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IEEE Transactions on Quantum Engineering

Mixed Grover: A Hybrid Version to Improve Grover's Algorithm for Unstructured Database Search

Romain Piron, Muhammad Idham Habibie, Claire Goursaud

Published Jan 1, 2025• 4 citations
Publisher

Abstract

In this article, we propose a new strategy to exploit Grover's algorithm for unstructured search problems. We first show that running Grover's routine with a reduced number of iterations but allowing several trials presents a complexity advantage while keeping the same success probability. Then, by a theoretical analysis of the performance, we provide a generic procedure to parameterize the number of iterations$k$within one shot of Grover's algorithm and the maximum number of trials$T$, given a targeted success$p$and the size of the database$N$. At the end, we highlight that this new approach permits to reduce the computational time by at least 10% for$p \geq 0.999$independently of the size of the database.

DOI
10.1109/tqe.2025.3555562

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In this article, we propose a new strategy to exploit Grover's algorithm for unstructured search problems. We first show that running Grover's routine with a reduced number of iterations but allowing several trials presents a complexity advantage while keeping the same success probability. Then, by a theoretical analysis of the performance, we provide a generic procedure to parameterize the number of iterationskkkwithin one shot of Grover's algorithm and the maximum number of trialsTTT, given a targeted successpppand the size of the databaseNNN. At the end, we highlight that this new approach permits to reduce the computational time by at least 10% forp≥0.999p \geq 0.999p≥0.999independently of the size of the database.
Grover's quantum algorithm was developed to solve the problem of searching an unstructured database for a certain unique element. In general, this problem can be formulated as follows: an unstructured database consists of elements and contains one unique element that has a certain property that can be tested with polynomial complexity, and which must be found with minimal time and complexity. Classical methods require querying the database to find the desired element, while Grover's algorithm allows you to perform only approximately steps, which are iterations of the procedure, to the database and be sure that the resulting element will be exactly the desired element with a probability close to 1. As in quantum algorithms aimed at cryptanalysis of symmetric transformations, Grover's algorithm repeats the procedure (Grover's iteration) to increase the probability of obtaining the correct result. Similarly to such algorithms in Grover's method, when iterations are continued after reaching the required number of iterations, the probability of obtaining the correct result decreases. This is due to the fact that during the execution of Grover's iteration, a rotation is performed in the complex space. Thus, each iteration, making a turn, brings the register closer and closer to the desired state, but at a certain point a maximum closeness is reached, at which the continued use of iterations will lead to a turn past the desired state, which will move the state of the system away from the desired state. The database for this algorithm can be any search space consisting of a certain number of elements. So, for example, it can be applied to find a secret key for a symmetric cryptograms formation or to find a collision for a hashing function.
In my project I would have considered how the old search algorithms that are used when working with large unstructured data set end up taking linear time (O(N)) that in effect is the time that scales with the size of the data. These old ways get gradually sluggish, as everything gets bigger. Then I read about quantum computing, and especially Search Algorithm that was developed by Grover which is capable of quadratic speed-up with a complexity of O(sqrt(N)) of unstructured data. I created a hybrid system which fuses classical and quantum search applied on Flask as the web front end and Qiskit as the quantum one. This will enable me to compare the simple linear algorithm of searching through the list item-by-item with the quantum algorithm of Grover which uses superposition, a marking oracle to identify the desired item, and multiple amplitude amplification diffusion operators. The web page also accepts a student ID, and actually searches both simultaneously and allows me to tweak a depolarizing noise model so as to more closely mimic real quantum hardware. I measured such variables as the number of calls to the oracle, the number of qubits that I require, the quantum run time, and the number of measurements that I receive out of the quantum measurement. As the results indicate, the algorithm developed by Grover reduces the number of operations significantly compared to the classical one, and it must be emphasized that the algorithm remains quite effective even in the case of simulated noise. On the whole, it demonstrates that the classical and quantum components may be an efficient method to deal with the large search in the database.
