Quantum Computing 

PauliX,Y,Z
The Pauli gates are a set of onequbit operations that play a fundamental role in the manipulation of quantum states. There are three Pauli gates, represented by the Pauli matrices: X, Y, and Z. These gates are essential building blocks for more complex quantum circuits and algorithms. They allow for state manipulation and provide the foundation for the implementation of larger quantum operations. There are three gates that belong to Pauli gates :
NOTE : flip vs phase flip
In the context of quantum computing, a flip and a phase flip refer to different types of transformations applied to qubit states. Flip (Bitflip): A flip, also known as a bitflip, is a transformation that changes the state of a qubit from 0> to 1> or from 1> to 0>, just like a classical NOT gate changes the state of a classical bit from 0 to 1 or 1 to 0. The PauliX gate performs this flip operation on qubits. When applied to a qubit, the PauliX gate essentially swaps the amplitudes of the 0> and 1> states. For example:
X 0> = 1> X 1> = 0>
Phase flip: A phase flip is a transformation that changes the relative phase of the 1> state while leaving the 0> state unchanged. In this case, only the phase of the qubit state changes, not the actual basis states themselves. The PauliZ gate is responsible for phase flips. When applied to a qubit, the PauliZ gate changes the sign of the amplitude of the 1> state, while the amplitude of the 0> state remains the same. For example:
Z 0> = 0> Z 1> = 1>
Following is the type of flip operation for X, Y, Z gate
The PauliX gate is also known as the bitflip gate and works on a single qubit. PauliX gate acts like a "quantum NOT gate." A classical NOT gate inverts a binary bit, changing 0 to 1 and 1 to 0. Similarly, the PauliX gate flips a qubit's state, switching 0⟩ to 1⟩ and 1⟩ to 0⟩. Here's an example: If you have a qubit in the state 0⟩ and apply the PauliX gate, the qubit's state changes to 1>. If you have a qubit in the state 1⟩ and apply the PauliX gate, the qubit's state changes to 0>. However, since qubits can exist in superpositions, the PauliX gate can also transform states that are combinations of 0> and 1>. For example, consider a qubit in the following superposition state: ψ> = a0> + b1>, where a and b are complex numbers representing the amplitudes of the qubit states. When the PauliX gate is applied to this qubit, the result will be: Xψ> = a1> + b0> As you can see, the PauliX gate has effectively swapped the amplitudes of the 0> and 1> states.
In symbol and mathematical form, it is presented as follows. It takes 1 bit as input and return 1 bit as output. If you look into the matrix form, you would notice that it takes 2x1 vector (representing 1 qubit) and swap the elements within the vector.
If I you plug in the state vector for 0> and 1> into X gate matrix equation, you can get the output as follows. You would see that the element in the statevector get swaped by the gate X matrix.
The PauliY gate is another fundamental operation in quantum computing, acting on a single qubit. The PauliY gate is a combination of the PauliX and PauliZ gates, meaning it applies both a bitflip and a phaseflip to the qubit. It mean that the PauliY gate not only changes the state of the qubit (like the PauliX gate), but it also changes the relative phase between the states 0> and 1> (like the PauliZ gate).
For example, if you have a qubit in the state 0> and apply the PauliY gate, the qubit's state changes to the state i1> , where i is the imaginary unit (i^2 = 1). If you have a qubit in the state 1⟩ and apply the PauliY gate, the qubit's state changes to the state i0>.
However, since qubits can exist in superpositions, the PauliY gate can also transform states that are combinations of 0> and 1>. For example, consider a qubit in the following superposition state: ψ> = a0> + b1>, where a and b are complex numbers representing the amplitudes of the qubit states. When the PauliY gate is applied to this qubit, the result will be: Yψ> = i b0> + i a1> As you can see, the PauliY gate has not only swapped the amplitudes of the 0> and 1> states but also added a relative phase of i to the resulting states.
In symbol and mathematical form, it is presented as follows. It takes 1 bit as input and return 1 bit as output. If you look into the matrix form, you would notice that it takes 2x1 vector (representing 1 qubit) and swap the elements within the vector.
If I you plug in the state vector for 0> and 1> into Y gate matrix equation, you can get the output as follows. You would see that the element in the statevector get swaped by the gate Y matrix.
The PauliZ gate is also known as the phaseflip gate working on a single qubit. Unlike the PauliX gate, which flips the state of a qubit (acting like a "quantum NOT gate"), the PauliZ gate changes the relative phase between the states 0> and 1>, without changing the basis states themselves. In layman's terms, the PauliZ gate modifies the "phase" of the qubit's state, which is an essential aspect of quantum mechanics.
For example, if you have a qubit in the state 0> and apply the PauliZ gate, the qubit's state remains unchanged, staying in the state 0>. If you have a qubit in the state 1> and apply the PauliZ gate, the qubit's state changes to the state 1>, which means the relative phase between the 0> and 1> states has been inverted.
Since qubits can exist in superpositions, the PauliZ gate can also transform states that are combinations of 0> and 1>. For example, consider a qubit in the following superposition state: ψ> = a0> + b1>, where a and b are complex numbers representing the amplitudes of the qubit states. When the PauliZ gate is applied to this qubit, the result will be: Zψ> = a0⟩  b1> As you can see, the PauliZ gate has changed the sign of the amplitude of the 1> state, while the amplitude of the 0⟩ state remains the same.
In symbol and mathematical form, it is presented as follows. It takes 1 bit as input and return 1 bit as output. If you look into the matrix form, you would notice that it takes 2x1 vector (representing 1 qubit) and swap the elements within the vector.
If I you plug in the state vector for 0> and 1> into Z gate matrix equation, you can get the output as follows. You would see that the element in the statevector get swaped by the gate Z matrix.

