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International Journal of Quantum Information

The addition and subtraction of quantum matrix based on GNEQR

Hai-Sheng Li, Yusi Xu, Yunbai Qin, Deli Fu, Haiying Xia

Published Oct 1, 2019• 4 citations
Publisher

Abstract

The efficient quantum circuits of arithmetic operations are important to perform quantum algorithms. To implement efficient matrix operations, we first modify the generalized model of the novel enhanced quantum representation of digital images (GNEQR) to store unsigned and signed integer matrices. Next, we design the circuits of the circuits of quantum addition, quantum modulo addition, quantum subtraction, and quantum modulo subtraction, these operations all keeping two operands unchanged. Then, we propose the circuits of quantum matrix addition, quantum matrix modulo addition, quantum matrix subtraction, and quantum matrix modulo subtraction for the first time. Furthermore, we present a simulation method to verify the correctness of the proposed arithmetic operations of matrix. The results of simulation experiment show that the propose arithmetic operations of matrix are efficient and correct.

DOI
10.1142/s0219749919500564

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The efficient quantum circuits of arithmetic operations are important to perform quantum algorithms. To implement efficient matrix operations, we first modify the generalized model of the novel enhanced quantum representation of digital images (GNEQR) to store unsigned and signed integer matrices. Next, we design the circuits of the circuits of quantum addition, quantum modulo addition, quantum subtraction, and quantum modulo subtraction, these operations all keeping two operands unchanged. Then, we propose the circuits of quantum matrix addition, quantum matrix modulo addition, quantum matrix subtraction, and quantum matrix modulo subtraction for the first time. Furthermore, we present a simulation method to verify the correctness of the proposed arithmetic operations of matrix. The results of simulation experiment show that the propose arithmetic operations of matrix are efficient and correct.
Arithmetic operations on images are important primary transformations that can be applied to many complicated image operations. We mainly study the intensity-based arithmetic operations on the QIRHSI quantum image representation model to facilitate the development of quantum image processing. The quantum circuits and unitary matrices for four arithmetic operations including reverse, complement, addition, and subtraction are proposed. Finally, the complexity analysis of quantum networks is illustrated to show the effectiveness of the proposed arithmetic operations.
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.
Image scrambling on classical computers has been widely studied, however, the corresponding quantum algorithms are rare. This paper proposes a block-based image scrambling scheme for the generalized model of novel enhanced quantum representation (GNEQR). It is not only suit for grayscale images and color images, but also can be used in 2n× 2mimages (i.e., rectangular images). To realize the image scrambling method, we use geometric transformations and an operation of bitplane scrambling to perform position and pixel scrambling, respectively. Furthermore, the complexity of the implementation circuit of the proposed image scrambling is only O(n2) for a 2n× 2nimage. Therefore, it shows our proposed quantum image scrambling is efficient. Then, the result analysis of simulation experiments demonstrates that our proposed image scrambling is effective. In addition, security analysis shows that the proposed scrambling algorithm is secure.
Abstract Linear gray enhancement is a spatial domain image enhancementtechnique commonly used in classical computers, mainly including image negative, image contrast stretching, and piecewise linear gray transformation. Inorder to realize these three linear gray enhancement techniques in the quantumcomputers, this paper proposes three types of linear gray transformationschemes for quantum images based on the generalized model of novel enhancedquantum image representation(GNEQR), and the quantum circuits that realizethese three transformation methods are constructed according to theschemes. The proposed circuits take advantage of efficient quantum arithmeticoperations and parallel Controlled-NOT modules to factor classical transformationsinto basic unitary operators such as the Controlled-NOT gates andthe Toffoli gates. The results show that the linear gray enhancement algorithmfor quantum images is better than the classical algorithm in both spatialcomplexity and time complexity.
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.
In classical computing, 2’s complement representation is essential for handling signed binary arithmetic, enabling efficient operations for both positive and negative numbers. However, many quantum multiplication circuits typically lack native support for 2’s complement, limiting their application to signed arithmetic tasks. Prior quantum multiplication algorithms capable of handling 2’s complement often demand an impractically large number of qubits. This paper introduces 3 novel quantum circuit designs for real-number 2’s complement multiplication, inspired by classical signed multiplication algorithms. These designs achieve a quadratic time complexity, comparable to RCA-based quantum unsigned multiplication algorithms, while significantly reducing qubit requirements compared to prior quantum signed multiplication schemes. The first model requires the largest number of qubits but applies to a broad range of cases; the second uses fewer qubits but requires specific multiplier conditions; and the third uses the fewest qubits but operates under stricter multiplier constraints and circuit positioning conditions. Benchmarking results demonstrate their ability to perform 2’s complement multiplication correctly. This work advances qubit-efficient solutions, laying a foundation for further development of 2’s complement-based quantum algorithms.