Q1L3d.txt

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A key concept in quantum error correction
is the ability to measure the effect of the environment
without disturbing the encoded quantum state.
This is realized using a method known as operator measurement.
Given a unitary operation u, which
has eigenvalues plus or minus 1 and eigenvectors plus or minus
u, the concept of measuring u is defined
by this quantum circuit.
First, we start out with a qubit in the 0 state.
We perform a Hadamard to put that into a superposition,
and then use that superposition to control the operation of u
onto another state.
Assume that state is a superposition of the plus
and minus eigenstates.
The controlled u operation is followed by a Hadamard
on the control qubit and then a measurement.
In addition to the measurement output,
this circuit also produces a output state psi.
Let's call the measurement result z, and also label
the intermediate states in the circuit
so that we may analyze its operation systematically.
Phi0, Phi1, and phi2 will be the states.
The initial state, phi0, is a two-register quantum state
with the first qubit in an equal superposition of 0 and 1,
and then the second qubit being a superposition
with unknown amplitude C0 and C1 of the two eigenstates of u.
Phi1 follows the application of the controlled u operation,
which flips the sign of the u minus term,
but only when the control qubit is in the 1 state.
So thus, for the 0 state, control, the bottom register,
stays unchanged, but when the control is 1,
the bottom register has u1... Phi2
then applies a Hadamard operation
to that first control qubit.
The result is easy to write down by flipping 0 for 0 plus 1,
and 1 to 0 minus 1, but it is meaningful to further simplify
this expression.
Specifically, we find that it is just
the superposition of two terms where
0 labels the case when the eigenstate is u plus,
and 1 when the eigenstate is u minus.
This then lets us collapse the input state into eigenstates
of u, and we find that the measurement result, z, tells us
when the output state, psi, is a plus u eigenstate or a minus u
eigenstate.
Operator measurement is thus seen as a projector
into the plus or minus 1 eigenvalue subspaces
of a unitary operator.
This is extremely useful to us for error correction.

