[MUSIC] The transformations of qubit gates, now multiple copies, can be in general described by a very big image transmission. And this is called quantum gate. So this is kind of diagram attic picture, so we prepare interstate. So interstate and it's an encrypted and they're all initialized in a very trivial state which is zero. And I mean the way we see this diagram is that here the direction is time and here is the number of cubits. So we have included here and it evolved in time in this direction. So as time goes by the U is applied to this into the system and we have a final state here and still we didn't perform measurement. And this is the state that we want to desire to have at the end of the process. And if you look at this transformation in more detail, in a closer look, then in general this can be decomposed into many small cube it. I mean the cubit on small number of the gates on small number of cubits. And this property is called in universality, so I suppose you have a very big unitary transformation. And using that transformation, we can transform interstate to arbitrary content state as we desire. So any transformation can be achieved by this military transformation. And the universality means we can actually construct or build this transformation using. A set of unitary transformation, this container, small numbers. So for instance, we can say, later we can explain and this will be we call c not gate. So control the not gate the messages that if this is to communicate, so we can collect this scene at gates and single cubit rotation. Then these guys so the set of gates and single cubit operations can approximate. Or can build very big quantum circuit or quantum gate with a high procedure in arbitrary procedure. And this is called universality, so the goal of quantum circuits. So this the sequence of circuits and by congratulating quantum gates and then you can have quantum circuit. The goal of this quantum circuit, is to realize arbitrary transformation on on on on the interstates. So it is very important and important message here again any quantum circuit can be factories. Or can be decomposed or constructed by these two cubic gates and and single gate single cubic, it's only. So we don't have to look at three cubic gates because it's sufficient to look at two cubit instructions and a single human rotations. In fact, this is a way to generate income state and it came to that, I will come to that point later. Okay, so then, we have a general form of single Cuban operation which is rotation within blocks here. And then now, we want to now describe these two cubic it then to keep the gate means interaction between the two cubit. So therefore this is a single cubic gate and then, we put here control. So this basically explain, so two cubic gates but control U gate, so this define and control the U gate. And this means we have an interstate A and B, whenever A is zero you will have the same states A and B. But if A is one then the issue is applied to the second. And the cubic in the second register then you have you be so I we can put here A. I mean so that whenever A zero this will be identity, so you will have the same state. But whenever A is one then you really act on on the second the Cuban in the second system, then this will work. So this is controlled U gate and a good example is controlled not operation and or simply because he not gate. And this be right down as X, so X for single tribute, often called the not gate, so that then, we can write in this way. So this C not get and this is I mean equivalently often people just write down this one is beat white edition. Because if you apply here A and B. What happened here is the Cuban, the first register will remain the same. The second what happened if you apply X here, okay then the cubit in the second register can be written as B. Plus A and plus here is a bit wide edition, so often we write down the gate in this form. And the useful between X and C is that X, how to my transformation H and X and H will be Z. Or equivalently, we can also drive X from how the transformation and see and how the transformation. So this guy, okay and then can be rewritten in terms of C. So and this one and we have this relation and therefore, the transformation. And then, we perform controlled not gate and had a transformation. And this is precisely controlled the city gate and this is called controlled phase gate. So this one is called see if educate then, how can we generate entire state using to communicate. So far the entire state, we found this guy and so you cannot generate and internal state from single cubic operation only. So if you imagine this local unitary gates, then the output states. If you apply zero and one and the rejecting state will be some vector again, this one, their local municipality. So you never have have F internal state, so it is necessary to perform two bit interactions to generate the entire state. And how can you prepare this guy, so we start from trivial state zero and zero. And we perform the heart of transformation the first QB and then C, not get here. And you can check out the researching state is site five plus. So we can simply write down this one as how the transformation the first register 200 and then, we perform you see not see notice. Basically this guy and the resulting state is five plus. So to buy the entire state it is necessary to consider interactions between two cubits. And this means, if there are two cubits and they should interact directly. And the alternative of way generating internal state is that we introduce an intimated particle here. And first this guy so I label you A and B, so the goal is to entangled, entangled to system A and B. And for that purpose I labor the so the one SAC and C, first interact with A and then, see sent it to be. And then, this guy will interact with be here interaction means we won should apply to communicate here for instance the nut. And then, he should perform two cubic it as well and then, this will be gone. And after all to stay two cubits A and B can be found in the entire state. So two ways or two distinct way of generating internal states. And the first one direct interaction, which described by this circuit two QB circuit gates. And the next one is to introduce some intermediate particle, this interact firstly with A and then B. So A and B, they interact, interact indirectly through a system, C and then A and B system can be entangled after all
