Elementary two-qubit gates and their implementation in NMR

As discussed in Chapter 3, two-qubit gates such as CNOT and Hadamard are fundamental for quantum information processing any experimental method that aims to be used as 

Spin with different spins effectively equal. For example, the pulse sequence corresponding to the operation bellow can be applied to a four spin system (lets say spins 1,2, 3 and 4) to make the effective coupling between all spins equal after a period of [18], 

let us return to the implementation of two-qubit gates. In Chapter 3 we saw that the action of the CNOT gate is invert one of the qubits (the target qubit) provided the other (the control qubit) is in the state 11) . In a two-qubit AB) system this is accomplished by following operators  

Note that these operators are not exactly equal to CNOT operators, but they act as CNOT gates for most of two qubit states. 

A two qubit-gate very used in quantum algorithms is the SWAP gate. It can be directly implemented by the pulses corresponding to three successive CNOT gates. 

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