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Quantum Information and Computation

Quantum-based algorithm and circuit design for bounded Knapsack optimization problem

Wenjun Hou, Marek A. Perkowski

Published Aug 1, 2020• 6 citations
Publisher

Abstract

The Knapsack Problem is a prominent problem that is used in resource allocation and cryptography. This paper presents an oracle and a circuit design that verifies solutions to the decision problem form of the Bounded Knapsack Problem. This oracle can be used by Grover Search to solve the optimization problem form of the Bounded Knapsack Problem. This algorithm leverages the quadratic speed-up offered by Grover Search to achieve a quantum algorithm for the Knapsack Problem that shows improvement with regard to classical algorithms. The quantum circuits were designed using the Microsoft Q# Programming Language and verified on its local quantum simulator. The paper also provides analyses of the complexity and gate cost of the proposed oracle. The work in this paper is the first such proposed method for the Knapsack Optimization Problem.

DOI
10.26421/qic20.9-10-4

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The Knapsack Problem is a prominent problem that is used in resource allocation and cryptography. This paper presents an oracle and a circuit design that verifies solutions to the decision problem form of the Bounded Knapsack Problem. This oracle can be used by Grover Search to solve the optimization problem form of the Bounded Knapsack Problem. This algorithm leverages the quadratic speed-up offered by Grover Search to achieve a quantum algorithm for the Knapsack Problem that shows improvement with regard to classical algorithms. The quantum circuits were designed using the Microsoft Q# Programming Language and verified on its local quantum simulator. The paper also provides analyses of the complexity and gate cost of the proposed oracle. The work in this paper is the first such proposed method for the Knapsack Optimization Problem.
The Multi-Dimensional Knapsack Problem (MDKP) is a strongly NP-Hard combinatorial optimization problem that extends the Knapsack Problem (KP) to multiple constraints. While effective heuristic classical MDKP algorithms have been studied, there remains no optimal exact classical algorithm and only few contributions using quantum algorithms exist. In this study, we leverage Grover's search and pre-processing techniques to propose an iterative quantum search algorithm and circuit design that integrates classical adjustments to the oracle, progressively refining solutions until optimal. Evaluations show that our method reduces depth, width, iterations, and runtime.
Abstract The knapsack problem is an important example of a combinatorial optimization problem with a wide range of applications in direct practical applications. In this article, the author considers the classical version of the problem (0-1 knapsack) and proposes a quantum algorithm for solving it based on a hybrid approach that combines the Grover quantum search algorithm with the procedure for finding the Durr–Hoyer minimum. The paper presents an analysis of the computational complexity of the proposed algorithm, and in addition to theoretical analysis, the article contains an explicit description of the quantum circuit implementing the proposed algorithm.
The K-SAT problem is a fundamental challenge in computational theory, widely used in artificial intelligence, cryptography, and optimization. Classical algorithms struggle with exponential time complexity, making them inefficient for large-scale instances. In this research, we propose an efficient quantum computing approach based on Grover’s algorithm to enhance the search process for satisfiable solutions in K-SAT problems. Our implementation leverages quantum superposition and amplitude amplification to explore multiple possible solutions simultaneously, reducing the search complexity from O (2ⁿ) in classical methods to O(√2ⁿ) in quantum computing. We design a quantum oracle that encodes the K-SAT clauses and integrates it into Grover’s iterative search framework. The performance is evaluated through Qiskit simulations, demonstrating a significant improvement in search efficiency compared to classical brute-force techniques. The results highlight the potential of quantum algorithms in solving complex combinatorial problems with enhanced speed and accuracy. This study contributes to the development of quantum-accelerated optimization methods, paving the way for real-world applications in machine learning, cryptanalysis, and large-scale data processing.
We observe an enormous increase in the computational power of digital computers. This was due to the revolution in manufacturing processes and controlling semiconductor structures on submicron scale, ultimately leading to the control of individual atoms. Eventually, the classical electric circuits encountered the barrier of quantum mechanics and its effects. However, the laws of quantum mechanics can be also used to produce computational devices that lead to extraordinary speed increases over classical computers. Thus quantum computing becomes a very promising and attractive research area. The Computer Aided Design for Quantum circuits becomes an essential ingredient for such emerging research which may lead to these powerful computers to be realized—an era of Quantum computing. This thesis presents an integrated theoretical study of software algorithms to design circuits of quantum oracles as well as methods for designing quantum oracles for Grover algorithm to solve combinatorial problems. An implementation of quantum algorithm involves the initialization of the input state and its manipulation with quantum gates followed by the measurements. In Grover algorithm the problem to be solved is specified by a permutative logic oracle – the fundamental problem is then how to build this oracle from quantum logic circuits and how to optimize these circuits. These problems are NP-hard and require search algorithms. In future, the search will be also done in quantum and this thesis leads to quantum algorithms to design quantum circuits more efficiently.
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.
We propose new quantum circuit designs for the Grover oracle, i.e., the oracle of Grover’s algorithm, to solve the vertex cover problem (VCP) efficiently. Grover’s algorithm is an unstructured quantum search algorithm to search a target data item out of N data pieces by calling the oracle O(N)\mathrm{O}(\sqrt{N})O(N​) times. It represents a quadratic speedup compared to the classical unstructured data sequential search algorithm, which requires O(N) oracle calls. We are in the noisy intermediate-scale quantum (NISQ) era in which quantum computers have a limited number of qubits, short decoherence time, and low gate fidelity. It is thus desirable to design quantum circuits with three properties: (1) a reduced number of qubits, (2) shorter quantum depth, and (3) fewer gates. We created novel designs of a quantum counter and a quantum semaphore to be applied to the Grover oracle to have the three properties. Through the IBM Quantum Lab service, we executed Grover’s algorithm with the novel quantum circuit designs to solve the VCP to validate the design correctness.