6.4.4 Quantum Computing Feature

Up to this point, we have already applied the Hadamard operator to put the qubits into a superposition state and then applied the (U_f) function. After the ( U_f) operation is completed, the Hadamard gate will be applied to the first qubit again. This second application of the Hadamard is a particularly important step in quantum computer algorithms. We will discuss this in detail here.

In this section, we will discuss in detail the second application of the Hadamard gate and how this application extracts information. First, let us remember that this process is fundamentally aimed at determining whether the underlying function is constant or balanced. Let us first summarize the operations we performed here.

In the first step, during the initial phase of the algorithm application on a quantum circuit, the Hadamard operator is applied to the first qubit. This operator creates an equal probability mixture of the (|0⟩) and (|1⟩) states by putting any qubit state into superposition.

In the second step, a specific function ( U_f ) is used. When applied to a qubit, this function produces a specific output depending on the input qubit state. The ( U_f ) function, for example, adds the value ( f(x) ) to the operation qubit (as a modulo 2 sum). This operation affects the superposition on the first qubit based on the values of ( f(0) ) and ( f(1) ).

In the third step, which is the main part of the discussion, we will apply the Hadamard operator for the second time. After applying the ( U_f ) function, the Hadamard operator is applied again to the first qubit. This second application of the Hadamard allows us to extract information by comparing the values ( f(0) ) and ( f(1) ). This second application of the Hadamard puts the qubits back into superposition, resulting in the following situation:

• If ( f(0) = f(1) ), meaning the function is constant, the process ends in the (|0⟩) state. This means ( f(0) ⊕ f(1) = 0 ).
• If ( f(0) ≠ f(1) ), meaning the function is balanced, the process ends in the (|1⟩) state. This means ( f(0) ⊕ f(1) = 1 ).

In the final step, only the first qubit is measured. The measurement result shows whether the function is constant or balanced:

• If the (|0⟩) state is measured, it is concluded that the function is constant.
• If the (|1⟩) state is measured, it is concluded that the function is balanced.

This process utilizes interference effects, allowing quantum computing to determine the nature of a function with a single measurement, unlike classical computing. This is an indication of how quantum algorithms can potentially solve certain problems much faster.