# Why can a ZY decomposition not decompose an arbitrary single qubit gate?

To decompose an arbitrary single qubit gate, we need to do a "$$e^{i \alpha}$$ZYZ" decomposition.

Intuitively however I do not understand why a "$$e^{i \alpha}$$ZY" decomposition is not sufficient:

An arbitrary single qubit state can be decomposed into its $$|0\rangle$$ and $$|1\rangle$$ components. We can move both $$|0\rangle$$ and $$|1\rangle$$ to any arbitrary state $$|\psi\rangle$$ on the Bloch sphere by first applying a $$Y$$ rotation by some angle $$\gamma$$ and then a $$Z$$ rotation by some angle $$\beta$$.

ZY gives you pitch and yaw but it doesn't give you roll. You made sure your plane's nose was pointing in the right direction, but you didn't check if it ended upside down or not.

You've shown that you can put the state $$|0\rangle$$ wherever you want. But you also need to put the $$|+\rangle$$ state in the correct location on the great circle of states perpendicular to where $$|0\rangle$$ ended up. This is the third degree of freedom in the problem, requiring a third degree of freedom in the solution.

Indeed, as Craig said - if your initial state is a Y eigenstate, the Y rotation won’t move it and then you’ll be confined to the equator. In fact, your argument shows why exactly 3 rotations are needed: If the state is an eigenstate of the first axis, you can still have 2 left to move wherever you need.

As you rightly say, a sequence $$e^{i\alpha}ZY$$ can prepare any single-qubit pure state. That is not the same as having an arbitrary single qubit unitary.

A unitary can be defined by its action on the basis states. So, you've already got arbitrary action on the $$|0\rangle$$ input, which is a great start. Moreover, the unitary must take $$|1\rangle$$ to a state orthogonal to the one that $$|0\rangle$$ is mapped to, which is incredibly limiting. However, there is still a global phase factor allowed in the possible definition of $$U|1\rangle$$ (it's an important global factor because when applied to an input in superposition, it's a relative phase between the two components). How do you incorporate that extra phase? Put an arbitrary phase rotation in first to add that phase to the $$|1\rangle$$ component.