All right, good morning.

So as always, I start with a brief recap

of what the last lecture was about.

So this was still or finished the part on applications

of phase estimation.

And the first of the two main things discussed in last lecture

was period finding, which is part of Shor's algorithm.

And this is a two factor large integer numbers.

And basically, what happens in there

is that a unitary, which has this effect,

one does phase estimation on the state 1, which

can also be written as a to the power 0,

and is an equal superposition of the eigenstates

of this unitary.

So and from this one can infer this period R.

This should be an S. And the second part

was solving systems of linear equations.

In a quantum formulation, this would be an operator A.

Acting on a state x is a state B. And we

would like to know what x is.

And the so-called HHL algorithm provides an answer in the sense

that it prepares the state x, which

is proportional to a to the minus 1 acting on B.

And the way this is done is using phase estimation.

So the phase estimation circuit applies a bunch of Hadamards

on qubits and a controlled unitary, which here

acts on the input state B, and then an inverse Fourier

transform. And to get this inverse of A,

we do here a controlled rotation,

controlled on the output of the phase estimation thing

with an ancilla qubit that starts in 0.

And so we're here R on the state y.

0 would be y root 1 minus y squared over 2, 0 plus 1 over y

the 1.

So y here, the role of y is played by the output

of the phase estimation circuit.

And we therefore need to measure to make sure

we pick the 1 here, which results

in the implementation of a 1 over y.

And this 1 over y is the thing we

want because we need to do the inverse rotation.

Because we need to do the inverse of this matrix.

What I didn't draw here is that the computation of the phase

estimation needs to be undone to erase all correlations

between these qubits and the measured qubit, which

would destroy the effect of the algorithm.

OK.

So in today's lecture, I now want

to start with a different topic, quantum error correction.

Before I do this, one organizational thing.