Okay, so we have discussed so far the field of circuit quantum electrodynamics that is

placing a superconducting qubit inside a superconducting microwave resonator and we have learned for

example how this can be employed to read out the state of the two-node system.

And now it's time to take a step back and to explain even briefly the general context

of all of this because a lot of this general field of adequate quantum systems has become

interesting precisely for the reason that you may think of a two-level system as a sort

of quantum bit that stores quantum information.

And so what we want to discuss today is some of the most basic principles of quantum computation

and how they relate to the particular example of circuit committee.

So the field as a whole is called quantum information processing because this includes

a little bit more than just doing computations on individual qubits.

Of course this can be the subject of a lecture, of a whole lecture and I will just establish

the very basics that you need to know.

The elementary idea is very simple, that a two-level system can be treated as a bit in

the sense that it has two different states and then we call this bit a quantum bit to

distinguish it from its classical analog.

So that would be the states say zero or one and in each particular physical realization

you would have to tell me which physical states these states correspond to.

This is the so-called computational basis.

Now this idea is simple enough but then one has to ask about the differences with respect

to classical bits and one of the most obvious differences is that a two-level system can

be brought into a superposition state.

So you would have two complex amplitudes say A and B. They have the property that their

squares add up to one because their squares are the probabilities of finding the bit in

state zero or one and apart from that they can vary continuously.

So one of the methods to visualize this we have already learned to know namely the so-called

Bloff factor that can lie anywhere on the Bloff sphere and so you can turn it continuously.

And so this is the first point where you might worry that quantum bits may not be as useful

as classical bits because you see this smells a lot like analog computation.

Before there was digital classical computers there were analog computers where for example

computations were done on the values of voltages or currents that could vary continuously but

one of the big troubles with analog computers is that if there is a tiny error, a tiny deviation

because of some noise it's very hard to tell and correct.

So analog information intrinsically is much less robust than digital information and that

was one of the problems back then and so the question is whether the same problem will

come up with a quantum computer.

And as we will see now there is something more digital about quantum bits but it's not

so easy to see at first sight.

So then if you have only one bit of course this doesn't make a computer.

A computer will have a large memory composed of many many bits and then if we are dealing

with quantum bits there is another thing we learn namely that the amount of information

we need to specify the state of the quantum computer at each instant of time is really

enormous.

So in other words we have an exponentially large Hilbert space if we are given the number

of qubits.

So for n qubits obviously the dimension of the Hilbert space is just 2 to the n because

each of them can have two different states.

And what this means is that if you represent the state of your quantum computer at any

instant of time in terms of a basis where you just specify the state of each qubit individually

then this becomes a very large linear superposition of states.