So good morning. Before I start, a happy new year to everyone. Let's hope it's a politically

calmer one than the last one. And the start wasn't good. Okay. In terms of the lecture,

before Christmas, in general terms, we discussed sort of what the future of quantum computing

could be like. And now I would like to come more to things that are possible today or

at least very soon. As always, I'll start with a short recap of what I discussed in

the last lecture and then turn to the new topics. So the last lecture, so basically

before Christmas, I discussed stabilizer codes for error correction.

And in particular, the so-called TORI code, which is a 2D grid of qubits

of physical qubits to be more precise, it has periodic boundary conditions.

in both directions. And this has four logical states. So this means two logical qubits.

So for that reason, I made this distinction between physical and logical qubits because

the number of physical qubits that I actually need to build can be much larger than this

two. That's the number of logical qubits. And then what you do there is that measurements

of stabilizer elements. So in this case, these are operators that are either a product

of 4x or 4z operators. Detect the errors. So an error means the state of the system

went out of this subspace of the four logical states.

Then there is also a planar version of that, which people often call the surface code.

So planar means it doesn't have this periodic boundary conditions. And that's of interest

because that's what you can actually build. And this is actually the code of choice

to implement error correction. So the estimates indicate that it needs about

a thousand physical qubits per logical qubit.

That of course means it's currently out of reach.

So that links back to what I said in the very beginning. The algorithms I discussed in the

lectures before Christmas need perfectly working gates. These are in practice not available.

So you could in principle run these if you were able to implement error correction. That

however requires a thousand physical qubits per logical qubit, which means instead if

you want to run like a 100 qubit algorithm you need 100,000 physical qubits. And that

is also something that is currently not available.

That of course immediately triggers the question like, okay, so what can we do currently?

And that's what I want to discuss next. And so the name that these things have is so-called

NISQ quantum computing. I explained what this means.

So that acronym is for noisy intermediate scale quantum computer.

So that's what NISQ stands for.

Okay so what's the scale of things? So there are several, so there's currently a development

in big companies. So this is mainly Google, IBM. I'm not claiming I cover everyone here.

Intel, also Alibaba, and there are probably more.

A number of startups. So for example, Rigetti, PsiQuantum. So I'm talking here only about

hardware development. INQ, AQT, let me also add IQM, and also University Labs.

So several of these startups are of course linked to University Labs. I just add here

one that is OpenSuperQ. So that is something that is supported by

European Union project, geographically hosted in Jülich here in Germany.

So maybe I should add here is hardware, because there are also a number of companies or startups

that just develop software. But of course the really challenging thing

is develop good hardware. That's why.

So what can current hardware do? So we have systems of 50 to 100 qubits, and

these can execute about 40 layers of gates. So basically, if I have a 2D grid of qubits,

then what layer means is I can of course run single qubit gates on all these qubits in

parallel. So of course, still experimentally this is

a challenge to control the crosstalk between the qubits. But what I want to say here is

that if I want to run two qubit gates, then I can run a two qubit gate on these two qubits