[MUSIC] Once we have identified the errors that we want to fix, we can move on to creating a quantum code that is resistant to them. As we found out the theory of quantum error correction codes, deals with the construction of code that are resistant to errors of a certain kind of from the subgroup E. In all such code, the key qubits that one wants to protect are encoded by M qubits. Moreover, the number of coding qubit is strictly greater than the number of qubits and we need that to be protected. As a result, one logical cubit is specified by several physical qubits. With such coding information exists as a redundant form and it is difficult to destroy it. To construct the theory of quantum error correction codes, we need to highlight the basic properties of the element of the incubate pauli group Pn. Each element of the group as a unitary. Indeed, since, well, the operators are unitary, the terms of product is also a unitary. Any square of the operator from n qubit pauli group n is equal to plus or minus the interpreter. Indeed, this property is obviously bear in mind that the one qubit pauli operator squared is equal to plus or minus the interpreter. In the two elements M and N from Pn either commute or anti commute. Again, these properties is to verify, were an amounted computation relations for one qubit power operators. How do we have identified these properties? Were then construct our code. To do this, let us consider some subgroup S as of the group Pn, which consists of operations community pay rise with each other. In other words, let us single out the operators that commuters each other. From the general properties of quantum mechanics were now the operators have a common set of ion state, if and only if they commute with each other. It follows from this that the operators from our children subgroup S have common eigenstate. This data set, to define a common proper space which is a subspace of an anti dimensional Hilbert space. Let's call this space HS. This space is called code space. It turns out that the basis waiter of given space are code word,s that are protected from errors. With the help of this basic states, information must be encoded, so that is resistant to certain errors. Later on, we will understand to what errors the code words of given space our resistance. However, for for now, let's talk a little more about the properties of results in space. If the group S, which we have selected have n-k generators. Than the space HS has dimension to the power of K. Generators of a group, are operators that can be used to express any elements of a given group. Here we will not dwell on the proof of this fact. It is only important for us to understand that key qubits can be encoded in the space as HS, we have obtained. Also, it is important to know that the operators of the group S are stabilizers of the basic states of the space HS. For this reason, it is argued that this code is called the stabilizing code. It's now important to understand that the space we have constructed is actually formed by quote words. And we need to figure out what errors this code can protect against. It is accurate that the code we have constructed is capable of correcting errors. Whose operators anti commute with operators from the group S. Let's shake it out. To do this take the operator capitalized EA. That can anti commute with operator from the set S. In this case, we can act by the operator ME. On the against that side of the space HS. Using the property of anti community activity of operators, we can see that the state EA ,site is an eigenstate of the operator M. In this case the eigenvalue of this state is equal to -1. This means that when we measure the operator M will be able to determine whether the operator EA ,acts on the state side or not. Thus, Ea is an error operator, then by measuring the operator M. We can determine whether the encoded stateside is affected by error or not. Summarizing the above, we can conclude that the code we have constructed can detect errors whose operators, anti compute with operators from the group S. Therefore, when constructing a quantum error correction code, we must proceed as follows. First, set these error operators that we want to correct. Then from all possible operator of the polar group, choose those that compute with each other and at the same time, intercumpute with the error operators. It remains for us to understand how many qubic should be sufficient to encode a state that is resistant to one or another error. We remember that in the classic example where three bit, we could protect information from errors in one bit. If the error occurrence two bits, then the encoded information can no longer be distinguished. It follows from this that in other €4 to not irrevocably spoils information, they should not not great desert our code words. Let's understand what this means. That is to say, let's define the criteria and that code words must satisfy. It is logical to assume that strong distortion occurs when one code word due to errors can be confused with another code word. In the classical example state 000, is affected by two bit error on the second and serve it than transitions state 011. The same stage is obtained if the state, 111 is affected by one error on the first bit. Thus errors should translate different code words into different states. In the quantum case it should be the same. Two different code states under the action of an error must change the different states. Mathematical this condition can be written in the form of a necessary sufficient condition presented on the slide. Here Jane i have two code words that belongs to the space HS. E A and EB are two different errors that we want to correct. Let's now look at a few examples. As the first example of constructing quantum error correction codes. Let's consider the case in which three physical qubits are eastern code, one logical qubit. Will consider qubit flip arrows. This subgroup of errors is determined by the set of XII, IXI and IIX. Selection from the three qubit pair of the group all operators of which commute with each other and the commute with the operator from E. We obtain a set of stabilizers S. This set consists of the operators ZZ, II,ZZ,IZ. It is not difficult to see that such a group is set by two generators ZZI,IZZ. This code defines one logical qubit with eigenstate, 0,L and1,L. This code is a quantum analogy of the classic Three Qubit Code. At the second example considers the construction of a quantum code resistance to Phase Errors. As we have already said, the phase error is determined by the operators that for the recording one logical qubit will use three physical ones. In this case an error will occur in one physical qubit. Furthermore errors are specified by a set of E1. Using these operators one can select the S1 group, the generator of which are shown on the slide. These generators defined the space with basis code words. This code words defined the state of the phase error protected qubit. The codes described here are called three cubits codes. These codes can corrupt either one qubit flip error or one qubit phase error. Unfortunately three qubits codes cannot correct any one qubit errors. This raises the question or determine how many qubits are required to construct a universal code. That protects against any single qubit errors. Or more generally, determine how many qubits are needed to correct any errors with a certain weight. To answer this question, we need to understand how many errors in general exist with a weight equal to T. If we have unphysical qubits and T errors occurring them, then the number of possible errors is given by the expression chi of T, as shown on the slide. It follows from that there is a total number of errors with the weight not exceeding T is given by the some end of T. The value M equal to 0 corresponds to the code where no errors occurred. Suppose now that we have K coded qubits. This means that with the help of three of these qubits it is possible to encode 2 to the power of K basis states. If you want to correct any errors with the weight not higher than T, then we must have a quantum space. It must accommodate all states N of T multiplied by 2 to the power K. This follows from the fact that all states modified by errors must transition to unique states. The condition for the capacity of all of the vectors with errors is set by one expression shown on the slide. Here 2 the power of n is the dimension of a Hilbert space which contains all vectors with errors. This condition is called Hamming Quantum Bound. Let's explore this boundary in the case of encoding a single qubit which is resistant to anyone qubit errors. That is for K equal to 1 and T equal to 1. In this case we're getting inequality for the number of qubits needed to encode states. The Senate quality is fulfilled. Starting from the end equals to 5. Very still correct any one qubit errors, we need to use a quantum code that is encoded with five or more qubits. In particular, the nine qubit code protects against both phase and inversion errors. The code voice this code as shown on the slide. Now we'll reach the end of this section. Let's once again talk about the algorithm of actions when constructing the quantum error correction code. At the first stage, one needs to decide on the errors that can occur in our quantum system. Whether there will be any specific errors or any errors with the weight not exceeding T. Next based on the Hamming boundary, one needs to understand how many physical qubits are enough to encode a logical qubit that is resistant to these errors. After that one needs to match operators to all errors. At the last stage, it is necessary to choose operator from the power group that compute with each other and anti commute with error operators. This operators defined the space of error resistance going words. Now that the method for construction stabilizing codes presented here is common for a unitary errors. Next will consider the application of the described methods for construction correction codes to continuous variable one week quantum computation.