Why do we start from the least significant bit in phase estimation algorithms?

I've been watching some videos and tutorials for quantum phase estimation. Here's a video I found helpful, which explains the phase estimation in general. I also learned the iterative phase estimation (IPE) from this Qiskit notebook.

It seems to me that no matter which version of phase estimation we implement (standard or iterative or others), we will always start by learning the least significant bit in the bit string, and perform the controlled unitary operation $$U^{2^{n-1}}$$. My question is why do we always start from the least significant bit?

In terms of the associated time-evolution, I think the controlled $$U$$'s should commute, so is it correct to say we only need to order the qubit correctly to learn the string, and the order of controlled $$U$$ doesn't really matter?

Thanks!!

• Considering the fact that $e^{(2\pi +\theta)i}=e^{\theta i}$ and in binary system $2*0.d_1d_2=d_1.d_2$ , it's just more natural and easier to devise algorithm this way. Commented Jun 16, 2022 at 5:24
• @narip Thanks for the comment! The video showed an equality $\exp(2\pi ib_1.b_2) = \exp(2\pi i0.b_2)$, could you explain a bit why that's the case?
– IGY
Commented Jun 16, 2022 at 5:40
• Think for examples. Let $b_1$ be $1$, then you have $e^{2\pi ib_1.b_2}=e^{2\pi i(1+0.b_2)}$, the answer is obvious now, I think. Commented Jun 16, 2022 at 5:42