Okay, so we are just discussing qubits, in particular in the context of circuit quantum

electrodynamics.

And we ended last lecture by pointing out that if you have two qubits sitting inside

a cavity, then the real point that both of them interact with the cavity will automatically

induce an interaction between the qubits.

And the interaction is of a very simple type, just exchanges and excitation between the

qubits.

And if you switch on this interaction for a sufficient amount of time, then you can

basically swap the quantum information between both qubits.

So of course it is evident that in order to do a real quantum computation such as carrying

out Schor's algorithm to factor a large number, you need more than two qubits.

And so it is an obvious question that comes up in any physical implementation of quantum

computation.

How would you lay out making qubits?

Because that's not so trivial in real space and physical space, say, on a chip.

And so that's something I want to discuss, especially in the context of circuit quantum

computation.

Okay, if you think about it, the simplest possibility would just be to arrange all of

those qubits in a chain or a one-dimensional array.

So that is one of the generic possibilities that are not respected, of course, to super

linear chain circuits.

And of course, since all of these qubits are just two-level systems, you can identify them

as spin-one-half systems and therefore you can just view this as a spin-one-half chain.

One of the points, though, is that, say, if you want to have this and that qubit interact

with each other, so you want to perform a two-qubit gate on these two qubits, then it's

usually not possible directly because the coupling terms will only affect nearest neighbors.

So what you really do then is you exploit these nearest neighbor coupling terms to do

a swap operation to swap the quantum information inside these two qubits and then you apply

it again.

And only then, if you brought together the two qubits on which you want to act with your

two-qubit gate, are you able to carry out the two-qubit gate.

And afterwards, if you desire, then you can swap back the quantum information.

But it becomes immediately clear that this is a kind of overhead in the computation.

That is, the further along on the chain these qubits are, the more steps you need just to

swap them back and forth.

So that is one of the problematic points.

An alternative, of course, is just to take a 2D layout, a 2D array, 2D grid, and the

simplest would be a square lattice.

And there are even physical implementations where this is the natural layout.

For example, if you trap code atoms in an optical lattice, then automatically they will

sit in such a grid.

And the question in any physical implementation, however, would be, again, how to let two qubits

interact that are not directly adjacent to each other.

So depending on the physical implementation, there may be different tricks you can play.

So what I want to do now is to discuss an example from circuit quantum electrodynamics.

It's an example from our own research from a few years back where we just tried to implement

a 2D grid.

And a 2D grid, notably, which does not have a large swapping overhead because it has,

so to speak, global interactions between qubits.

And you will see why.