# 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
• Euler angles should be part of FAQs on Qiskit doc. A lot questions on this topic. +1 Oct 13, 2022 at 4:35
• @RajeshSwarnkar. This really is one of the most poorly documents parts of Qiskit. There seems to be very little documentation as to the geometric meaning of U(theta, phi, lambda). According to the documentation for OneQubitEulerDecomposer, it hints that it is defined as: Z(phi) Y(theta) Z(lambda). Is That accurate?? Oct 22, 2022 at 3:44
• I believe those derivations are from heavy math resources. Hence might not be included. But would really help if they did ELI5. Oct 22, 2022 at 4:56

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.