In my project I would have considered how the old search algorithms that are used when working with large unstructured data set end up taking linear time (O(N)) that in effect is the time that scales with the size of the data. These old ways get gradually sluggish, as everything gets bigger. Then I read about quantum computing, and especially Search Algorithm that was developed by Grover which is capable of quadratic speed-up with a complexity of O(sqrt(N)) of unstructured data. I created a hybrid system which fuses classical and quantum search applied on Flask as the web front end and Qiskit as the quantum one. This will enable me to compare the simple linear algorithm of searching through the list item-by-item with the quantum algorithm of Grover which uses superposition, a marking oracle to identify the desired item, and multiple amplitude amplification diffusion operators. The web page also accepts a student ID, and actually searches both simultaneously and allows me to tweak a depolarizing noise model so as to more closely mimic real quantum hardware. I measured such variables as the number of calls to the oracle, the number of qubits that I require, the quantum run time, and the number of measurements that I receive out of the quantum measurement. As the results indicate, the algorithm developed by Grover reduces the number of operations significantly compared to the classical one, and it must be emphasized that the algorithm remains quite effective even in the case of simulated noise. On the whole, it demonstrates that the classical and quantum components may be an efficient method to deal with the large search in the database.
Grover's algorithm is a quantum algorithm that interrogates an N-element unstructured database inN\sqrt{N}N​searches as opposed to the best classical algorithm, which requiresN/2N/2N/2[1]. The algorithm is non-deterministic, with a significant probability of failing that varies with the size of the database and the number of marked elements. For example, with a single marked element, the success probability is 100% forN=4N=4N=4, drops to 87% forN=7N=7N=7, is near 100% forN=9N=9N=9, and is below 94% forN=15N=15N=15. This can be improved by several repetitions of the algorithm, which, however, lowers its quadratic quantum advantage.
Target item searching in large-scale, unstructured text databases is a major computational task for traditional systems and usually requires algorithms whose time complexity is on the order ofO(N)O(N)O(N). Quantum computing presents a strong competitor by using Grover's algorithm, where theoretically the search complexity comes down toO(N)O(\sqrt{N})O(N​), making faster retrieval possible. This research considers the design and assessment of Grover's algorithm as it is used to find a target word in a paragraph. The process entails encoding a group of unique words into a quantum register and implementing a circuit that incorporates a bespoke oracle and a diffusion operator to amplify the probability amplitude of the target state. Performance was evaluated in two situations: an ideal noise-free simulation with Qiskit's AerSimulator and a real-world run on IBM's quantum hardware. Simulation results demonstrated always-high accuracy in identifying the target word. But tests on real quantum machines indicated that the reliability of the algorithm decreases with an increase in qubits. Though circuits of up to five qubits had tolerable accuracy, larger configurations were plagued by noise, gate errors, and decoherence effects. These results highlight both the potential of quantum search algorithms for text processing and the limitations of NISQ today.
Now-a-days, the use of No-SQL databases is one of the potential options for storing and processing big data lakes. However, searching for large data in No-SQL databases is a complex and time-consuming task. Further, information retrieval from big data management suffers in terms of execution time. To reduce the execution time during the search process, we propose a fast and suitable approach based on the quantum Grover algorithm, which represents one of the best-known approaches for searching in an unstructured database and resolves the unsorted search query in O (√n) time complexity. To assess our proposal, a comparative study with linear and binary search algorithms was conducted to prove the effectiveness of Grover's algorithms. Then, we perform extensive experiment evaluations based on ibm_qasm_simulator for searching one item out of eight using Grover’s search algorithm based on three qubits. The experiments outcomes revealed encouraging results, with an accuracy of 0.948, well in accordance with the theoretical result. Moreover, a discussion of the sensitivity of Grover's algorithm through different iterations was carried out. Then, exceeding the optimal number of iterations round (π/4 √N), induces low accuracy of the marked state. Furthermore, the incorrect selection of this parameter can outline the solution.