# Simplify the tensor product of two exponentials

If I have a 2-qubits circuit with a Ry rotation gate acting on each one : My unitary transformation performed on the 2-qubits state is written as :

$$e^{-i\theta_{1} \sigma_{y}} \otimes e^{-i\theta_{2} \sigma_{y}}$$

I was wondering if I could simplify this product in order to have only one exponential(thus making the tensor product disappear) and what would the result be ?

• Since the transformation is unitary, this is possible- any unitary can be written as exp(iH) with H an Hermitian operator. Apr 19 at 19:50
• Can you detail how you would do it ? I am still confused about those kind of calculations
– Duen
Apr 19 at 20:51

The two gates act on separate qubits, so their generators commute and you can use the identity $$e^A \otimes e^B = e^{A \otimes \mathbb{I} + \mathbb{I} \otimes B}$$ to get $$e^{-i\theta_1 \sigma_y} \otimes e^{-i\theta_2 \sigma_y} = e^{-i(\theta_1 \sigma_y \otimes \mathbb{I} + \mathbb{I} \otimes \theta_2 \sigma_y)}$$ if that helps simplify things.
• The generator is the operator in the exponent; in your case, the generator of $R_y(\theta)$ is $-\theta \sigma_y$. Apr 20 at 9:18
• Oops, nice catch; I've added the $i$ to the answer! The $i$ is not considered part of the generator though. Apr 20 at 9:22