Learning fundamental single qubit quantum gates

Various quantum gates exist for a single qubit. They are integral to the quantum circuits that make up quantum applications. There are many manuals, but I created my own material that lists the functions of basic quantum gates. This will be of great help when creating the necessary quantum circuits. This material was created with reference to the quantum simulator Qni [1], which has an excellent GUI.

First, Fig. 1 is a compact summary of the basic quantum gate functionality. This figure is a collection of descriptions of quantum gates in Qni. The results of applying quantum gates to qubit initial vectors "|0>" and "|1>" are shown. In other words, it shows what kind of super-positioning state is created by "|0>" and "|1>". 

Next, Fig.2 is a hand-made model of Bloch sphere. Using φ and θ corresponding to the state vector of the qubit shown in this figure, Fig.3 and Fig.4 show the working of the quantum gate in more detail. These figures display the complex coefficients (ie, probability magnitude and phase) that indicate the quantum superposition after application of quantum gates.

 

There is something worth noting here. For example, for the quantum state vector "|0>", both Pauli X and Pauli Y gates result in "|1>" with 100% probability. However, their amplitudes and phases are different. The phase is zero for Pauli X, but π/2 for Pauli Y. Such a phase difference is important later in causing wave interference. Moreover, the application result of the Hadamard gate is characterized by being "|0>" or "|1>" with a probability of 50% each. The probability (in this case 50%) is proportional to square of amplitude.

Reference

[1] Qni Tutorial: https://qniapp.net