6.8.3. Grover’s Algorithm Working Principle

6.8.3.1. Problem Definition

Grover’s algorithm is one of the advantages quantum computers have over classical computers in solving search problems, specifically used for quickly finding a particular item in a database. This algorithm alone provides a significant speedup over classical algorithms.

A classical computer must check each item one by one in a database of \(N\) items to find the correct item. If the items are unsorted, it would take, in the worst case, \(N\) steps to find the correct item, meaning the classical search algorithm has a time complexity of \(O(N)\). Grover’s algorithm operates with square root time complexity, i.e., \(O(\sqrt{N})\), to find the correct item, which is much faster than the classical search method.

Unlike classical computers, quantum computers can process multiple states in parallel using the property of superposition. The key quantum techniques in Grover’s algorithm are superposition and phase transformation, allowing all items in the database to be queried simultaneously.

6.8.3.2. Quantum Circuit Design

We have discussed these many times up to this section of the study, and we will briefly discuss them again here.

Grover’s algorithm implementation involves three main steps: (1) Superposition Creation, (2) Oracle Application, and (3) Diffusion Operator (also known as the Grover Operator).

• Superposition Creation: The algorithm creates a quantum superposition state representing all \(N\) states with equal probability. This is achieved using Hadamard gates.
• Oracle Application: The Oracle is a quantum subcircuit designed to mark the correct item. The Oracle applies a phase shift to the state to be marked. Oracle function: \[ O |x\rangle = (-1)^{f(x)} |x\rangle \] Here, a negative phase is applied to the states where \(f(x) = 1\).
• Diffusion Operator (Grover Operator): After the phase shift, a diffusion operator that redistributes the amplitudes within the system is applied. This increases the amplitude of the correct state while decreasing the amplitudes of the others. Diffusion process: \[ \mathbf{D} = 2 |\psi\rangle \langle \psi| – \mathbf{I} \] This operator performs a reflection about the average amplitude.