# One Qubit Gates

we need to organise bits into sections better, eg see also the evolutions and operations catagory, and there seem to be two subcatagories that are closely related under computation

The pure states of a qubit can be parameterised by two quantitiies, typically denoted θ and ϕ,

|\psi>=\cos(\theta/2)|0>+\sin(\theta/2)e^{i\phi}|1>

These have a representation as the surface of a sphere, known as the Bloch Sphere. The aim of a one qubit gate is to rotate one state to another. Using the Bloch Sphere picture, it is readily observed that any such rotation can be composed of rotations about any two (non-parallel) axes. We typically select the x- and z- axes,

U=e^{i\alpha}R_z(\beta)R_x(\gamma)R_z(\delta)

Hence, if we can demonstrate rotation about two different axes in a Physical Realisation, we know that it is possible to create arbitrary one qubit rotations.

It is common to reduce this set of gates even further by introducing the Hadamard gate,

H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)

since this allows us to write that

R_x(\gamma)=HR_z(\gamma)H.

Hence, we only require an arbitrary rotation about a single axis, combined with one extra, fixed gate.

If we are willing to accept a one qubit rotation to arbitrary accuracy, as opposed to a perfect gate, then this can be achieved with a discrete set of gates, at the cost that we will require more applications of them the more accurate we desire the rotation to be. This leas to the idea of Approximate Universality.

Category:Handbook of Quantum Information