Quantum Feature Map


In a second-order Pauli-Z evolution circuit for the ZZ feature map with ϕ_{i} (x)=x_i and ϕ_{1,2} (x)=(π-x_1 ) (π-x_2 ) (Havlíček et. al., 2019), the circuit design represents these functions with specific gates. Specifically, ϕ_{i} (x) is implemented by the gate U1(2 * X[0]), and ϕ_{1,2} (x) is implemented by U1(2 ( π- X[0]) ( π- X[1])).

I have observed a multiplication by ‘2’ in these gate parameters when designed and implemented. I am curious about the rationale behind this factor ‘2’. Is it a constant factor, and why is it specifically ‘2’? Additionally, can this factor be any real number, and should it be the same for all wires in the circuit?

Thank you in advance!

Good question! It is common to find the 2 when applying rotations since the matrix of the rotations are defined with the half angle as you can see here:

Captura de pantalla 2024-02-06 a las 18.22.13

Sometimes is just more confortable avoid that fraction.
Not sure if that the case here but could be related :smile:

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