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announcement. So there will be a talk next Monday at 2 p.m. in the room is 2 729 and

that's going to be by Adam Smith who's at Technical University in Munich and he'll

talk about using the IBM quantum computer so that's the cloud available

one to simulate quantum dynamics. So maybe this should be a quite interesting

talk about things that one can do today on existing quantum computers and it

will touch upon topics that we are going to discuss later in these lectures as

well. Okay, so then as always let me start with a short recap of what we did last

lecture and the main topic of last lecture was quantum teleportation.

So here the sender has an arbitrary quantum state which I denote here as

alpha 0 plus beta 1 and then sender and receiver initially share some maximally

entangled state. The version that we discussed it's this particular Bell

state.

In this protocol the sender then does a CNOT gate between the two quantum

qubits he owns and then a Hadamard gate on the first qubit. Receiver does

nothing all the time and then the sender measures the two qubits they own and

communicates the results to the receiver.

The receiver does gates depending on the measurement outcomes of the sender.

So in the first instance depending on the measurement outcome M2

applies an X gate or not in the second instance receiver applies a Z gate or

not and then the resulting state and the receiver qubit will be exactly the same

state is initially owned by the sender. I want to remind you of two important

things. So prior to the classical communication of the measurement result

here and the density matrix for the qubit at the receiver side is maximally

mixed meaning the receiver cannot get any information at all about the state.

So the consequence of that is that the information flows with the classical

communication.

Meaning in particular it cannot flow faster than the speed of light which

means there is no conflict with the theory of relativity here which one

initially might have guessed because the separation between the sender and the

receiver side can be arbitrarily high and everything that happens in terms of

gates happens locally. Of course both sender and receiver share this resource

of a maximally entangled state. And then in the last part of the lecture I

introduced two types of gates we hadn't discussed before the so-called T gates

and S gates

where S can be decomposed of two T gates meaning a T gate is more powerful in the

sense what you can do with it because if you have T gates at your disposal you

can do algorithms that have an odd number of T gates where if you only have

S gates available then you can only do algorithms that correspond to an even

number of T gates whereas if you have T gates calibrated you can of course do

both types of algorithms. Okay so with that let me start with the topics of

today's lecture. And this is the question can a quantum computer

a posture register

So basically in today's lecture I want to address the two questions.

So how hard is it for a quantum computer to simulate classical computations, to do classical

computations and how hard is it for a classical computer to do quantum computations?

And so for this first question the answer is yes.

And the way to see this is to translate classical so-called Boolean circuit

into a quantum circuit.