Quantum walks (QWs) are the quantum analogue to classical random walks. We explore various implementations of QWs and create visualizations for each, with the goal of increasing educational accessibility. A new QW, called sticky walk, is introduced, where the walker sticks to targets once found. Corresponding pseudocode and Qiskit implementations are provided. Sticky walk helps bridge the gap between non-searching QWs and Quantum Walk Search (which may be an unintuitive search algorithm for those new to QWs).Quantum walks (QWs) are the quantum analogue to classical random walks. We explore various implementations of QWs and create visualizations for each, with the goal of increasing educational accessibility. A new QW, called sticky walk, is introduced, where the walker sticks to targets once found. Corresponding pseudocode and Qiskit implementations are provided. Sticky walk helps bridge the gap between non-searching QWs and Quantum Walk Search (which may be an unintuitive search algorithm for those new to QWs).

IEEE International Conference on Quantum Computing and Engineering (QCE)Sep 1, 20223 citations

Quantum walk search algorithm based on phase matching and circuit cmplementation

Hanwu Chen, Ke Li, Shengmei Zhao

As a new quantum computing model, quantum walk has been widely studied in recent years. It consists of discrete time quantum walk and continuous time quantum walk. Discrete time quantum walk includes coin quantum walk and scattering quantum walk. Meanwhile as a quantum search algorithm, Grover algorithm can search an unsorted database in time complexity of O(√N) . Recent years quantum walk has been used to solve search problems and some of them have been proved to be as efficient as Grover algorithm. Making full use of the novel properties of quantum walk, the quantum walk search algorithms on the 2D grid and hypercube have been proposed. Inspired by phase matching condition of Grover algorithm, we propose a new quantum walk search algorithm which is based on coin quantum walk. Firstly we give the graph to the quantum walk, and then describe the algorithm in detail. Algorithm uses different coin operators and shift operators for two different cases and draws the corresponding iteration operators. Then we prove that iteration operators used in the algorithm are unitary operators. Then we analyze the time complexity and probability of success of the algorithm. Analysis indicates that the time complexity of the algorithm is the same as that of Grover algorithm, however when the targets to be searched are more than 1/3 of the total targets, the algorithm probability of success is greater than that of Grover algorithm. Finally we give the circuit implementation of the algorithm.As a new quantum computing model, quantum walk has been widely studied in recent years. It consists of discrete time quantum walk and continuous time quantum walk. Discrete time quantum walk includes coin quantum walk and scattering quantum walk. Meanwhile as a quantum search algorithm, Grover algorithm can search an unsorted database in time complexity of O(√N) . Recent years quantum walk has been used to solve search problems and some of them have been proved to be as efficient as Grover algorithm. Making full use of the novel properties of quantum walk, the quantum walk search algorithms on the 2D grid and hypercube have been proposed. Inspired by phase matching condition of Grover algorithm, we propose a new quantum walk search algorithm which is based on coin quantum walk. Firstly we give the graph to the quantum walk, and then describe the algorithm in detail. Algorithm uses different coin operators and shift operators for two different cases and draws the corresponding iteration operators. Then we prove that iteration operators used in the algorithm are unitary operators. Then we analyze the time complexity and probability of success of the algorithm. Analysis indicates that the time complexity of the algorithm is the same as that of Grover algorithm, however when the targets to be searched are more than 1/3 of the total targets, the algorithm probability of success is greater than that of Grover algorithm. Finally we give the circuit implementation of the algorithm.

Acta Physica SinicaJan 1, 20150 citations

Quantum Walk Based Search Algorithms

Miklós Sántha

arXiv:0808.0059Springer eBooksApr 29, 2008182 citations

Quantum Search Algorithms

Uwe Schöning, Randall J. Pruim

Springer eBooksJan 1, 19980 citations

Robust quantum walk search without knowing the number of marked vertices

Yongzhen Xu, Delong Zhang, Lvzhou Li

There has been a very large body of research on searching a marked vertex on a graph based on quantum walks, and Grover's algorithm can be regarded as a quantum walk-based search algorithm on a special graph. However, the existing quantum walk-based search algorithms suffer severely from the souffl\'e problem, which mainly means that the success probability of finding a marked vertex could shrink dramatically, even to zero, when the number of search steps is greater than the right one, thus heavily reducing the robustness and practicability of the algorithm. Surprisingly, while the souffl\'e problem of Grover's algorithm has attracted enough attention, how to address this problem for general quantum walk-based search algorithms is missing in the literature. Here we initiate the study of overcoming the souffl\'e problem for quantum walk-based search algorithms by presenting a quantum walk-based search framework that achieves robustness without sacrificing the quantum speedup. In this framework, for any adjustable parameter $\ensuremath{\epsilon}$, the quantum algorithm can find a marked vertex on an $N$ -vertex complete bipartite graph with probability at least $1\ensuremath{-}\ensuremath{\epsilon}$, whenever the number of search steps $h$ satisfies $h\ensuremath{\ge}ln(\frac{2}{\sqrt{\ensuremath{\epsilon}}})\sqrt{N}+1$. Note that the algorithm need not know the exact number of marked vertices. Consequently, we obtain quantum search algorithms with stronger robustness and practicability.There has been a very large body of research on searching a marked vertex on a graph based on quantum walks, and Grover's algorithm can be regarded as a quantum walk-based search algorithm on a special graph. However, the existing quantum walk-based search algorithms suffer severely from the souffl\'e problem, which mainly means that the success probability of finding a marked vertex could shrink dramatically, even to zero, when the number of search steps is greater than the right one, thus heavily reducing the robustness and practicability of the algorithm. Surprisingly, while the souffl\'e problem of Grover's algorithm has attracted enough attention, how to address this problem for general quantum walk-based search algorithms is missing in the literature. Here we initiate the study of overcoming the souffl\'e problem for quantum walk-based search algorithms by presenting a quantum walk-based search framework that achieves robustness without sacrificing the quantum speedup. In this framework, for any adjustable parameter $\ensuremath{\epsilon}$, the quantum algorithm can find a marked vertex on an $N$-vertex complete bipartite graph with probability at least $1\ensuremath{-}\ensuremath{\epsilon}$, whenever the number of search steps $h$ satisfies $h\ensuremath{\ge}ln(\frac{2}{\sqrt{\ensuremath{\epsilon}}})\sqrt{N}+1$. Note that the algorithm need not know the exact number of marked vertices. Consequently, we obtain quantum search algorithms with stronger robustness and practicability.

arXiv:2111.09012Physical Review ANov 14, 202216 citations

N

-vertex complete bipartite graph with probability at least $1\ensuremath{-}\ensuremath{\epsilon}$, whenever the number of search steps

h

satisfies $h\ensuremath{\ge}ln(\frac{2}{\sqrt{\ensuremath{\epsilon}}})\sqrt{N}+1$. Note that the algorithm need not know the exact number of marked vertices. Consequently, we obtain quantum search algorithms with stronger robustness and practicability.