A quantum particle evolving by Schr\"odinger's equation in discrete space constitutes a continuous-time quantum walk on a graph of vertices and edges. When a vertex is marked by an oracle, the quantum walk effects a quantum search algorithm. Previous investigations of this quantum search algorithm on graphs with cliques have shown that the edges between the cliques can be weighted to enhance the movement of probability between the cliques to reach the marked vertex. In this paper, we explore the most restrictive form of this by analyzing search on a weighted barbell graph that consists of two cliques of the same size joined by a single weighted edge or bridge. This graph is generally irregular, so quantum walks governed by the graph Laplacian or by the adjacency matrix can differ. We show that the Laplacian quantum walk's behavior does not change, no matter the weight of the bridge, and so the single bridge is too restrictive to affect the walk. Similarly, the adjacency quantum walk's behavior is unchanged for most weights, but when the weight equals the size of a clique, the probability does collect at the clique containing the marked vertex, and utilizing a two-stage algorithm with different weights for each stage, the success probability is boosted from 0.5 to 0.996, independent of the size of the barbell graph.A quantum particle evolving by Schr\"odinger's equation in discrete space constitutes a continuous-time quantum walk on a graph of vertices and edges. When a vertex is marked by an oracle, the quantum walk effects a quantum search algorithm. Previous investigations of this quantum search algorithm on graphs with cliques have shown that the edges between the cliques can be weighted to enhance the movement of probability between the cliques to reach the marked vertex. In this paper, we explore the most restrictive form of this by analyzing search on a weighted barbell graph that consists of two cliques of the same size joined by a single weighted edge or bridge. This graph is generally irregular, so quantum walks governed by the graph Laplacian or by the adjacency matrix can differ. We show that the Laplacian quantum walk's behavior does not change, no matter the weight of the bridge, and so the single bridge is too restrictive to affect the walk. Similarly, the adjacency quantum walk's behavior is unchanged for most weights, but when the weight equals the size of a clique, the probability does collect at the clique containing the marked vertex, and utilizing a two-stage algorithm with different weights for each stage, the success probability is boosted from 0.5 to 0.996, independent of the size of the barbell graph.

arXiv:2408.08244Physical Review AOct 15, 20242 citations

Quantum Circuit Compilation for the Graph Coloring Problem

Angelo Oddi, Riccardo Rasconi, Marco Baioletti +2 more

Lecture notes in computer scienceJan 1, 20233 citations

Quantum circuit synthesis with diffusion models

Florian Fürrutter, Gorka Muñoz-Gil, Hans J. Briegel

arXiv:2311.02041Nature Machine IntelligenceMay 20, 202432 citations

Searching via walking: How to find a marked clique of a complete graph using quantum walks

Mark S. Hillery, Daniel Reitzner, Vladimír Bužek

We show how a quantum walk can be used to find a marked edge or a marked complete subgraph of a complete graph. We employ a version of a quantum walk, the scattering walk, which lends itself to experimental implementation. The edges are marked by adding elements to them that impart a specific phase shift to the particle as it enters or leaves the edge. If the complete graph has $N$ vertices and the subgraph has $K$ vertices, the particle becomes localized on the subgraph in $O(N/K)$ steps. This leads to a quantum search that is quadratically faster than a corresponding classical search. We show how to implement the quantum walk using a quantum circuit and a quantum oracle, which allows us to specify the resources needed for a quantitative comparison of the efficiency of classical and quantum searches---the number of oracle calls.We show how a quantum walk can be used to find a marked edge or a marked complete subgraph of a complete graph. We employ a version of a quantum walk, the scattering walk, which lends itself to experimental implementation. The edges are marked by adding elements to them that impart a specific phase shift to the particle as it enters or leaves the edge. If the complete graph has $N$ vertices and the subgraph has $K$ vertices, the particle becomes localized on the subgraph in $O(N/K)$ steps. This leads to a quantum search that is quadratically faster than a corresponding classical search. We show how to implement the quantum walk using a quantum circuit and a quantum oracle, which allows us to specify the resources needed for a quantitative comparison of the efficiency of classical and quantum searches---the number of oracle calls.

arXiv:0911.1102Physical Review AJun 17, 201036 citations

Circuit Synthesis of Maximum Clique Problem using Combinatorial Approach of Classical-Quantum Hybrid Model.

Arpita Sanyal, Amit Saha, Banani Saha +1 more

In the Maximum Clique Problem, the objective is to find a clique whose size is maximum among all the cliques of an arbitrary undirected and unweighted graph. Maximum Clique Problem resembles with minimum vertex cover problem, independent set problem. As it is an NP-hard problem, no polynomial time algorithms can be found. As these problems have several important practical applications such as information retrieval, community detection in network, spatial data mining etc, it is of great interest to try to synthesis the circuit of Maximum Clique Problem. In classical computing any brute-force solution to the Maximum Clique Problem requires an exponential increase of time with the size of the problem (i.e., with time complexity of $O(2^n)$ ). In this paper, Maximum Clique Problem has been solved using combinatorial approach of Grover's search algorithm. An algorithm has been proposed that auto-generates the circuit for any given undirected and unweighted graph which makes the approach generalized in nature for Maximum Clique Problem. Computational speed up is achieved by using our approach with the help of quantum mechanical effect that is quantum superposition. We have simulated the proposed circuit using IBM's QISkit platform and verified its correctness.In the Maximum Clique Problem, the objective is to find a clique whose size is maximum among all the cliques of an arbitrary undirected and unweighted graph. Maximum Clique Problem resembles with minimum vertex cover problem, independent set problem. As it is an NP-hard problem, no polynomial time algorithms can be found. As these problems have several important practical applications such as information retrieval, community detection in network, spatial data mining etc, it is of great interest to try to synthesis the circuit of Maximum Clique Problem. In classical computing any brute-force solution to the Maximum Clique Problem requires an exponential increase of time with the size of the problem (i.e., with time complexity of $O(2^n)$). In this paper, Maximum Clique Problem has been solved using combinatorial approach of Grover's search algorithm. An algorithm has been proposed that auto-generates the circuit for any given undirected and unweighted graph which makes the approach generalized in nature for Maximum Clique Problem. Computational speed up is achieved by using our approach with the help of quantum mechanical effect that is quantum superposition. We have simulated the proposed circuit using IBM's QISkit platform and verified its correctness.

Mar 10, 20200 citations

N

vertices and the subgraph has

K

vertices, the particle becomes localized on the subgraph in

O(N/K)

steps. This leads to a quantum search that is quadratically faster than a corresponding classical search. We show how to implement the quantum walk using a quantum circuit and a quantum oracle, which allows us to specify the resources needed for a quantitative comparison of the efficiency of classical and quantum searches---the number of oracle calls.

O(2^n)

). In this paper, Maximum Clique Problem has been solved using combinatorial approach of Grover's search algorithm. An algorithm has been proposed that auto-generates the circuit for any given undirected and unweighted graph which makes the approach generalized in nature for Maximum Clique Problem. Computational speed up is achieved by using our approach with the help of quantum mechanical effect that is quantum superposition. We have simulated the proposed circuit using IBM's QISkit platform and verified its correctness.