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Russian Microelectronics

Analysis of Quantum Computing Resources in the Implementation of the Quantum Fourier Transform and Grover Algorithm on Qudit Systems

A. S. Chudakov, N. A. Borschevskaia, Dmitriy V. Fastovets, А.Ю. Чернявский, B. I. Bantysh, Yu. I. Bogdanov

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract The effectiveness of the implementation of the quantum Fourier transform (QFT) and the Grover algorithm on qudit systems is investigated. A comparative analysis of computing resources for quantum states of N ≥ 1024 dimension is carried out. It is shown that using qudits can significantly reduce the number of quantum operations in Grover’s algorithm compared to its qubit counterparts, while for QFT the main advantage is to reduce the number of physical carriers. The effect of noise using the Monte Carlo method on the accuracy of algorithms for qubit systems of various dimensions is analyzed.

DOI
10.1134/s1063739725601584

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Abstract The effectiveness of the implementation of the quantum Fourier transform (QFT) and the Grover algorithm on qudit systems is investigated. A comparative analysis of computing resources for quantum states of N ≥ 1024 dimension is carried out. It is shown that using qudits can significantly reduce the number of quantum operations in Grover’s algorithm compared to its qubit counterparts, while for QFT the main advantage is to reduce the number of physical carriers. The effect of noise using the Monte Carlo method on the accuracy of algorithms for qubit systems of various dimensions is analyzed.
The method of noisy multiqubit quantum circuits modeling is proposed. The analytical formulas for the dependence of quantum algorithms accuracy on qubits count and noise level are obtained for Grover algorithm and quantum Fourier transform. It is shown that the proposed approach is very much in line with results obtained by Monte Carlo statistical modeling method. The developed theory makes it possible to predict the influence of quantum noise on the accuracy of the forward-looking multiqubit quantum systems under development.
The Quantum Fourier transform (QFT) is a key ingredient in most quantum algorithms. We have compared various spin-based quantum computing schemes to implement the QFT from the point of view of their actual time-costs and the accuracy of the implementation. We focus here on an interesting decomposition of the QFT as a product of the non-selective Hadamard transformation followed by multiqubit gates corresponding to square- and higher-roots of controlled-NOT gates. This decomposition requires only O(n) operations and is thus linear in the number of qubits n. The schemes were implemented on a two-qubit NMR quantum information processor and the resultant density matrices reconstructed using standard quantum state tomography techniques. Their experimental fidelities have been measured and compared.
The Quantum Fourier transform (QFT) is a key ingredient in most quantum algorithms. We have compared various spin-based quantum computing schemes to implement the QFT from the point of view of their actual time-costs and the accuracy of the implementation. We focus here on an interesting decomposition of the QFT as a product of the non-selective Hadamard transformation followed by multiqubit gates corresponding to square- and higher-roots of controlled-NOT gates. This decomposition requires only O(n) operations and is thus linear in the number of qubits n. The schemes were implemented on a two-qubit NMR quantum information processor and the resultant density matrices reconstructed using standard quantum state tomography techniques. Their experimental fidelities have been measured and compared.
This survey presents a comprehensive analysis of the quantum Fourier transform (QFT), a fundamental tool in quantum computing, in comparison to its classical counterpart, the fast Fourier transform (FFT). The study begins with an introduction to the classical Fourier transform, which is widely used in different disciplines such as signal and image processing. The article introduces the QFT as a quantum version of the Fourier transform, detailing how it leverages quantum parallelism and superposition to reduce the time complexity of the operation, and highlighting its crucial role in quantum algorithms like Shor’s algorithm for integer factorization. The analysis also addresses the mathematical foundations of the QFT, its implementation in quantum circuits, and the key advantages and challenges associated with its use, such as measurement precision and quantum decoherence. Finally, it concludes with an exploration of the current and potential applications of QFT in quantum computing.
Quantum computing with qudits, quantum systems with d>2d > 2d>2 levels, offers a powerful extension beyond qubits, expanding the computational possibilities of quantum systems, allowing the simplification of the implementation of several algorithms and, possibly, providing a foundation for optimised error correction. In this work, we propose a novel formulation of qudit gates that is universally applicable for any number of levels ddd, without restrictions on the dimensionality. By extending the mathematical framework of quantum gates to arbitrary dimensions, we derive explicit gate operations that form a universal set for quantum computation on qudits of any size. We demonstrate the validity of our approach through the implementation of the Quantum Fourier Transform (QFT) for arbitrary ddd, verifying both the correctness and utility of our generalized gates. This novel methodology broadens the design space for quantum algorithms and fault-tolerant architectures, paving the way for advancements in qudit-based quantum computing.
For searching an item in unstructured databases, Grover's quantum search algorithm offers quadratic speedup overclassical search algorithms. This paper reports 2 to 5 quantum-bit (Qubit) implementations of Grover's search algorithmusing the phase-flip method for oracle function realization without any extra ancilla qubit. A comprehensive estimation andanalysis of the theoretical and physical accuracies of the algorithm have been presented. The impact of increasing qubits onaccuracy has been computed and analyzed. The metrics delineated for comparison are the number of qubits and gates, depthof the circuit, execution time, and theoretical/physical accuracy. The results revealed a greater disparity between theoreticaland physical accuracy using a higher number of qubits perceived to be caused by noisy qubits utilized in computations. Thenovelty of the work is the investigation of variations caused by the noise in the accuracy and execution time of Grover'ssearch algorithm. The results indicate that because of noise, the accuracy of 2- and 3- qubit implementations declined by14.49% and 33.86%, whereas the execution time increased by 50% and 80%; respectively.