Okay, up to this moment, we have discovered two major problems with the state-of-the-art classical computer. The first one is that the base element of modern computer, the pair injunction, becomes small and is ready to reach a big molecular size where pair injunction cannot be implemented no more. So, we have to base our computer or the next generation on something as small or even smaller as an atom, where there is quantum effects. The second is that there are problems for which the complexity of the solution grows exponentially with the size of the problem. Eventually, one of these hard or intractable problems is the problem of emulating of quantum system. As the number of particles in the quantum system grows linearly, the complexity of its classical description grows exponentially. This fact was noted by American Physicist and Nobel Prize winner Richard Feynman, which in 1982, he suggested that since these quantum systems are so hard to model, maybe they will be good in modeling themselves or maybe it is good to base the new generation of computers on quantum systems. Since, in this case, they will effectively emulate themselves and maybe they will be able to solve other intractable problems. So, these two facts, that the base element is going to be quantum for the next generation of computers and the system with several quantum elements is hard to model, leads us to believe that the next generation of computers is going to be quantum. Only in three years after this famous suggestion, in 1985, the David Deutsch developed a mathematical model of universal quantum computer, which leads us to the structure of our course. Now, the first week, we are struggling through the introduction, and the purpose of this week is to convince you, since I'm already convinced, that quantum computers are unavoidable and future belongs to them. On the second week, we are going to learn this mathematical model of the David Deutsch, like quantum data and quantum gates, and how quantum algorithms are represented mathematically. The whole course material will be based on this mathematical model, so this second week will be crucially important. On the third week, we will build with our bare hands, our own very basic and small quantum computer with two quantum bits. We will implement the most basic quantum algorithm on this computer, and we are going to learn some other basic algorithms during the third week. The week four and the week five are dedicated to the more useful but more complicated algorithms, that of Peter Shor, the component for factoring composite numbers, and Grover's algorithm for solving any problem from the complexity class and B in general. Now, it's time to talk about these quantum effects and why they frighten us so much when we scale down to the atom size. We already know that to start one bit of information, we need the system with those states. A hydrogen atom, at first glance, has these two states. This may be the orbitals of its electron. But it has already become tradition, when you first meet these quantum effects to start not with a hydrogen atom but with this famous two-slit experiment, and we are going to follow this tradition. So, instead of an atom, we have this complex set. A photon emitter, which can fire one photon, for example, pair of [inaudible] and it shoots photons in this direction here. On the way of the photons, we have a screen with two narrow slits in.. The slits must be really narrow because if we have like several meters here, like three, five meters, then the width of the slit and of this partition between them has to be something like one-tenth of a millimeter. Okay, if we put the photon detectors here and here, then we can gladly affirm that we have this system with two states. If photon comes through the upper slit it hits this detector and if it comes through the lower slit it hits this second lower detector. We can assign our favorite numbers, like one and zero, to these states. The first problem is, if you ever have seen light coming out of flashlight, you probably have noticed that the light does not propagate in the same direction. The spot of light becomes wider as light travels further. Why is that? One can believe that it is because of these small particles of gas and dust or something else which alter the light direction, propagation. Or photons can have small angles with this axis here. So, this is maybe the reason why this a spot of light becomes wider as it goes further. But even if you have a very directed radiation coming out from a laser, and there's nothing here like vacuum, we still have this effect. The reason of this is the Heisenberg's uncertainty principle which tells us that we cannot have a perfect measurement of both the momentum and coordinate of the particle. For the photons coming out of the laser, we perfectly know their momentum because we know their wavelength. So, we must have problems with the coordinates and for photons here when they reach this screen with two slits, we of course we know the x coordinate. So, we must have problems with y coordinate and z coordinates here. The consequences of this is that we actually don't know which slit the photon is going to choose. So, instead of classical system with two states, we have some probabilistic measurement outcome like with probability one half photon goes through the upper slit and probability of one half it goes through the lower slit. Now, it's very important to understand that photon never travels through both slits at once. It's always choose only one slit and only one detector. Here, only one with this one detector, or this zero detector works and detects the photon. Okay. They are not frightened and yet we go further. We remove one of the detectors. For example, this lower detector and we place another screen somewhere here, and this second screen can detect photons in any point over it. So, if you can continue this experiment, half approximately half of the photons will be stopped by this upper detector because they have chosen this upper slit and half of the photons will reach this second screen and give us a flash somewhere on this screen. We are going to count the flashes for each point we are going to draw a graph. So, for each point, how many flashes we have seen in this board, and the graph is going to be something like this. So, right in front of this second open slit, we will have some kind of maximum and some distribution of flashes which go to zero, then it goes further from this open slit. Now, what can we expect if we now remove this upper detector? The half of the photons were stopped by it, and so now, when there is no detector here, this half of the photons which we have chosen the upper slit and can reach the screen and add a sound flashes so we might expect that there will be, now the maximum here in front of the first of the upper slit and the number of flashes here will also increase. But in practice, in relative, we have a very different picture. We have an interference picture which is some maximum right between the slits and several mansions here and here. The interesting thing is that we are going to have this black spots or black rings. So, if we look at this screen from this point of view, from the front, this will be rings, rings of light and rings of dark spots. So, again, we had one detector here which stopped approximately half of the photons and another half of the photons was reached in this second screen and showed us this graph when we counted flashes. We allow another half of photons to reach the screen, and remember we fire one photon per minute, and instead of adding more light, we obtain black spots or black rings there were before was light. So, we don't have no more flashes here. This quite stretch and this black spots, black rings are the geometrical place of points where photons from different slits come in interface. So, if there were more than one photon at a time, more than one photon a minute, then we could explain this by this interference of different photons, but we cannot because we fire only one photon per minute and we are sure that every photon choose only one slit. In the places before we had this flashes we now don't have any flash. If you remember the law of big numbers, they can be sure that if you run this experiment long enough, and there will be a very long series where photons will choose, for example, only the lower slit. So, at least sometimes we have to see this picture. But we don't, we always see this interference.