The Quantum Fourier Transform (QFT) grants competitive advantages, especially in resource usage and circuit approximation, for performing arithmetic operations on quantum computers, and offers a potential route towards a numerical quantum-computational paradigm. In this paper, we utilize efficient techniques to implement QFT-based integer addition and multiplications. These operations are fundamental to various quantum applications including Shor's algorithm, weighted sum optimization problems in data processing and machine learning, and quantum algorithms requiring inner products. We carry out performance evaluations of these implementations based on IBM's superconducting qubit architecture using different compatible noise models. We isolate the sensitivity of the component quantum circuits on both one-/two-qubit gate error rates, and the number of the arithmetic operands' superposed integer states. We analyze performance, and identify the most effective approximation depths for quantum add and quantum multiply within the given context. We observe significant dependency of the optimal approximation depth on the degree of machine noise and the number of superposed states in certain performance regimes. Finally, we elaborate on the algorithmic challenges - relevant to signed, unsigned, modular and non-modular versions - that could also be applied to current implementations of QFT-based subtraction, division, exponentiation, and their potential tensor extensions. We analyze performance trends in our results and speculate on possible future development within this computational paradigm.
The Quantum Fourier Transform (QFT) grants competitive advantages, especially in resource usage and circuit approximation, for performing arithmetic operations on quantum computers, and offers a potential route towards a numerical quantum-computational paradigm. In this paper, we utilize efficient techniques to implement QFT-based integer addition and multiplications. These operations are fundamental to various quantum applications including Shor's algorithm, weighted sum optimization problems in data processing and machine learning, and quantum algorithms requiring inner products. We carry out performance evaluations of these implementations based on IBM's superconducting qubit architecture using different compatible noise models. We isolate the sensitivity of the component quantum circuits on both one-/two-qubit gate error rates, and the number of the arithmetic operands' superposed integer states. We analyze performance, and identify the most effective approximation depths for quantum add and quantum multiply within the given context. We observe significant dependency of the optimal approximation depth on the degree of machine noise and the number of superposed states in certain performance regimes. Finally, we elaborate on the algorithmic challenges - relevant to signed, unsigned, modular and non-modular versions - that could also be applied to current implementations of QFT-based subtraction, division, exponentiation, and their potential tensor extensions. We analyze performance trends in our results and speculate on possible future development within this computational paradigm.
Abstract The Quantum Fourier Transform (QFT) grants competitive advantages, especially in resource usage and circuit approximation, for performing arithmetic operations on quantum computers , and offers a potential route towards a numerical quantum-computational paradigm. In this paper, we utilize efficient techniques to implement QFT-based integer addition and multiplications. These operations are fundamental to various quantum applications including Shor’s algorithm, weighted sum optimization problems in data processing and machine learning, and quantum algorithms requiring inner products. We carry out performance evaluations of these implementations based on IBM’s superconducting qubit architecture using different compatible noise models. We isolate the sensitivity of the component quantum circuits on both one-/two-qubit gate error rates, and the number of the arithmetic operands’ superposed integer states. We analyze performance, and identify the most effective approximation depths for unsigned quantum addition and quantum multiplication within the given context. We then perform a similar analysis of signed addition and compare to the unsigned results. We observe significant dependency of the optimal approximation depth on the degree of machine noise and the number of superposed states in certain performance regimes. Finally, we elaborate on the algorithmic challenges - relevant to signed, unsigned, modular and non-modular versions - that could also be applied to current implementations of QFT-based subtraction, division, exponentiation, and their potential tensor extensions. We analyze the performance trends in our results and speculate on possible future developments within this computational paradigm.
The quantum Fourier transform (QFT) brings efficiency in many respects, especially usage of resource, for most operations on quantum computers. In this study, the existing QFT-based and non-QFT-based quantum arithmetic operations are examined. The capabilities of QFT-based addition and multiplication are improved with some modifications. The proposed operations are compared with the nearest quantum arithmetic operations. Furthermore, novel QFT-based subtraction, division and exponentiation operations are presented. The proposed arithmetic operations can perform nonmodular operations on all signed numbers without any limitation by using less resources. In addition, novel quantum circuits of two’s complement, absolute value and comparison operations are also presented by using the proposed QFT-based addition and subtraction operations.
