Q2L7a.txt

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Quantum teleportation is a remarkable and important
primitive for quantum information science.
We may understand it better by utilizing a stabilizer
picture to analyze it.
Recall that the traditional circuit representation
of teleportation involves the qubit psi which
we wish to teleport, an entangled pair of qubits in the state 0,
0 plus 1, 1, a bell basis measurement,
and two classically controlled Pauli operators, X and Z.
The original state psi is reproduced
at the output, and the Bell basis measurement
destroys the original psi.
Let us describe this now using operator measurement,
instead of the bell basis measurement.
That means that we measure in X X and in Z Z.
And here in this circuit, I include the fix
up operators in a particular way,
whose meaning will become clear soon.
Let us analyze this by writing down
the stabilizer starting from S1 at the beginning going
to S3 at the end.
The initial state has stabilizer IXX and IZZ.
And, in addition to this, I would
like to specify the normalizer, which is given by X and Z,
acting on the first arbitrary qubit input state psi.
The first part of the bell basis measurement projects onto XXI.
And thus, the stabilizer immediately
after this projection, is using rules about measurement
on stabilizers minus 1 to the a XXI.
That is the measurement operator with its measurement result
a, and IXX.
Because IXX commutes with the measurement.
IZZ does not commute.
The X normalizer stays unchanged,
but the Z normalizer has to change.
And it becomes ZZZ, which is the product of ZII,
its original value, and IZZ.
Note that the normalizer commutes with the stabilizer
elements at all times.
The application of a ZZ controlled on a,
remove this sign minus 1 to the a, leaving the stabilizer S2,
given by XXI and IXX.
The normalizers stay unchanged.
The second part of the bell basis measurement
projects along ZZ.
This projection leaves us with a stabilizer at minus 1
to the b, ZZI, and XXI.
This, after correcting for the sign with an XX, gives us S3.
And the normalizers, X3, are given by XXX.
And Z3, a ZZZ.
The interpretation of these normalizers
becomes clear by multiplying them by the stabilizer
generators giving IIX and IIZ.
This describes one freely encoded qubit.
Namely, the input state psi.
Thus, we see the input qubit is teleported to the output
according to the stabilizer formalism.
The stabilizer description allows
us to quickly see what happens when we teleport
with a modified ancilla state.
Suppose that we have the same circuit as before, but now
hadamard is applied to the lower half of the entangled pair.
Now, instead of using ZZ, and XX for the fix up operators,
let us use ZX and XZ.
What then is the output state of this circuit?
Well, let us look at the stabilizers step by step.
The initial stabilizer is IXX IZZ.
The stabilizer of the state after the hadamard
is thus, IXZ, IZX.
It has normalizers XII and ZII just as before.
Next, the stabilizer after the first bell basis measurement
and fix up is IXZ, XXI.
The measurement operator XXI replaces the stabilizer
generator IZX, and thus, the normalizers become XII and ZZX.
Finally, the output state stabilizer
is generated by ZZI, XXI.
And we had to replace the generator IXZ.
Thus, the normalizers become XXZ and ZZX,
which we may interpret like before as being single qubit
normalizers.
Here, X4 is IIZ, and Z4 is IIX.
The roles of X and Z have been swapped.
This indicates that the qubit has been
transformed by a hadamard gate.
So the output of this modified teleportation is H times psi.
This was accomplished, again, by encoding
Hadamard with this entangled pair, and then changing
the fix up operators, showing how we may teleport a Hadamard gate.