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Quantum Studies Mathematics and Foundations

Applying Grover’s algorithm to implement various numerical and comparison operations

Arnab Roy, Joseph L. Pachuau, Gopal Krishna, Anish Kumar Saha

Published Mar 12, 2024• 4 citations
Publisher
DOI
10.1007/s40509-024-00323-w

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Grover's search algorithm was a groundbreaking advancement in quantum algorithms, providing a quadratic speedup of querying for items. Since the creation of this algorithm it has been utilized in various ways, including in preparing specific states for a general circuit. However, as the number of desired items increases, so does the gate complexity of the sub-process that conducts the query. To counter this complexity, an extension of Grover's search algorithm is derived where the focus of the query is on the undesirable items in order to suppress the amplitude of the queried items. To display the efficacy of this extension, the algorithm is implemented as a sub-process into QAOA and applied to a traveling salesman problem. For a basis of comparison, the results are compared against QAOA.
Grover’s algorithm harnesses the power of quantum computing to swiftly locate specific elements in an unstructured database, outperforming classical computers in tasks like database searching. This algorithm capitalizes on the unique ability of qubits to be in both 0 and 1 states simultaneously (superposition), allowing it to scan the entire search space at once. It then boosts the probability of the target element, making it more prominent. While the foundational concepts of Grover’s algorithm are well-documented, practical implementation using quantum operators, especially for large search spaces, remains less explored beyond basic examples with a small number of qubits. Existing general synthesis techniques often involve numerous operators or are time-consuming. Our proposed methods specifically address the amplitude-amplification component of Grover’s algorithm for any size of search space. These methods detail the required types and quantities of qubits and operators, emphasizing minimal usage and efficient assembly. Developed and evaluated in Python, our methods consistently identified target elements with over 95% accuracy and achieved configurations comparably compact as those in the existing literature but at a faster pace. We anticipate that these methods facilitate practical implementations of Grover’s algorithm across various domains.
The Grover algorithm is a crucial solution for addressing unstructured search problems and has emerged as an essential quantum subroutine in various complex algorithms. By using a different approach with previous studies, this research extensively investigates Grover's search methodology within non-uniformly distributed databases, a scenario frequently encountered in practical applications. Our analysis reveals that the behavior of the Grover evolution differs significantly when applied to non-uniform databases compared to uniform or 'unstructured databases'. Based on the property of differential equation, it is observed that the search process facilitated by this evolution does not consistently result in a speed-up, and we have identified specific criteria for such situations. Furthermore, we have extended this investigation to databases characterized by coherent states, confirming the speed-up achieved through Grover evolution via rigorous numerical verification. In conclusion, our study provides an enhancement to the original Grover algorithm, offering insights to optimize implementation strategies and broaden its range of applications.