Q2L7c.txt

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Quantum teleportation turns out to be
a very useful, basic primitive.
In particular, gates can be teleported.
To see this, first, let us look at a representation
of teleportation.
Recall the standard teleportation circuit,
which employs a Bell basis measurement,
producing two classical bits X and zed, which are then
used to do some control.
Imagine, though, that one has a unitary operation applied
to half of the entangled pair.
This partial, modified teleportation circuit
is equivalent to this following circuit
where one takes the input state, qubit, applies
a controlled zed, a controlled X, and then
the unitary operator u.
We may prove this equivalence by returning
to the traditional circuit for teleportation.
Recall that teleportation employs a single qubit, psi,
a entangled state, 00 plus 1 1, and a Bell basis measurement
implemented here using a controlled NOT and Hadamard
gate together with two measurements
in the computational basis.
The outputs of the measurements produce bits X and zed.
And these bits are then used to perform controlled X and zed
gates.
Psi is then recovered.
Denote this state as phi.
We note that phi is this state right here
before the fix up operators.
And phi is found to be a Pauli X or Pauli zed, acting
on psi in this particular order, zed first.
X and zed, here in exponents, are 0s or 1s.
And these Paulis are undone by the fix up operations.
This proves the desired equivalence, since U is then
just appended to the end.
We now use this equivalence to show an interesting application
of teleportation, namely, teleporting gates.
Consider the case when u is the Hadamard gate.
Then psi goes through the controlled zed, the controlled
X, and the Hadamard gate.
This circuit is equivalent to one
where we have pushed the Hadamard gate backwards,
conjugating X to become zed and zed to become X. The output is
H times psi by inspection.
Thus, we can conclude that teleportation,
if implemented with a Hadamard on the entangled pair
and implemented flipping the fix up operations
so that X controls zed and zed, the bit measurement zed,
controls X, then we find that the teleported output is
H times psi.
And we can interpret this entangled state,
modified entangled state, as a kind of resource state.
This teleportation is actually built out
of two more primitive steps known
as one-qubit teleportation.
We may construct this by considering
the swap gate constructed out of three controlled NOT gates.
This two qubit circuit swaps any two states, phi and psi.
Now consider the case when phi is the state zero.
That means that we can drop the first controlled NOT
gate because its control is zero,
leaving us with a swap circuit that has two controlled NOT
gates, again, for the special case, when one of the qubits
is zero.
This circuit simplifies.
We may first make it a little more complicated
by replacing the second controlled NOT
gate with a controlled NOT gate, flipped
in the opposite direction and four Hadamard gates
using a well-known identity.
Moreover, since H X H is zed, we may
replace this second controlled NOT
gate and two of the Hadamards with a controlled zed gate.
Note then that the lower qubit at the output
is in the plus state.
And the control is a computational basis control.
So we can actually measure that qubit
and then perform the zed operation classically.
So it's a classically-controlled Pauli gate.
This circuit, which moves psi from one qubit to another
is called Z-teleportation.
And it is counterintuitive because a measurement
happens on the qubit, which encodes psi, or appears
to encode psi.
Another such construction maybe also constructed
from the swap gate.
But now instead of starting with zero in one qubit,
let's start with the plus state, that is 0 plus 1 in one
of the qubits.
This state is an eigenstate of the X operator.
And thus, the third controlled NOT gate
can be dropped, leaving us again with two controlled NOT gates.
Let us explicitly write out the plus state
as being created by Hadamard acting on zero.
Note the second controlled NOT can
be replaced by a classical measurement and a classically
controlled X gate just as before.
We, thus, construct a circuit known as X-teleportation.
The X and zed teleportation circuits are quite useful,
and I encourage you to remember them and play with them.
Here is a useful example.
Recall the T gate is a square root of the S gate
or the fourth root of the zed gate.
It has matrix elements 1 0 0 square root of I.
And the neat thing is that together with CNOT, H,
and S, which are Clifford group gates, the T gate, which is not
a Clifford group gate, generates universal quantum computation.
That is any u can be well-approximated by these four
primitives.
The T gate is like a rotation about the zed axis,
and, thus, it is unsurprising that when
it commutes with the X gate, it adds an additional phase
factor of minus i and an additional S gate.
Thus, if we look at the X-teleportation circuit, which
takes zero and psi as input, utilizes a Hadamard
and a controlled NOT gate and a measurement
and classically-controlled X gate.
And if we add to this an extra T gate,
the output should be T times psi.
But we may push the T backwards by commuting it
with the X gate.
And this gives us a modified one qubit teleportation circuit
where we now have a controlled minus ISX gate, instead
of a controlled X gate.
The T commutes with the control of the controlled NOT gate.
And thus, we may push it back to the input,
giving us this transformed X-teleportation circuit.
The structure is the same, but we have a different input state
here, which has H T acting on zero.
And we may consider this again to be a resource state.
And we have a Clifford operation, minus ISX
instead of just the X operation.
This means that given this resource state, T H zero,
universal quantum computation is possible fault tolerantly
on many stabilizer codes such as the Steane code,
using this teleported T gate construction.