2.12 Composition of rotations

Unitaries as three rotations

We are now in a position to understand the circuit in Figure 2.3 in geometric terms. It is a very useful fact of geometry (which we shall take for granted) that any rotation in three-dimensional Euclidean space can be performed as a sequence of three specific rotations: one about the $z$-axis, one about the $x$-axis, and one more about $z$-axis. The circuit does exactly this:

The first phase gate effects rotation by $\alpha$ about the $z$-axis, the second phase gate is sandwiched between the two Hadamard gates, and these three gates together effect rotation by $\varphi$ about the $x$-axis, and, finally, the third phase gates effects rotation by $\beta$ about the $z$-axis. So we can implement any unitary $U$ by choosing the three phase shifts, $\alpha$, $\varphi$, and $\beta$, which are known as the three Euler angles.