Introduction to Grover's Algorithm

Another algorithm that has an important place among quantum algorithms is the Grover algorithm. It was developed by the Indian physicist Lov Grover in 1997.

I described the Deutsch-Jozsa algorithm algorithm in my previous articles. Although the Deutsch-Jozsa algorithm helps us find the characteristics of the functions and gives us good information about the working principles of the quantum algorithm, it has no application to a real physical problem.

Grover's algorithm allows us to solve real physical problems.

In this article, I will give information about the working principle of the Grover algorithm and its application as an example.

Let's imagine two cities, A and B!

Suppose there is more than one way to get from A to B. What we want is to find the shortest path among the existing path options.

When we want to solve this problem through a classical algorithm, it will deal with each option one by one with iterations and reach the result. However, Grover's algorithm will do this with iteration. Thanks to this situation, we will reach the desired result in a shorter and faster way.

Grover's algorithm passes the problem through two stages while doing this.

Stage 1: It evaluates how many qubits we have and makes the amplitude of the probable result negative, but leaves the amplitudes of the qubits with no real solution positive.

Stage 2: It will increase the amplitude of the qubit that it considers as its real solution, that is, the qubit that has negative amplitude, and will make it positive again. The amplitudes of the other qubits will be shorter than in their original form.

Let's take a look at the details of this algorithm.

Considering our example again, let's have x₁, x₂, …… x*, .…..x_N path options between cities A and B, and x* represents the shortest path.

Performing step 1 first will make the actual solution qubit x* change, which seems possible.

Our operator that will change the amplitude of the x* solution is as follows;

Now let's look at how to increase the amplitude of the x* qubit.

The operator to increase the size of the x* amplitude is given below.

In this way, the amplitude of x* is increased and the amplitudes of other qubits are shortened. The Grover algorithm can perform this process with as many iterations and reach the solution in a short time.

Reference

https://www.youtube.com/watch?v=jhmaI4Hm444&list=PLqNc_xpYGu775P7iJA7Kvfxmv_Fm9wwHj&index=4

https://quantum-computing.ibm.com/composer/docs/iqx/guide/grovers-algorithm

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