## 2.9 Any unitary operation on a single qubit

There are infinitely many **single-qubit unitaries**, i.e. unitary operations that can be performed on a single qubit.
In general, any complex ^{57}
The unitarity constraint removes

This sort of argument — counting how many parameters determine a family of matrices — is really an example of calculating the dimension of a vector space.
More generally, saying things like “imposing a polynomial equation condition on the coefficients lowers the number of (complex) parameters necessary by

In particular, we need *four* real parameters to specify a *any* unitary on a single qubit in some simple way?

Delightfully, the answer is *yes, we can.*

Any unitary operation on a qubit (up to an overall multiplicative phase factor) can be implemented by a circuit containing just two Hadamards and three phase gates, with adjustable phase settings, as in Figure 2.3.

If we multiply the matrices^{58} corresponding to each gate in the network we obtain the single matrix

Any complex number

z is uniquely specified by two real parameters, writingz=x+iy orz=re^{i\varphi} , for example. This is an instance of the fact that\mathbb{C} is a two-dimensional vector space over\mathbb{R} .↩︎Remember that the order of matrix multiplication is reversed when compared to reading circuit diagrams.↩︎