantum computing (and in the Q# language) there are so many gates that seemingly do very similar things.

For example: Assuming we have a qubit $$\psi$$ in state $$|+\rangle$$, and we apply the Pauli Z gate. This would flip the state vector to state $$|-\rangle$$, a rotation of $$\frac{\pi}{2}$$ radians about the Z axis.

We also have the $$R_z$$ gate, which is a rotation by an arbitrary amount, so we could just supply $$\frac{\pi}{2}$$ to get the same result as Pauli Z

We also have the R1 gate, which would rotate the vector about the $$|1\rangle$$ state. Is the one state not just a vector along the Z axis, thus equivalent to a $$R_z$$?

And finally we have the S gate, which is the phase shift gate. My understanding is that this introduces yet another rotation about the $$Z$$ axis, by $$90^\circ$$ or $$\frac{\pi}{2}$$ radians. This gate should introduce an imaginary element to the $$|0\rangle$$ component, which is represented by a nonzero value on the $$Y$$ axis of the Bloch sphere. How is this different than $$R_z(\frac{\pi}{2})$$?

I am also a bit confused how the angles $$\phi$$ and $$\theta$$ on the bloch sphere translate to phase and amplitude of a quantum system, as well as exactly what those terms mean.

• If you are thinking about this as computer components then all the redundancy you point out is correct but there are differences in the meaning between Pauli operators and rotations. For that you need to see how this operators come up in quantum theory. Pauli operators generate rotations and also can be thought of as measurement operators. – oleg Jul 23 at 15:53
• Phi is usually the phase. Amplitude usually refers to a basis element like 0,1,+,- etc and not a "quantum system". I suggest you focus on the quantum mechanics of a single qubit before you proceed with quantum computing (it is only going to get worse if you skip on the formalism of quantum theory). – oleg Jul 23 at 16:00
• @oleg "I suggest you focus on the quantum mechanics of a single qubit". This is exactly why I asked this question. I have been unable to find any resources that truly explain the link between the math, physics, and computation theory of a single qubit. The quantum katas I am going through provide code solutions, but no mathematical insight as to why these are solutions. Is there any resources you can recommend to get a good understanding of the qubit? I have been reading Microsoft's guide on the qubit but I still have questions. – Raijin_ Jul 23 at 22:22
• Chapter 2 in "Quantum Computation and Quantum Information" by Nielsen Chuang is still a good and popular reference for introductory quantum mechanics for quantum computing and information.You will probably learn more QM than you wanted to but I think there is no way around that. – oleg Jul 24 at 23:23
• @oleg You're too late, I downloaded it yesterday and am working my way towards Ch 2! This seems like exactly what I need, thanks! And I want to learn all the QM so there's no need to worry :) – Raijin_ Jul 24 at 23:30