Q2L8b.txt

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# Captions for 8.421x module
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What happens in quantum teleportation
if the measurement is generalized?
Well, recall z-teleportation, this
is the one-qubit teleportation picture
where we have a qubit to teleport, psi, and a state 0.
We perform a controlled not followed by hadamard.
Then we measure the qubit being teleported
and do a controlled z to the other qubit
and observe that our state emerges.
A simple transformation of this circuit
reveals something interesting.
Replace the controlled not with a controlled z
gate and two hadamard gates.
Keep the other hadamard and the measurement-controlled z
just as they were before.
Again, we're just rewriting the controlled not gate
as a controlled z with two hadamards.
Label the measurement result m and note
that we may rewrite the 0 with hadamard as a plus state.
So now we have psi and plus going into a controlled z,
hadamard and measurement, and leaving out the controlled z
from the measurement and the lower hadamard,
we find the output is h z to the m times psi.
This just rewrites the fact that the controlled z gives us
z to the m, and the hadamard here just follows on.
Now we generalize by introducing another rotation gate.
Let this box with a z theta in it
represent a rotation about the z-axis
acting on a single qubit.
That is, the unitary operation e to the minus i theta
divided by 2 times Pauli z.
Suppose the input were z theta acting
on psi instead of just psi.
Then looking at our teleportation circuit
and observing now that h may be pushed through the z
to m over here to get x to the m h psi instead of h
z to the m psi, we find that the output just
has a z theta inserted after the hadamard.
We may also push the z theta gate from the input
past the controlled-z gate.
This lets us then combine the z theta
with the hadamard as a new gate before the measurement.
Think of this part as a measurement change.
The output is the same as we just had, x to the m
h z theta acting on psi.
This new circuit is a very useful one
for measurement-based quantum computation.
It represents a measurement in a non-stabilizer basis used
to, in a sense, teleport the z theta gate onto the qubit
we are teleporting.