One of the major promises of quantum computing is the realization of SIMD (single instruction - multiple data) operations using the phenomenon of superposition. Since the dimension of the state space grows exponentially with the number of qubits, we can easily reach situations where we pay less than a single quantum gate per data point for data-processing instructions which would be rather expensive in classical computing. Formulating such instructions in terms of quantum gates, however, still remains a challenging task. Laying out the foundational functions for more advanced data-processing is therefore a subject of paramount importance for advancing the realm of quantum computing. In this paper, we introduce the formalism of encoding so called-semi-boolean polynomials. As it turns out, arithmetic Z/2nZ ring operations can be formulated as semi-boolean polynomial evaluations, which allows convenient generation of unsigned integer arithmetic quantum circuits. For arithmetic evaluations, the resulting algorithm has been known as Fourier-arithmetic. We extend this type of algorithm with additional features, such as ancilla-free in-place multiplication and integer coefficient polynomial evaluation. Furthermore, we introduce a tailormade method for encoding signed integers succeeded by an encoding for arbitrary floating-point numbers. This representation of floating-point numbers and their processing can be applied to any quantum algorithm that performs unsigned modular integer arithmetic. We discuss some further performance enhancements of the semi boolean polynomial encoder and finally supply a complexity estimation. The application of our methods to a 32-bit unsigned integer multiplication demonstrated a 90% circuit depth reduction compared to carry-ripple approaches.
In this research, we create a scalable version of the quantum Fourier transform-based arithmetic circuit to perform addition and subtraction operations on N n-bit unsigned integers encoded in quantum registers, and it is compatible with d-level quantum sources, called qudits. We present qubit- and ququart-based multi-input QFT adders, and we compare and discuss potential benefits such as circuit simplicity and noise sensitivity. The results show that a ququart-based system significantly reduces gate count and improves computational efficiency compared to qubit-based systems. Overall, the findings presented in this study represent a promising step forward in the development of efficient quantum arithmetic circuits, particularly for multi-input operations, with clear advantages for ququart-based systems in reducing gate count, decoherence, and circuit complexity.
Quantum computing is a transformative technology providing exponential speedup over classical computing for specific computational problems. At the foundation of computing lies arithmetic, with addition as a fundamental operation for subtraction, multiplication, division, and other operations. In this paper, we evaluate the performance of quantum arithmetic circuits – including half adders, full adders, Toffoli-based adders, Quantum Fourier Transform (QFT)-based adders, and quantum comparators – on IBM’s Eagle Processor, a 127-qubit superconducting Quantum Processing Unit (QPU) accessible through the IBM Quantum cloud platform. These circuits are implemented using IBM Qiskit and tested on both ideal and noisy simulators to validate their functionality before execution on quantum hardware. Experimental results show that the probabilities of accurate computations are notably low, even for smaller input sizes, due to hardware imperfections such as shorter qubit coherence times and limited qubit stability. A comparative analysis of Toffoli-based and QFT-based adders demonstrates that Toffoli-based adders have higher accuracy, making them more suitable for precise quantum arithmetic operations. These findings highlight the significant impact of quantum hardware imperfections on the reliability of arithmetic operations.
In recent decades, the field of quantum computing has experienced remarkable progress. This progress is marked by the superior performance of many quantum algorithms compared with their classical counterparts, with Shor's algorithm serving as a prominent illustration. Quantum arithmetic circuits, which are the fundamental building blocks in numerous quantum algorithms, have attracted much attention. Despite extensive exploration of various designs in the existing literature, researchers remain keen to develop novel designs and improve existing ones. In this review article, we aim to provide a systematically organized and easily comprehensible overview of the current state of the art in quantum arithmetic circuits. Specifically, this study covers fundamental operations such as addition, subtraction, multiplication, division and modular exponentiation. We delve into the detailed quantum implementations of these prominent designs and evaluate their efficiency considering various objectives. We also discuss potential applications of the presented arithmetic circuits and suggest future research directions.This article is part of the theme issue 'Emerging technologies for future secure computing platforms'.