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Chinese Journal of Physics

Novel optimization of quantum search algorithm to minimize complexity

Tarun Kumar, Dilip Kumar, Gurmohan Singh

Published Mar 9, 2023• 7 citations
Publisher
DOI
10.1016/j.cjph.2023.03.008

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A single machine total weighted tardiness minimization (TWTM) problem in operational planning is considered. The problem is formulated as an NP-hard constrained combinatorial problem, which has no known deterministic polynomial complexity solution using classical computing. Based on efficient Grover's quantum search and Trugenberger's quantum optimization algorithms, a novel efficient quantum optimization algorithm is proposed to solve the NP-hard single machine TWTM problem, which makes the desired solution satisfying the searching constraints and showing the minimal TWT value be measured with the highest probability.
A quantum computer encodes information in quantum states and runs quantum algorithms to surpass the classical counterparts by exploiting quantum superposition and quantum correlation. Grover's quantum search algorithm is a typical quantum algorithm that proves the superiority of quantum computing over classical computing. It has a quadratic reduction in the query complexity of a database search and is known to be optimal when no a priori information about the elements of the database is provided. In this work we consider a probabilistic Grover search algorithm allowing nonzero probability of failure for a database with a general a priori probability distribution of the elements and minimize the number of oracle calls by optimizing the initial state of the quantum system and the reflection axis of the diffusion operator. The initial state and the reflection axis are allowed to not coincide and thus the quantum search algorithm rotates the quantum system in a three-dimensional subspace spanned by the initial state, the reflection axis, and the search target state in general. The number of oracle calls is minimized by a variational method, and formal results are obtained with the assumption of low failure probability. The results show that for a nonuniform a priori distribution of the database elements, the number of oracle calls can be significantly reduced given a small decrease in the success probability of the quantum search algorithm, leading to a lower average query complexity to find the solution of the search problem. The results are applied to a simple but nontrivial database model with two-value a priori probabilities to show the power of the optimized quantum search algorithm. The paper concludes with a discussion about the generalization to higher-order results that allows for a larger failure probability for the quantum search algorithm.
Abstract One of the challenging scientific computing problems is topology optimization (TO), where the two tasks of searching through the combinatorially complex configurations and solving the constraints of partial differential equations need to be done simultaneously. In this paper, a novel variational quantum algorithm for constrained TO is proposed, which allows for the single-loop parallel search for the optimal configuration that also satisfies the physical constraints. The optimal configurations and the solutions to physical constraints are encoded with two separate quantum registers. A constraint encoding scheme is also proposed to incorporate volume and connectivity constraints in optimization. The gate complexity of the proposed quantum algorithm is analyzed. The algorithm is demonstrated with compliance minimization problems including truss structures and Messerschmitt–Bölkow–Blohm beams.