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arXiv:2207.05309

Quantum Algorithm based on Quantum Fourier Transform for Register-by-Constant Addition

Filipe C. Ferraz

Published Jul 12, 2022• 0 citations
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Abstract

Since Shor's proposition of the method for factoring products of prime numbers using quantum computing, there has been a quest to implement efficient quantum arithmetic algorithms. These algorithms are capable of applying arithmetic operations simultaneously on large sets of values using quantum parallelism. Draper proposed an addition algorithm based on the quantum Fourier transform whose operands are two quantum registers, which I refer to as register-by-register addition. However, for cases where there is the need to be added a constant value to a target register, Draper's algorithm is more complex than necessary in terms of number of operations and number of qubits used. In this paper, I present a more efficient addition algorithm than Draper's for cases where there needs to be added just a constant to a target register.

DOI
10.48550/arxiv.2207.05309

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Since Shor's proposition of the method for factoring products of prime numbers using quantum computing, there has been a quest to implement efficient quantum arithmetic algorithms. These algorithms are capable of applying arithmetic operations simultaneously on large sets of values using quantum parallelism. Draper proposed an addition algorithm based on the quantum Fourier transform whose operands are two quantum registers, which I refer to as register-by-register addition. However, for cases where there is the need to be added a constant value to a target register, Draper's algorithm is more complex than necessary in terms of number of operations and number of qubits used. In this paper, I present a more efficient addition algorithm than Draper's for cases where there needs to be added just a constant to a target register.
It has been proven that a quantum computer is superior to an electronic com puter in solving some NP problems. Based on quantum operations, this article proposes a new quantum sum and quantum multiplication, and then the floating point quantum multiplier and quantum sum are created on the basis of fixed point number operations. These studies lay the foundation for the quantum implementation of digital filters. This article provides a new way to calculate the summator on a quantum computer. This method uses the quantum Fourier transform (QFT) and reduces the number of qubits needed to be added by eliminating the need to temporarily transfer bits. This approach also allows you to add a classical number to a quantum superposition without encoding a classical number in a quantum register. This method also allows for mass parallelization during its execution. Adding and multiplying capabilities based on QFT are improved with some changes. The proposed operations are compared with the operations of close quantum arithmetic.
Quantum circuits for elementary arithmetic operations are important not only for implementing Shor's factoring algorithm on a quantum computer but also for understanding the computational power of small quantum circuits, such as linear-size or logarithmic-depth quantum circuits. This paper surveys some recent approaches to constructing efficient quantum circuits for elementary arithmetic operations and their applications to Shor's factoring algorithm. It covers addition, comparison, and the quantum Fourier transform used for addition.
We present a novel and efficient in terms of circuit depth design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a generic modular quantum multiplier which is the fundamental block for modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about 2000n22000n^{2}2000n2 and requires 9n+29n+29n+2 qubits, where nnn is the number of bits of the classical number to be factored. The total quantum cost of the proposed design is 1600n31600n^{3}1600n3. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.
We present a novel and efficient, in terms of circuit depth, design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the Quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a quantum modular multiplier which is the fundamental block for the modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about 2000n22000n^22000n2 and requires 9n+29n+29n+2 qubits, where nnn is the number of bits of the classic number to be factored. The total quantum cost of the proposed design is 1600n31600n^31600n3. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.
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.