I'm having trouble understanding his explanation of interference. I understand applying the 45 degree counter-clockwise rotation twice would transform the qubit from |0> to |1>. I don't understand how this implies that there are two paths to state |0>. How could those two rotations could get you anywhere besides |1>?
> How could those two rotations could get you anywhere besides |1>
Obviously they can't, which is why the math shows that they don't.
After the first rotation, you are the state (|0>+|1>)/sqrt(2). The physical interpretation of this this state is that it represents a 50% chance of being in |0>, and a 50% chance of being in |1>. If you apply this rotation to either of those two possibilities, you arrive back at (|0>+|1>)/sqrt(2), which still has a 50% chance of being in the state |0>. The two paths leading to this are when the intermediate state is |0> or |1>.
When you actually do the math (in which "rotation" is just a name we give to multiplying by a unitary matrix, U". You find that you end up in the state (.5-.5)|0>+(.5+.5)|1> = .5|0>-.5|0>+.5|1>+.5|1> = |1>.
Here, we can again see the semblance of 2 paths leading to zero (the two |0> terms), however they have opposite signs, so cancel out.
Aha, thanks to you and ufo for the explanation. I think the problem was that I was thinking about it in complex exponential notation. If I had done it in matrix form, the cancellation of the two amplitudes of the |0> state would have been much more obvious.
Its not two paths to state |0>. Its one path to 1/sqrt(2)|0> and one to -1/sqrt(2)|0> and those cancel out.
If you start at |0> and do a 45 degree rotation you end up at (|0> + |1>)/sqrt(2). If you start at |1> and do a 45 degree rotation you end up on (-|0> + |1>)/sqrt(2) (note that the coefficient on the |0> is negative now).
Now the trick is that rotations are a linear transformation so rotate(a(x+y)) = a(rotate(x) + rotate(y)). In our case, when we rotate |0> twice, we first end up at (|0> + |1>)/sqrt(2), then we can use linearity to split that into a |0> and a |1> component, rotate them individually and then add up the the results. When we rotate the |0> component we get a +|0> and when we rotate the |1> we get a -|0> and those cancel out (destructive interference).
