Simplifying quantum circuits

Question

I am very new to quantum circuits and am unsure how to simplify them. Say I'm given a very simple circuit:

and I want to simplify it using basic quantum gates. I'm not looking for an answer, but more just how to go about it. I started by computing the outputs with $|0\rangle$, $|1\rangle$, $|+\rangle$ and $|-\rangle$ states and thought that a simple NOT gate would suffice because $|0\rangle \rightarrow |1\rangle$, $|1\rangle \rightarrow |0\rangle$, $|+\rangle \rightarrow |+\rangle$. However, when it comes to the $|-\rangle$ state, the output is $-|-\rangle$, and I don't know how to account for that when doing the simplification.

Very basic question but any help is appreciated.

Answer 1

Computing the effects of the sequence of gates on the basis states and comparing the results to those of the known gates is indeed a common technique for circuit simplification. If you're careful about not discarding any global phases acquired when the gates are applied, you don't even need to compute the effects on $|+\rangle$ and $|-\rangle$ states, since the effects on a linear combination of basis states can be computed as a linear combination of effects on basis states, so if the results on $|0\rangle$ and $|1\rangle$ match, so will the results on any superposition states.

(Not sure why you call out getting $-|-\rangle$ state when applying the sequence of gates to $|-\rangle$, that's exactly the effect the NOT gate has on the $|-\rangle$ state: $NOT(|0\rangle - |1\rangle) = |1\rangle - |0\rangle = -(|0\rangle - |1\rangle)$.)

Another technique is building a visual of the circuit using ZX-calculus and transforming it using the visual rules. I've never used it myself, only heard a tutorial about it (based on the book "Quantum in Pictures"), but it sounded pretty convincing :-)