Q1L3e.txt

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A fundamental difference between quantum error correction
and classical error correction is
that there are far more different kinds of quantum
errors than classical errors.
The bit flip channel, which we have just studied,
where a bit flip happens with probability p on a single qubit
is really very much a classical kind of error.
It's an analog of a classical bit flip.
The quantum bit flip code which we
have seen though has many useful lessons
for our construction of quantum error correction codes.
A more quantum mechanical kind of error
is given by the phase flip channel.
The phase flip channel acts on the input density matrix,
flipping the qubit with probability p,
but in phase and not in amplitude.
Recall that the Z operator, which
has 1 0 0 -1 as the matrix elements,
acts on 0 to stay unchanged, but adds a minus sign to the 1
state.
How does one correct phase flip errors?
Well, a good starting point is to review the bit flip quantum
code, which has, as its code word states, a 000 plus b 111.
Here, logical 0 and logical 1 are repetitions
of the three basis states.
Can we correct phase flip errors in the same way?
Well, recall the Hadamard gate H.
It is kind of like a square root of NOT gate.
It creates a superposition state from basis states.
And also it conjugates X to become Z and Z to become X.
Thus, we may observe a very interesting duality,
namely the bit flip channel with Hadamard's applied to the input
and the output is actually the phase flip channel.
This allows us to use some intuition
and consider the plus minus states which
are superpositions of 0 and 1, that
is Hadamard acting on 0 and 1.
And employ those as a phase flip code,
namely let us encode as our qubit a plus,
plus, plus, plus b minus, minus, minus,
where logical 0 is a plus, plus, plus state, and logical 1
is the minus, minus, minus state.
This is in direct analogy to the bit flip code.
Note that the phase flip channel flips plus and minus,
just like the bit flip channel flips 0 and 1.
And this is why, in many ways, intuitively,
the phase flip code works.
Just like we did for the bit flip code,
let us now look at a quantum circuit for the phase
flip code.
We start with the same kind of construction as with the bit
flip code, namely an arbitrary qubit state a0 plus b1 is
encoded using two controlled NOT gates to become an output 000
plus 111.
Then, we transform the 000, 111 basis to the plus, plus, plus,
minus, minus, minus using three Hadamard operators.
The result is then sent through the phase flip channel,
acting independently and identically
on each of the three qubits.
If we then transform the output back into the 01 basis with
three more Hadamard operations, we may then apply the same
syndrome operation measurement and recovery operations as we
found for the 3 qubit bit flip code.
And thus, complete the error correction process.
This example leads to an interesting and important
claim, that is that arbitrary qubit errors can
be represented as X, Z, and XZ, namely Y type errors.
We may show this formally by looking again
at the operator sum representation of a quantum
channel.
Recall that this form has operators E sub k rho Ek
dagger summed over k.
Now, let this represent the Pauli matrices as sigma j,
including the identity.
So, j goes from 0 through 3.
To connect with the claim, recall that the Pauli y-matrix
is 0 -i i 0, and that is related to X and Z
by a factor of i.
That is, Y is equal to i times XZ.
That, you can show with a little bit of matrix algebra
reproduced here.
The sigma j form an additive basis for any 2 by 2 matrix.
And thus, E sub k can be written as a linear combination
of the sigma j matrices.
Using this expression, we may thus
re-express the operator sum notation
in terms of sigma j's and their coefficients C sub kj.
We find that E of rho is given by this expression
here, which has some cross terms with sigma j
acting on rho and sigma j prime, where j and j prime are not
necessarily equal to each other.
In front of this is a matrix of coefficients chi jj prime.
This expression is known as a chi-matrix representation
of a quantum operation.
It is, in fact, fascinating, because chi jj prime
is a density matrix formally.
Let us return to our example of a small rotation error.
Recall that the rotation about the x-axis by angle 2 epsilon
transforms the input state psi into a superposition of psi
and with epsilon amplitude, a bit-flipped version of psi.
Recall we said error correction worked
because syndrome measurements collapse the error
into a definite state.
Well, now using the chi-matrix representation,
we can see explicitly what's happening.
That is, the off diagonal term here
disappears when the syndrome measurement is performed.
In particular, what the syndrome measurement does in general
removes all these off diagonal terms
in the chi-matrix representation.
The syndrome measurement projects the environment
into a definite action given by a Pauli matrix, namely an X, Y, Z,
or identity as the error.