# What is the $\lambda$ parameter in the $U3$ gate used for?

The most general single qubit gate is $$\mathrm{U3}$$ given by matrix

$$\mathrm{U3}= \begin{pmatrix} \cos(\theta/2) & -\mathrm{e}^{i\lambda}\sin(\theta/2) \\ \mathrm{e}^{i\phi}\sin(\theta/2) & \mathrm{e}^{i(\phi+\lambda)}\cos(\theta/2) \end{pmatrix}.$$

If the gate is applied on qubit in state $$|0\rangle$$ again the most general description of quantum state is obtained, i.e.

$$|\varphi_0\rangle = \cos(\theta/2)|0\rangle + \mathrm{e}^{i\phi}\sin(\theta/2)|1\rangle,$$

where angles $$\phi$$ and $$\theta$$ describe position of the state on Bloch sphere.

When the gate is applied on qubit in state $$|1\rangle$$, the result is

$$|\varphi_1\rangle = \mathrm{e}^{i\lambda}(-\sin(\theta/2)|0\rangle + \mathrm{e}^{i\phi}\cos(\theta/2)|1\rangle)$$

Obviously term $$\mathrm{e}^{i\lambda}$$ can be ignored because it is the global phase of a state.

I can imagine that global phase can be useful for constructing contolled global phase gate but it can be implemented as $$\mathrm{Ph}(\lambda) \otimes I$$, where

$$\mathrm{Ph}(\lambda) = \begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{i\lambda}. \end{pmatrix}$$

My question is: What a parameter $$\lambda$$ in $$\mathrm{U3}$$ is used for?

• where is this notation (the "U3" gate) from?
– glS
Jan 28, 2020 at 12:42
• @glS: It is based on IBM Q user manual from 2018. Jan 28, 2020 at 12:55

The parameter $$e^{i\lambda}$$ is only a global phase if it's acting on the basis state $$|1\rangle$$. Act on any superposition of states, and it's a relative phase, and absolutely critical.