6.8.2.3. Phase Transformation

Phase transformation is a critical component of Grover’s algorithm and is used to mark the correct solution. This process is particularly related to the properties of superposition and interference found in quantum computers. The Oracle uses phase transformation to take advantage of these properties to find the correct solution.

The purpose of the phase transformation is to invert the amplitude (probability) of the correct solution. That is, the state representing the correct solution, \(|w\rangle\), is multiplied by a negative phase, while all other states remain unchanged.

6.8.2.3.1. Why Do We Perform Phase Transformation?

The reason for using phase transformation is to make the correct solution more likely by leveraging superposition and interference. In classical search algorithms, the probability of finding the correct solution is equal in every attempt. However, in Grover’s algorithm, the amplitude of the correct solution is turned negative by the phase transformation, which leads to a higher probability of being “elevated” during the amplitude amplification process.

6.8.2.3.2. Geometric Interpretation

Each operation used in Grover’s algorithm represents geometric movements among quantum states. To better understand this situation, we can examine what kind of movement occurs in the Hilbert space (the space of all probabilities in a quantum computer).

The initial state \(|\psi\rangle\) is a superposition representing all possibilities with equal probabilities and is a vector in an N-dimensional Hilbert space. This vector has the same amplitude for each possible solution and exists together (equally likely).

The Oracle operation turns the amplitude of the correct solution state, \(|w\rangle\), negative. Geometrically, this means rotating the vector of the correct solution into a negative direction (half-space). In other words, the vector where the correct solution is located moves in the opposite direction due to a transformation in space.

The geometric reflection of this process is as follows:

• Initial State: The initial state is represented as a vector in Hilbert space. Since all probabilities are represented with equal likelihood, this vector is evenly distributed in an N-dimensional space.
• After the Oracle Operation: The Oracle operation only affects the state \(|w\rangle\) of the correct solution and multiplies this state by a negative phase.
• This transformation pulls the vector of the correct solution into negative directions in Hilbert space.

6.8.2.3.3.The Effect of Phase Transformation

The Oracle operation, by making the phase of the correct solution negative, distinguishes it from other solutions. This condition will be utilized interactively during the amplitude amplification process (diffusion operator) to make the correct solution more likely. That is, after the correct solution \(|w\rangle\) is marked with a negative phase transformation, the amplitude amplification operator will increase the probability of this solution.

Amplitude amplification changes the amplitudes of all states, but the amplitude of the correct solution will increase more significantly. The Oracle operation and the diffusion operator work together to increase the likelihood of finding the correct solution more effectively.

In summary:

Phase transformation is an operator used in Grover’s algorithm to mark the correct solution by making its amplitude negative. The Oracle operation marks the correct solution in this manner while leaving the amplitudes of other states unchanged. This creates an interference effect that increases the likelihood of the correct solution being chosen. Geometrically, multiplying the correct solution vector by a negative phase displaces it in the opposite direction in Hilbert space, and the amplitude amplification process makes the correct solution more likely.