6.8.1. Introduction and Basic Concepts

In this section of the study, we will examine four main topics. These topics are as follows:

1. What is Grover’s Algorithm? Starting with this question, we will explain the fundamental principles and operation of the algorithm.
2. How Does It Differ from Classical Search Algorithms? Under this heading, we will compare the advantages of Grover’s algorithm over classical search algorithms and in which situations it is more effective.
3. Quantum Parallelism and Grover’s Advantage: We will explain the concept of quantum parallelism and detail how Grover’s algorithm benefits from this feature, as well as the gains it provides.
4. The History and Significance of the Algorithm: We will conclude this section by highlighting the development of Grover’s algorithm from its inception to the present, its impact in this field, and its importance.

6.8.1.1. What is Grover’s Algorithm?

Grover’s Algorithm is a quantum algorithm developed for a specific search problem on a quantum computer, demonstrating quantum supremacy. This algorithm reduces a problem that can be solved on classical computers with O(N) complexity to O(√N) complexity on quantum computers, providing a significant acceleration. Grover’s Algorithm is typically introduced through the “database search for a target item” problem; however, it can more generally be used to find the input to a function that yields a specific output.

6.8.1.2. How Does It Differ from Classical Search Algorithms?

In classical computers, a linear search method is generally used to find a specific item in a dataset. In this method, items in the dataset are checked one by one until the target item is found. Therefore, in the worst case, finding the target item in a dataset with N items requires O(N) operations. However, Grover’s Algorithm can perform this process in O(√N) iterations thanks to quantum parallelism.

For example:

In the classical case: Searching a dataset of 1,000,000 items typically requires an average of 500,000 steps. The keyword here is “average”. If the element we are searching for is in the middle of the dataset, it could be found in 500,000 steps. In the worst case, it might take 1,000,000 steps because there are 1,000,000 elements. In the best case, it could be found in just 1 step.

With Grover’s Algorithm: To find the target item in the same dataset, only about 1,000 steps are needed. The number 1,000 here represents the worst-case scenario. Assume the item we are searching for is the last one in a list of 1,000,000 items. Because Grover’s Algorithm provides O(√N) complexity, it can find the item in O(√1 000 000 =1,000 steps.

6.8.1.3. Quantum Parallelism and Its Advantage

Quantum parallelism is a fundamental property of quantum computers and a critical concept in understanding the power of Grover’s Algorithm. Quantum computers can represent many different states at once thanks to superposition, and process these states in parallel. Grover’s Algorithm uses this feature to accelerate the search process. We have discussed the topic of superposition in detail in many contexts. We will not dwell on this topic here but will define it simply.

• Superposition: Quantum bits (qubits) can be in both 0 and 1 states simultaneously. This is very important in quantum computing and plays a key role.
• Grover’s Advantage: The algorithm also uses phase transitions and amplification techniques along with superposition to increase the probability of the correct answer and suppress unnecessary probabilities.

These topics of phase transitions and amplification have been discussed in detail in previous studies. Since these two topics are key components of the algorithm’s efficiency and speed, we can briefly discuss them again here.

a. Phase Transformations

One of the key components of Grover’s algorithm is a function called the “Oracle”, which identifies the searched item. The Oracle applies a specific phase transformation to the quantum state corresponding to the searched item. This is generally achieved by adding a -1 phase factor to the searched state, effectively flipping the amplitude of that state. This process serves to distinguish the searched item’s quantum state from all others “by phase”.

After passing through the Oracle, we return to a starting state where all amplitudes are equally superposed. However, at this stage, the phase of the searched item has been altered relative to the others.

b. Amplitude Amplification

After the Oracle process is completed, the amplitude amplification step is applied. This step consists of two phases:

1. Phase Inversion: First, an average of all states is taken, and a phase inversion is performed around this average. This operation is generally done using a function known as the “diffusion operator”, which transforms the amplitudes of all states. In practice, this process subtracts amplitudes from their average and then adds this result back to the average. This significantly increases the amplitude of the searched item while decreasing the others.
2. Amplitude Increase: As a result of phase inversion, as the amplitude of the searched state increases, so does the probability of the algorithm finding this state in the next measurement. This step continuously increases the amplitude of the correct answer while decreasing the amplitudes of incorrect ones.

These two processes, dependent on the iterative application of the algorithm, step-by-step maximize the measurement probability of the searched item. Each iteration enhances the amplitudes further, thus increasing the likelihood of finding the correct item. The power of Grover’s algorithm stems from the combined effect of these two processes, making it an effective search tool for large datasets.