This article discusses the four most common types of quantum walks used in quantum algorithms and computing, along with their applications.

# 4 Most Common Types of Quantum Walks

Quantum Walks is a mathematical concept that has emerged as a crucial building block in quantum algorithms, quantum computing, and quantum information theory. It is a quantum version of classical random walks, which allows a quantum particle to move from one vertex to another in a graph. There are several types of quantum walks, but we will discuss the four most common types in this article.

## 1. Coined Quantum Walks

Coined Quantum Walks were introduced in 1993 and are the simplest form of quantum walks. They are similar to classical random walks with the exception that a coin flip is included to determine the direction in which the walker moves. In a Coined Quantum Walk, the walker is in a superposition of states, each of which corresponds to a vertex in the graph. The walker then applies a quantum coin operation, which is a unitary operator acting on the coin state, followed by a conditional shift operation based on the outcome of the coin flip. Coined Quantum Walks are used to explore the graph structure efficiently, and it has been shown that they provide exponential speedup in certain algorithms.

## 2. Discrete-Time Quantum Walks

Discrete-Time Quantum Walks (DTQW) were first introduced in 2001 as a natural extension of Coined Quantum Walks. DTQW is a unitary operator that alternates between a coin flip and a shift operation. The coin operator is typically a two-dimensional unitary matrix, and the shift operation is a conditional operation that moves the particle from the current vertex to one of its neighboring vertices based on the coin state. DTQW is a powerful tool for exploring graph properties and is used in various quantum algorithms, including the element distinctness problem and the graph isomorphism problem.

## 3. Continuous-Time Quantum Walks

Continuous-Time Quantum Walks (CTQW) were first introduced in 2003 as a natural generalization of classical random walks in continuous time. In CTQW, the particle moves continuously, and the coin operator is replaced by a time-dependent Hamiltonian. The Hamiltonian describes the evolution of the particle over time and governs the probability