8th August 2020 >_

Ok so this is a bit of a new format I’m playing with as well. I think I’ll call this written intuition but it’s suppose to feel almost like you and me are just chatting about a topic and trying to figure things out as we speak.

So I’m currently trying to figure out Whether I can explain phase kickback from a computational branching perspective. We can start with what I think it’s the canonical example where you got a qubit in the + state , another in the - state, you apply a CX gate and we then ens up with the same system but the top qubit has now turned into the - state as well. It has taken a phase related to the X gate when applied to the - state target qubit.

Lets see how can we think about it so you strat with a + state and apply a CX to a - state. This creates two branches where one is in the - state (because the control is 0) and the other is in a state like -0+1 right because we’ve applied an X gate to the minus state. The relative phase between the two components is still 180degrees and so this state is physically equivalent to the - state.

Aha maybe a way to see this thought is that in order to rlly make them equivalent you gotta minus the whole state in order to get the actual - state nevertheless this is in a computational branch! so feels like there should be a rule telling me that if I wanna truly make this equivalent I also have to negate the control qubit is these are somewhat entangled or correlated maybe. And actually this means to turn the 1 into a -1 which when merged with the other branch root (0) makes the 0-1 state and so here you have phase kickback!!!

Thats actually pretty cool but it feels a bit more like a rule to fit the data rather than a standalone truth and that bothers me a bit Why are such states equivalent in the first place? I mena the states which are equal but got a different global phase? I’ve just been reading it as a fact rather than a well founded explanation. Because to be honest we could just keep the computational branches untouched and then we wouldn’t have the phase kickback…

9th August 2020 >_

I just realized that the reason we gotta do this is maybe because we gotta define a fix phase for the 0 component across the system and in this case one of the branches violates this and it just, you gotta have both branches following the same rules because they r part of the same system he ce u gotta correct for that global phase difference which in this case it impacts the control qubit as well because they are ‘multiplied’ or correlated and can’t be separted

10th August 2020 >_

I also wanted to update you guys on something else I found out while playing with it a bit more. If you use an X-Axis control (See here for reference) You actually observe the “phase kickback” affecting the amplitude rather than the phase of the control qubit!

I think that’s pretty awesome! Probably wrong to call this “amplitude kickback” though as we are not kicking back any amplitude but nevertheless cool.

I wonder what if we use a Y-axis control. Should do a mixture: affecting phase and amplitude?