[MUSIC] This week we will talk about continuous variable one way quantum computation. And then the discrete case, the entire computation process is divided into several stages. The first stage prepares a cluster state. The second stage mixes physical systems encoded in input states with some of the cluster nodes. The final stage involves local measurements of the resulting state. And the correction of unmeasured nodes of cluster states, taking into account the measurement results. Thus, as an indiscreet case, the main resource for one-way computations is the continuous variable or multiple tied to entangled cluster state. But now in continuous variables. To create such states, physical system described by continuous variables are used. The main question now, is what states such systems should have in order to use them to generate cluster states? In addition, it is necessary to understand how the entanglement process takes place. In other words, we need to understand what operations are needed to turn independent systems into a single cluster state. Before moving on to the definition of the continuous variable cluster states, let us remember how cluster states are created on qubits. And the first stage we prepare n qubits and the plus state, is of which we associate with a node of the graph G. The stabilizer of such a qubit state is a Pauli operator X. Furthermore, 2 qubits cPhase operations are applied to players of these qubits, so that the state corresponding to a certain graph G is obtained. To construct a continuous variable cluster state, we need to act similarly. First, it is necessary to select physical systems in certain status, and then apply a two multi operation to them, which will entangle them in pairs. To understand what these states should be and by what operations they should be entangled, let's look at the auxiliary operators X(s) and Z(s). These operators are called Heisenberg-Weyl operators and are analogs of Pauli operators in the infinite dimensional case. The introduced operators as well as the Pauli operators do not commute with each other. The eigenvectors of the Heisenberg-Weyl operators confute with the eigenvectors of the quadrature operators. This follows from the fact that these operators are functions of quadrature operators. In addition, using the relation between the basic states q and p, one can show that the continuous and aux of the Pauli operators, acts as displacement operators on the conjugate states. Employing the introduced continuous Pauli operators, it is possible to determine the state of the physical systems, which we should use for cluster state generation. To do this, we need to find the state for which the operator X(s) is a stabilizer. This state is the eigenstate of the y quadrature operator with 0 eigenvalue. In this state, the variance of the y quadrature is 0, since the state is strictly defined by the y quadrature. The variance of the x quadrature in this state is equal to infinity. On the phase plane, such a state is depicted as a line that has a 0 y coordinate and an infinite extension along the x coordinate. This means that this state is a welcome state infinitely squeezed in y quadrature. Unfortunately, such a state cannot be realized in practice since such a process requires infinite energy. In real conditions, only squeezed states with the finite squeeze in degree can be used. This imposes a limitation on cluster state and the computation implemented with their help. These limitations will be discussed in the next lesson of this work. But now, for simplicity sake, we can consider infinitely squeezed states as building blocks for generating cluster states and continuous variables. In the next stage of cluster state generation, we must determine what transformations were used to entangle the squeezed state with each other. To do this, we also need to define the transformation that is analog to the cPhase transformation. Such a transformation is two more transformation CZ. The Hamilton along which is proportional to the product of the ith and jth x quadrature. As we have already mentioned, this transform is a Gaussian transform. It adds the x quadrature of one oscillator to y quadrature of the other. The x quadrature are not transformed by CZ transformation. It performs an entanglement of two quantum physical systems in x and y quadratures. The y quadrature of the two oscillators are the target, and the x quadratures are controlling. Moreover, with this, the operator CZ contains a constant G ij, which depends on the interaction time of the system. And also on the coupling constants in the Hamiltonian. This constant directly affect the entanglement process. As we can see, this constant in present in the output y quadrature as a multiplier. As a result of the sequential application of CZ operators to pairs of squeezed oscillators, we're going to need to obtain a cluster state correspondent to a certain graph G. It is important to know that any two CZ operators commute with each other. This means that the nodes of the cluster state can be entangled in arbitrary order. In addition, each CZ operator contains the constant G ij, which as we have already noted, affects the entanglement process. This leads to the fact that clustered states in continuous variables will have a weighted graph. That is to say, a graph that has a constant G ij corresponding to each edge, which is called the weight coefficient of the graph. In other words, the weighted graph of a cluster state in continuous variables, reflect not only the fact of entanglement, but also which specific operator has created this entanglement. Such weights of cluster states are an additional degree of freedoms and can be controlled when generating a cluster state. And we will show later, this weights coefficient also affect the process of one-way quantum computation. Thus, considering all of the above, it is possible to formally define cluster state in continuous variables as a state with a certain weight function G. After we have defined the continuous variable cluster state, let's look at a few examples. Firstly, let's consider the simplest cluster state consistent of two-nodes. In this case, we will assume that the graph of a given cluster state is weighted, with the weighting coefficient equal to j. To create such a cluster state, we must associate a squeezed welcome state with each node with an entangled W and certain CZ operator. To describe this entanglement, let's rewrite the states as the first system in terms of the eigenstates of the x quadratures. To do this, we use the Fourier transform relating to conjugate basis. With such a decomposition, we can write an expression for the two-node cluster state in continuous variables. Here we have used the definition of the CZ transformation. From the presented expression, we can see that an entangled state is obtained. Moreover, we'd say that the resulting state depends explicitly on the weight coefficient, g. Let us now consider a three-node cluster state with a weight linear graph determined by the adjacency matrix A3. The weighted coefficients for this state will be g 12 and g 23. To get this state, we need to take three squeeze systems and entangle the first with the second, and the second with the third using the CZ transformation. To write down the wave function of this cluster state, it is easiest to rewrite the state of the second system in terms of the basis of eigenstates of the x quadrature operator. Having such a decomposition, it is not difficult to obtain the wave function G3 of the cluster state. In the obtained state, one can see the presence of correlations between the system as well as the dependence on the weight coefficients.