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International Conference on Software, Telecommunications and Computer Networks (SoftCOM)

The GR3 Algorithm for Parallel Quantum Searching of Subgraph Isomorphism

Radu-Iulian Gheorghica

Published Sep 23, 2021• 1 citation
Publisher

Abstract

The purpose of the research presented in this paper is finding the best execution times for searching subgraph isomorphisms. For this was created the GR3 Algorithm, its design consisting of a combination of Parallel Programming and Quantum Computing. The obtained execution times are significantly small. Thus can be deduced that an approach such as this is superior in obtaining the required results. The original contribution consists in studying a Parallel Programming approach combined with Quantum Computing Circuits and Grover’s Algorithm for searching subgraph isomorphisms using a laptop and real quantum computers [16].

DOI
10.23919/softcom52868.2021.9559108

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The purpose of the research presented in this paper is finding the best execution times for searching subgraph isomorphisms. For this was created the GR3 Algorithm, its design consisting of a combination of Parallel Programming and Quantum Computing. The obtained execution times are significantly small. Thus can be deduced that an approach such as this is superior in obtaining the required results. The original contribution consists in studying a Parallel Programming approach combined with Quantum Computing Circuits and Grover’s Algorithm for searching subgraph isomorphisms using a laptop and real quantum computers [16].
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In this work we study Grover's algorithm for quantum computers. This algorithm promises to search in an unstructured database in time comparable with a square root of the number of objects. A description of basic quantum computer principles is followed by the original Grover's formulation and a proof of the algorithm, as well as a later geometrical description of the algorithm. We also mention their suitability for educational purposes. We also include a description of the Grover's algorithm in a programming language, which makes it possible to simulate a quantum computing on classical computers.
It is widely believed that quantum computers (if realized) will be more powerful than today’s computers from the viewpoint of computational complexity . However, it is not obvious how to utilize quantum computers in practical situations. For practical purposes, a Grover search may be one of the most promising quantum algorithms known so far. Thus, in this paper, we propose an efficient framework where we can use Grover search for general programming. Our framework has the following steps. (1) A programmer writes a program by using standard C++ programming language. (2) Some if expressions within for loops in the C++ source code are chosen as candidates to be performed as a Grover search on a quantum computer. (3) The framework automatically generates a corresponding quantum circuit for each Grover search chosen in (2). Unlike the existing quantum circuit design methods, we can treat large problems in our quantum circuit design. (4) By evaluating the number of primitive quantum gates in the quantum circuit generated in (3), the framework determines whether the processing time of the quantum circuit is faster than the processing time of the corresponding if expression on a classical computer. If the framework determines that the quantum circuit is faster, it generates some interface source codes for a classical computer. Thus, in our framework, a programmer can use a Grover search with almost no effort.
We study a variant of the subgraph isomorphism problem that is of high interest to the quantum computing community. Our results give an algorithm to perform pattern matching in quantum circuits for many patterns simultaneously, independently of the number of patterns. After a pre-computation step in which the patterns are compiled into a decision tree, the running time is linear in the size of the input quantum circuit. More generally, we consider connected port graphs, in which every edge eee incident to vvv has a label Lv(e)L_v(e)Lv​(e) unique in vvv. Jiang and Bunke showed that the subgraph isomorphism problem H⊆GH \subseteq GH⊆G for such graphs can be solved in time O(∣V(G)∣⋅∣V(H)∣)O(|V(G)| \cdot |V(H)|)O(∣V(G)∣⋅∣V(H)∣). We show that if in addition the graphs are directed acyclic, then the subgraph isomorphism problem can be solved for an unbounded number of patterns simultaneously. We enumerate all mmm pattern matches in time O(P)P+3/2⋅∣V(G)∣+O(m)O(P)^{P+3/2} \cdot |V(G)| + O(m)O(P)P+3/2⋅∣V(G)∣+O(m), where PPP is the number of vertices of the largest pattern. In the case of quantum circuits, we can express the bound obtained in terms of the maximum number of qubits NNN and depth δδδ of the patterns : O(N)N+1/2⋅δlog⁡δ⋅∣V(G)∣+O(m)O(N)^{N + 1/2} \cdot δ\log δ\cdot |V(G)| + O(m)O(N)N+1/2⋅δlogδ⋅∣V(G)∣+O(m).
Abstract Solving combinatorial optimization problems represents a challenge for researchers, especially hard NP-complete problems that are commonly handled through heuristic methods providing approximate solutions. The emergence of quantum computing, which enables performing parallel computation, raises hope for solving such problems within a reasonable time frame. This paper proposes a quantum global optimization algorithm for solving combinatorial optimization problems with improved performance and runtime approaching Grover’s optimal time of quantum search. The algorithm determines the global optimum of the objective function defining the addressed problem using an enhanced application of Grover’s search by simultaneously exploring all comparison cases of possible solutions without requiring iterative operations, unlike previous approaches, which affect performance and runtime. The paper also introduces a version of the proposed algorithm for solving the knapsack problem, along with its execution results and a comparison with other competing algorithms.