Specifically, we will employ operator measurement
to obtain the error syndrome.
The error syndrome tells us what error
has occurred that is the effect of the environment.
It will be determined by syndrome operators.
For the 3-qubit bit flip code, these
are the two syndrome operators, namely ZZI and IZZ.
Recall our table of output probabilities.
Here are the measurement results obtained
when the syndrome operators are measured.
We can read out what the measurement results
are by realizing that 0 0 0 is a plus 1 eigenstate of ZZI.
The 1 1 1 state is also a plus 1 eigenstate state of u1
because it has two 1's, both of which contribute a minus sign
and thus cancel out.
Therefore, the input superposition
state a 0 0 0 plus b 1 1 1 is a plus 1 eigenstate of u1.
Therefore, the measurement result z1 will be equal to 0.
Remember, 0 corresponds to a plus 1 eigenstate,
and 1 corresponds to a minus 1 eigenstate.
This state is also a plus 1 eigenstate of u2.
Let us move on to the case when a single qubit
error has occurred.
Notice that 0 0 1 is a minus 1 eigenstate of u2,
and 1 1 0 is also a minus 1 eigenstate of u2.
In fact, in general, what we find is that ZZ applied
to a 2-qubit state, xy, is going to be a plus 1 eigenstate
if they're an even number of 1's, and a minus 1 state is xy
has an odd number of 1's.
Therefore, it is easy to read off by inspection
that a 0 0 1 plus b 1 1 0 is a minus 1 eigenstate of u.
Look at where the 01's exist.
Thus, by inspection, we find that a 0 0 1 plus b 1 1 0
has measurement results 0 and 1 for the z1 and u2 syndrome
operators.
The next case has syndrome measurement results 1 1
and then 1 0.
These four cases are the cases we're
interested in for error correction.
The second step after syndrome measurement
is to apply a recovery operation.
The recovery operator R takes the syndrome measurement result
and conditionally applies an appropriate unitary transform
to recover the original state.
Here we find by inspection that the operator needed
is a bit flip on qubits 3, 2, or 1 as appropriate,
labeled by z1 and z2.
The result of this recovery operation
is a recovery of the original encoded qubit state, namely
a 0 0 0 plus b 1 1 1.
Of course, this is only the case when a single qubit error
or no qubit error has occurred.
If two qubit errors occur, then we also find syndrome results.
For example, here we find 0 1, in the next case we find 1 1,
and then here we find 1 0, and then 0 0.
Of course, here, when the recovery operators
are applied, namely x3, x2, x1, and identity, we
find not the original qubit, but rather a 1 1 1 plus b 0 0 0,
an inverted qubit.
This is the wrong state.
It's an error.
We may use this to answer the question how good is
the error corrected state?
Let us compute the fidelity of the output state.
Recall that the fidelity of a quantum state, rho,
with a pure state, psi, is given by the square root
of the overlap of psi and rho.
This can be given a lower bound by taking the probabilities
of errors occurring.
Namely, we find the fidelity is at worst 1 minus 3p squared
plus 2p cubed, namely 1 minus a term which
is of order p squared.
Without quantum error correction,
this fidelity would have gone as 1 minus p instead of 1
minus p squared.
This is thus a significant improvement,
and that improvement is a signature
of quantum error correction.
So how did we get this improvement?
Well, let us look at the error correction procedure in total
by writing down, for this example
with the 3-qubit bit flip quantum error correction
code, the quantum circuit.
The quantum circuit model involves an encoding step,
the errors, and then the recovery and syndrome
measurement operation.
Here's the encoding circuit for the 3-qubit code.
We start with the single qubit unencoded.
We then perform two controlled NOTs
with zero state qubits as input.
This 3-qubit state then goes through three
separate, independent, and identically acting bit
flip channels, which produce outputs
that then go to our operator measurement circuit.
The operator's measurement circuit
measures the syndrome of the qubit.
The two operators measured are the parities
on the first two qubits, here done using two controlled Z
gates.
Remember, the Hadamard gates are used in the operator
measurement construction.
This measurement result here gives us
one of the syndrome bits.
The measurement result here is z1.
And for the second syndrome operator,
we measure the parity on the middle and the bottom qubit,
again with two controlled z operations.
These two controlled z operations
between the two Hadamard operators
gives us, with operator measurement, the measurement
result z2.

Following the syndrome measurement operations,
we perform the recovery operator.
The recovery operator is classically
controlled by the measurement results of z1 and z2.
The output of the recovery operator
is then, with high fidelity, the input qubit state,
and that is a 3-qubit error correction circuit.
Now, something that is very interesting
to realize about this circuit is that it doesn't just
correct for bit flip errors.
I claim that it also corrects for small rotation errors, that
is for continuous errors, not just
this extreme case of a complete bit flip.
We may see that this is true by analyzing
the effect of a small rotation.
Consider a channel which is a small rotation by angle
2 epsilon about the x-axis.
This rotation applied to a pure state
may be expanded to leading order in epsilon, giving us the two
terms psi, which is the original state, minus i epsilon x1 psi.
Here, X1 is the Pauli X operator applied to the first qubit.
When no error correction is applied,
the fidelity of this output is the overlap
of this perturbed state with the initial state,
and that is of order 1 minus epsilon.
When error correction is applied,
a funny thing happens due to the syndrome measurement.
The syndrome measurement causes the superposition of error
and no error to collapse.
Thus, we obtain either psi or epsilon x1 psi,
where epsilon x1 psi has a bit flip error applied to it.
That is, there is no continuous error case here.
You either have the error or no error.
And thus, the recovery operator applied to the error channel
output state gives us a fidelity which, to leading order,
goes as 1 minus epsilon squared.
This is, again, the same improvement
that we saw with the full quantum error correction
circuit.
Thus, we see that the scheme also corrects for a small rotation error.