## 2.13 A finite set of universal gates

The Hadamard gate and the phase gates, with adjustable phases, allow us to implement an arbitrary single-qubit unitary *exactly*.
The tacit assumption here is that we have *infinitely many* phase gates: one gate for each phase.
In fact, we can pick just one phase gate, namely any phase gate ^{61} with

If you want to be ^{62} *just one* phase gate to *approximate* the *three* phase gates in the circuit in Figure 2.3.

There are other ways of implementing irrational rotations of the Bloch sphere.
For example, take the Hadamard gate and the *finite* precision, and the phase gates will deviate from implementing the required *irrational* rotations.
It turns out, however, that we can tolerate minor imperfections; the final result will not be that far off.

That is, there do

*not*exist anym,n\in\mathbb{Z} such thatm\alpha=n\pi . For example, it suffices to take\alpha to be rational, since\pi is irrational.↩︎The notation

x\gg y is rather imprecise, but it basically means “x is much much larger thany , and, in particular, large enough for whatever we are claiming to be true”.↩︎