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Why are there such different UGate and U3Gate implementations in the Qiskit documentation?

Quantum Computing Asked on December 21, 2020

The documentation for the new version Qiskit 0.20.0 states that:

  • UGate is "Implemented using two X90 pulses on IBM Quantum systems:

$U(θ,ϕ,λ)=RZ(ϕ−π/2)RX(π/2)RZ(π−θ)RX(π/2)RZ(λ−π/2)$"

  • U3Gate is "Implemented using two X90 pulses on IBM Quantum systems:

$U3(θ,ϕ,λ)=RZ(ϕ)RX(−π/2)RZ(θ)RX(π/2)RZ(λ)$"

It looks like only the latter matches the known rotation sequence for U3:
$z$-rotation ($lambda$), $y$-rotation ($theta$), $z$-rotation ($varphi$)

Although their presentation matrices completely coincide:
$$
mathrm{U3}=
mathrm{U}=
begin{pmatrix}
cos(theta/2) & -mathrm{e}^{ilambda}sin(theta/2)
mathrm{e}^{iphi}sin(theta/2) & mathrm{e}^{i(phi+lambda)}cos(theta/2)
end{pmatrix}.
$$

Is this an inaccuracy in the documentation or am I missing something and these gates are actually different?

Just in case, the aforementioned difference stated in the documentation refers to the hardware (not software) implementation of these gates on IBM Q systems.

One Answer

If you look at the source code then you will see that the UGate is defined as an alias for the U3Gate. As to why do they need this alias, I do not know for sure. But if I were to hazard a guess, then it would be because in most quantum computing literature, $U$ is used to refer to an arbitrary unitary gate/operator. Since in qiskit U3Gate is the most generic single qubit unitary gate, it makes sense to identify it with $U$ from quantum computing literature.

Answered by e-eight on December 21, 2020

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