Three-Qubit Implementation of Quantum Fourier Transform for Shor’s Algorithm

This chapter deals with some of the most prominent successes of quantum computing. The most well-known quantum computing algorithm, Shor’s algorithm for factoring a number in its prime factors, is discussed in details. The key to Shor’s algorithm is the quantum Fourier transform that is explained with the help of simple examples. The role of quantum entanglement is also discussed. The next important quantum computing algorithm is Grover’s algorithm that helps in searching an item in an unsorted database. This algorithm is motivated by first discussing a quantum shell game in which a pea hidden under one of the four shells is found in one measurement with certainty each time. This amazing result is then generalized to an arbitrary number of objects and Grover’s algorithm.

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Quantum Fourier Transform and Its Application in Shor’s Algorithm

2020 4th International Conference on Computer Engineering, Information Science & Application Technology (ICCIA 2020) ◽

10.23977/iccia2020047 ◽

2020 ◽

Keyword(s):

Fourier Transform ◽

Quantum Fourier Transform ◽

Shor's Algorithm

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Halving the cost of quantum addition

Quantum ◽

10.22331/q-2018-06-18-74 ◽

2018 ◽

Vol 2 ◽

pp. 74 ◽

Cited By ~ 28

Author(s):

Craig Gidney

Keyword(s):

Fourier Transform ◽

Quantum Computation ◽

Modular Arithmetic ◽

Quantum Fourier Transform ◽

Integer Arithmetic ◽

Shor's Algorithm ◽

Surface Code ◽

The Cost ◽

Do So

We improve the number of T gates needed to perform an n-bit adder from 8n+O(1) to 4n+O(1). We do so via a "temporary logical-AND" construction which uses four T gates to store the logical-AND of two qubits into an ancilla and zero T gates to later erase the ancilla. This construction is equivalent to one by Jones, except that our framing makes it clear that the technique is far more widely applicable than previously realized. Temporary logical-ANDs can be applied to integer arithmetic, modular arithmetic, rotation synthesis, the quantum Fourier transform, Shor's algorithm, Grover oracles, and many other circuits. Because T gates dominate the cost of quantum computation based on the surface code, and temporary logical-ANDs are widely applicable, this represents a significant reduction in projected costs of quantum computation. In addition to our n-bit adder, we present an n-bit controlled adder circuit with T-count of 8n+O(1), a temporary adder that can be computed for the same cost as the normal adder but whose result can be kept until it is later uncomputed without using T gates, and discuss some other constructions whose T-count is improved by the temporary logical-AND.

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Quantum Fourier Operators and Their Application

10.5772/intechopen.94902 ◽

2021 ◽

Author(s):

Eric Sakk

Keyword(s):

Fourier Transform ◽

Quantum Computation ◽

Quantum Algorithms ◽

Quantum Fourier Transform ◽

Discrete Logarithms ◽

Universal Computation ◽

Shor's Algorithm ◽

Proper Perspective ◽

Classical Computer ◽

Novel Applications

The application of the quantum Fourier transform (QFT) within the field of quantum computation has been manifold. Shor’s algorithm, phase estimation and computing discrete logarithms are but a few classic examples of its use. These initial blueprints for quantum algorithms have sparked a cascade of tantalizing solutions to problems considered to be intractable on a classical computer. Therefore, two main threads of research have unfolded. First, novel applications and algorithms involving the QFT are continually being developed. Second, improvements in the algorithmic complexity of the QFT are also a sought after commodity. In this work, we review the structure of the QFT and its implementation. In order to put these concepts in their proper perspective, we provide a brief overview of quantum computation. Finally, we provide a permutation structure for putting the QFT within the context of universal computation.

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Entanglement and its role in Shor's algorithm

Quantum Information and Computation ◽

10.26421/qic6.7-6 ◽

2006 ◽

Vol 6 (7) ◽

pp. 630-640

Author(s):

V.M. Kendon ◽

W.J. Munro

Keyword(s):

Numerical Simulation ◽

Fourier Transform ◽

Quantum Information ◽

Quantum Information Processing ◽

Quantum Computer ◽

Quantum Algorithms ◽

Quantum Fourier Transform ◽

Shor's Algorithm ◽

Critical Resource ◽

Factoring Algorithm

Entanglement has been termed a critical resource for quantum information processing and is thought to be the reason that certain quantum algorithms, such as Shor's factoring algorithm, can achieve exponentially better performance than their classical counterparts. The nature of this resource is still not fully understood: here we use numerical simulation to investigate how entanglement between register qubits varies as Shor's algorithm is run on a quantum computer. The shifting patterns in the entanglement are found to relate to the choice of basis for the quantum Fourier transform.

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Fast quantum modular exponentiation architecture for Shor's factoring alogrithm

Quantum Information and Computation ◽

10.26421/qic14.7-8-8 ◽

2014 ◽

Vol 14 (7&8) ◽

pp. 649-682

Author(s):

Archimedes Pavlidis ◽

Dimitris Gizopoulos

Keyword(s):

Fourier Transform ◽

Modular Exponentiation ◽

Quantum Cost ◽

Quantum Fourier Transform ◽

Complex Arithmetic ◽

Factorization Algorithm ◽

Shor's Algorithm ◽

Modular Multiplier ◽

Circuit Depth

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 $2000n^2$ and requires $9n+2$ qubits, where $n$ is the number of bits of the classic number to be factored. The total quantum cost of the proposed design is $1600n^3$. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.

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