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International Conference on Electrical, Communication, and Computer Engineering (ICECCE)

Matrix Representation of Quantum Addition Using QFT

Pedro Juan Roig, Salvador Alcaraz, Katja Gilly, Cristina Bernad, Blas Trigueros

Published Aug 27, 2025• 0 citations
Publisher

Abstract

Quantum computing allows for undertaking certains problems in an efficient manner, although they may be considered intractable in classic computing. This is possible due to the specific features of quantum computing, such as superposition and entanglement. In this paper, a matrix representation of quantum addition using QFT has been described, focused on adding 2 -qubit strings and 3 -qubit strings. In each case, a schematic diagram is shown, along with two alternative implementations, followed by methods to find out QFT and IQFT matrices, and eventually, a step-by-step instance of an addition.

DOI
10.1109/icecce67514.2025.11258033

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Quantum computing allows for undertaking certains problems in an efficient manner, although they may be considered intractable in classic computing. This is possible due to the specific features of quantum computing, such as superposition and entanglement. In this paper, a matrix representation of quantum addition using QFT has been described, focused on adding 2 -qubit strings and 3 -qubit strings. In each case, a schematic diagram is shown, along with two alternative implementations, followed by methods to find out QFT and IQFT matrices, and eventually, a step-by-step instance of an addition.
Addition is a key arithmetic operation in computing, as the rest of arithmetic operations are somewhat based on it. Focusing on quantum computing, addition can be implemented in different ways, such as adapting the ripple-carry adder from classical computing, or using a QFT-based adder, which is more efficient than the former. However, a Toffoli-based adder is in fact an alternative way to optimize QFT-based adders by merging rotation gates into a string of controlled-NOT and Toffoli gates with different control qubits. In this paper, the basics of matrix analysis of quantum circuits are described, which in turn are used to perform a matrix analysis of Toffoli adder circuits.
In this study, we propose an efficient quantum multiplication approach based on a QFT-assisted parallelized addition scheme. The multiplication stage is implemented using a structure composed entirely of Toffoli gates, which generate partial products. In the second stage, these partial results are accumulated using a QFT-based adder. Unlike conventional QFT-based arithmetic circuits, the proposed design eliminates the repeated application of QFT and inverse QFT (IQFT) operations during intermediate summation processes. This leads to a significant reduction in the total gate count and circuit complexity, enabling a more resource-efficient implementation. To demonstrate the feasibility of the proposed approach, a quantum circuit that performs the multiplication of two 3-bit numbers is designed. The circuit is tested and validated using IBM quantum simulators. The results indicate that the proposed method provides a more efficient alternative to traditional quantum multiplication techniques in terms of gate cost and circuit depth.
Quantum computing leverages quantum-mechanical phenomena such as superposition, entanglement and interference to solve classes of problems that are intractable on classical machines.This paper unifies three complementary perspectives: (i) the central role of complex matrices in modelling quantum states, gates and error-correction protocols; (ii) the pervasive use of trigonometric functions in qubit representation, phase manipulation and the quantum Fourier transform (QFT); and (iii) a systematic comparison of classical and quantum computing paradigms with illustrative circuits and real-world applications.By integrating these dimensions, the study provides a consolidated reference that elucidates the mathematical foundations, circuit constructions, algorithmic advantages and practical challenges that define contemporary quantum information processing.
Quantum entanglement offers a fundamentally different approach to matrix computation, potentially surpassing classical methods in certain scenarios. This paper proposes a novel optimization algorithm specifically tailored for quantum entanglement, aiming to enhance the efficiency of matrix operations. We introduce a quantum state manipulation scheme leveraging entanglement to achieve a significant speedup in calculations, addressing limitations inherent in current quantum computing architectures. The core mechanism centers around strategically manipulating entangled qubit states to represent and manipulate matrix elements, offering a pathway to dramatically accelerate matrix computations. The algorithm's novelty lies in its incorporation of quantum entanglement as a key computational resource, providing a unique advantage over existing optimization strategies. We detail the algorithm's architecture, the quantum circuit implementation, and preliminary results demonstrating improved performance compared to classical methods. This work explores the potential of entanglement to fundamentally alter matrix computation, presenting a promising avenue for advancing quantum computing capabilities.
Linear matrix systems represent a fundamental class of problems in mathematics, physics, and engineering, and their efficient solution is crucial for various applications. Traditional methods often rely on computationally intensive algorithms, limiting scalability. This paper explores the potential of leveraging the inherent non-linear properties of linear matrices to design novel quantum algorithms for solving linear systems. We propose a quantum algorithm predicated on the principle that by mapping a linear system into a non-linear representation via a specific matrix transformation, we can achieve speedups that surpass classical approaches. The core mechanism involves constructing a quantum circuit that systematically explores this non-linear space, leveraging quantum phenomena like superposition and entanglement to enhance the search process. The paper details the design of a quantum algorithm for solving linear systems, emphasizing the benefits of this approach and its potential to unlock new possibilities in quantum computation.