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?

  • $\begingroup$ where is this notation (the "U3" gate) from? $\endgroup$ – glS Jan 28 '20 at 12:42
  • $\begingroup$ @glS: It is based on IBM Q user manual from 2018. $\endgroup$ – Martin Vesely Jan 28 '20 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.

  • $\begingroup$ Thanks, I see now. $\endgroup$ – Martin Vesely Jan 28 '20 at 10:48

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