Elementary single-qubit gates and their implementations using RF pulses

1 Elementary single-qubit gates and their implementations using RF pulses 

Single-qubit unitary operations play an important role in QIP. Using the nuclear spins as qubits, the most elementary single-qubit operations are those that perform a rotation of a single spin and can be represented by the following rotation operator  

It is obvious that rotations around the z-axis cannot be implemented by single RF pulses. A way of achieving that is introducing a resonance offset to shift the reference frame by an angle 9. Thus, all subsequent pulses and also the receiver will be seen in the new reference frame, which is equivalent to a rotation of 6 around the z-axis. Another way of implementing z-rotations is to make use of the property that allows us to write a rotation around z as rotations about the x or y axes. For example, a rotation of an angle 9 around the z-axis can be written as  

The answer is simple if we compare the matrix representation for the RF pulse (0) and the gate operator. For example, consider r-pulses around the x and y axis, respectively. The matrix representation of the corresponding pulse operators are  

Another fundamental single-qubit gate is the Hadamard gate. As it can be observed from Equation (4.2.7), a (f) pulse along the y-axis closely resembles a Hadamard matrix H. 

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