Hello everyone. Welcome to the second week of the course. During this week, we are going to learn the mathematical model of quantum computing, which will allow us to understand and design quantum algorithms without any background in quantum mechanics or of physical implementation of quantum hardware. Understanding of this week's material is crucially important for understanding all other models of the course. So, probably, this is the most significant week and the whole course. I strongly advise you to do all the exercises for this week, and if you don't understand something, you can always ask a question on the forum thread dedicated to this week. So, let's go. The most significant basic part of the whole theory of quantum computing is qubit, which is short for quantum bit, the minimal unit of information of quantum information. Just like this one bit describes the state of the simplest classical system like source engine here, one qubit describes the state of the minimal quantum system. As you remember in this source engine, one bit was recorded by a camera in which one molecular of gas resides, so the back camera. The left part, we call it 0, and the right part, we call it 1. Just the same one qubit of information can be encoded by polarization of a photon or the orbital of electron in the hydrogen atom or any other characteristic of simple quantum system. As you probably remember from school, electromagnetic waves are transverse waves, which means that the direction of the oscillation is orthogonal to the direction of the propagation. So, if wave propagates here, then oscillations can take place in this plane, which is orthogonal to the direction of propagation, and we can call the vertical oscillations. We can assign a value 1, and for the horizontal oscillations, we can assign a value 0 and voila. One photon now carries one bit of information. But you might notice that in this example here, the oscillations can take place not only vertically or horizontally but in any other possible direction on this plane. Moreover, these directions depend on buses that we choose in this space. So, remember we say that we assign 1 to the vertical oscillations. We first need to define what is vertical in this space, and you will be absolutely right saying this. But please note that for classical systems, we have exactly the same situation. For example, in the source engine here, this molecular can reside in any of these points. They have actually an almost number of possible states for this system. But we choose to distinguish only two of them because it is convenient to us, and we can measure these two states. In classical systems, this reducing of the number of distinguished states is called digitization, and we have a very similar notion for quantum systems, which we will discuss very soon. Now, it's time to give a strict mathematical definition of the term qubit. Qubit is a unitary vector in a two-dimensional Hilbert space. So, here is some Hilbert space. The number of dimensions is two. The length of vector Phi in it is 1. So, Phi is a qubit. Some of us may need to recall what is it a Hilbert space at all. Well, Hilbert space is a linear space over some field with a dot or inner product. In this course, we are going to discuss only the finite dimensional spaces over the field of complex numbers. So, if you have two vectors in some Hilbert space dimensionality n here draw them as columns. So, they both have n coordinates, which are complex numbers. Then the dot product or inner products, product of x and y, we will denote like this, and this will be this sum x_j, conjugated multiplied by y_j. This definition of the dot product satisfies all the conditions for these type of products because it has conjugate symmetry, positive definiteness and the linearity on the second argument. We have x Alpha y plus Beta z, where y and z are vectors and Alpha and Beta are some scalars. We have this Alpha xy plus Beta xz. Please note that sometimes the linearity on the first argument is needed for definition of the dot product. But here we will need this. Okay. Now, let's take a look at this notation here. This type of notation we will use to denote a vector and it is called ket and it is a vector column. So, it is a column. This is called bra and this is the transposit column. Let's take it for x again. This is the transpose column, so it is a row of conjugates, conjugated coordinates. Now, we can understand this notation for the dot-product. So here, we have bra for x and here we have ket for y and it is row of conjugates multiplied by a column and it is just a simple matrix multiplication which gives us this sum which gives us one scholar. So, just a number. Now, the advantage of having this dot-product of two vectors is the possibility to define the angles and the space. Probably remember that for Euclidean space, this expression, the dot-product of two vectors divided by their lengths is the cosine of angle between these two vectors. If you will try to define angles in the Hilbert space over their complex numbers, then it might happen then this dot-product is a complex number and we don't want complex cosines. So, we're going to place absolute value or modular here to define the cosine of an angle between two vectors. Since in Euclidean spaces view you don't have this absolute value, we could have there the negative cosines, so the angles in the Euclidean spaces vary from zero to b. Here, replace this absolute value to avoid complex cosines. Now, we also avoided the negative cosines. So, in our spaces here, all angles vary only from zero to pi divided by two. So, there's no obtuse angles there. Since we are going to discuss only the unitary vectors which have length one, they're going to remove this expression here. So, the cosine of the angle between two qubits will be the absolute value of dot-product of these qubits. Okay, this will allow us to define which angles are orthogonal. X is orthogonal to y if and only if dot-product of x and y is zero. Now, why we use only unitary vectors. Actually, for the state of a quantum system mathematically vertex corresponds and direction in the Hilbert pace or array. But, ray is infinite class of vectors which differ only with their length and manipulations of this infinite classes is not very convenient, so we choose a representative of each class. It's very natural to choose this representative to half the length one. Well, now we have this notion of the orthogonal vectors which allows us to define the orthonormal basis in this Hilbert space. Like this. Here, we have this vertical vector which we will call one and the horizontal vector which we call zero. The columns for this will be zero, one and one, zero and zero one. These vectors are of course, orthogonal. They're unitary, so they form the orthonormal basis in our space. They're depicted here on this picture. Actually, this picture is not good enough because each axis here represent a complex plane and to draw it right I would need to draw a four-dimensional picture and I wasn't provided with four-dimensional tablet yet. This unitary circle here is the all possible values of one qubit. So, you can see that one qubit can be not only these vectors, one and zero which we do not at least wait to make the connection with the values of classical bit. But, it can be any vector on this unitary circle like vector phi here which equals to alpha zero plus beta one. Alpha and beta both are complex numbers. So, we can see that qubit can have much more values than just ordinary bit. Actually, it can have any of these infinite number of values. This is what we already have seen before for the polarization of a photon since we discussed it, the vertical polarization, we can denote as one and the horizontal as zero. But, we can also have many other different polarizations like this and where it leads us, let's see in the next episode.
