# Quantum arithmetic with the quantum Fourier transform

@article{RuizPerez2017QuantumAW,
title={Quantum arithmetic with the quantum Fourier transform},
author={Lidia Ruiz-Perez and Juan Carlos Garc{\'i}a-Escart{\'i}n},
journal={Quantum Information Processing},
year={2017},
volume={16},
pages={1-14}
}
• Published 21 November 2014
• Physics
• Quantum Information Processing
The quantum Fourier transform offers an interesting way to perform arithmetic operations on a quantum computer. We review existing quantum Fourier transform adders and multipliers and comment some simple variations that extend their capabilities. These modified circuits can perform modular and non-modular arithmetic operations and work with signed integers. Among the operations, we discuss a quantum method to compute the weighted average of a series of inputs in the transform domain. One of the…
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## References

SHOWING 1-10 OF 44 REFERENCES
Quantum networks for elementary arithmetic operations.
• Computer Science
Physical review. A, Atomic, molecular, and optical physics
• 1996
This work provides an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation, and shows that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorized.
A new method for computing sums on a quantum computer is introduced. This technique uses the quantum Fourier transform and reduces the number of qubits necessary for addition by removing the need for
Quantum computation and quantum information
• T. Paul
• Physics
Mathematical Structures in Computer Science
• 2007
This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal
Quantum Carry-Save Arithmetic
This paper shows how to design efficient arithmetic elements out of quantum gates using "carry-save" techniques borrowed from classical computer design, which reduces the quantum gate delay from O(N^3) to O( N log N) at a cost of increasing the number of qubits required.
Quantum Arithmetic on Galois Fields
• Computer Science
• 2003
The controlled-multiplication operation, which is the only group-specific operation in Shor's algorithms for factoring and solving the Discrete Log Problem, is described, and the detailed size, width and depth complexity of such circuits are given, which ultimately will allow us to obtain detailed upper bounds on the amount of quantum resources needed to solve instances of the DLP.
Fast quantum modular exponentiation architecture for Shor's factoring algorithm
• Computer Science
Quantum Inf. Comput.
• 2014
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
Quantum arithmetic and numerical analysis using Repeat-Until-Success circuits
• Computer Science, Physics
Quantum Inf. Comput.
• 2016
We develop a method for approximate synthesis of single--qubit rotations of the form $e^{-i f(\phi_1,\ldots,\phi_k)X}$ that is based on the Repeat-Until-Success (RUS) framework for quantum circuit
Arithmetic on a distributed-memory quantum multicomputer
• Computer Science
JETC
• 2008
It is shown that the teledata approach performs better, and that carry-ripple adders perform well when the teleportation block is decomposed so that the key quantum operations can be parallelized.
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• P. Shor
• Computer Science
SIAM Rev.
• 1999
Efficient randomized algorithms are given for factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems.
Circuit for Shor's algorithm using 2n+3 qubits
A circuit which uses 2n + 3 qubits and 0(n3lg(n)) elementary quantum gates in a depth of 0( n3) to implement the factorization algorithm using Shor's algorithm on a quantum computer.