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Applied and numerical harmonic analysis

The Quantum Fourier Transform

Bao Luong

Published Jan 1, 2009• 1 citation
Publisher

Abstract

The FT has many applications, particularly in the fields of quantum computation and quantum information. In these fields, the FT is often called the quantum Fourier transform . Traditionally, notation used in quantum physics (i.e., the Dirac notation) to denote vectors is different from that used in mathematics. In this chapter, we introduce the Dirac notation and describe the FT in terms of the new notation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-0-8176-4916-6_8

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The FT has many applications, particularly in the fields of quantum computation and quantum information. In these fields, the FT is often called the quantum Fourier transform . Traditionally, notation used in quantum physics (i.e., the Dirac notation) to denote vectors is different from that used in mathematics. In this chapter, we introduce the Dirac notation and describe the FT in terms of the new notation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this chapter we introduce the Hilbert space which plays the central role in quantum mechanics. For a more detailed discussion of this subject we refer to the books of Balakrishnan [5], Richtmyer [137], Sewell [146], Stakgold [152], Steeb [163], Weidmann [181], Yosida [185]. Moreover the proofs of the theorems given in this chapter can be found in these books. We assume that the reader is familiar with the notation of a linear space. First we introduce the pre-Hilbert space. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
To begin to describe quantum algorithms, we need an understanding of the physical phenomena that motivate quantum mechanics, a simple mathematical model of a potential quantum storage device and the notation to describe and manipulate such a model. In this chapter we present the context in which we will work, motivate the need for quantum mechanics and provide a quick introduction to the essential mathematical tools and notation to be used throughout the book. As part of the price of brevity, we will work with a static model of quantum mechanics and will defer additional mathematical definitions until they are needed. The reader who is already familiar with this material can easily skip to Chapter 2, giving only a cursory glance at Chapter 1 to identify any idiosyncratic notation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Quantum computing (QC) is an exciting new technology that completes the text with an application that relies heavily on the mathematics of linear algebra. There is more to explain than the conversion of common matrix notation into Dirac bra-ket notation. There is also the bizarre world of quantum physics that is inspiring a range of quantum information technologies. Qubits and their entangled superposition states are introduced to show that the “quantum advantage” stems from qubits storing exponentially more information that can be processed at exponentially faster rates. The process of preparing, modifying, and measuring qubit states is explained. Linear algebra skills are exercised by representing QC coding through the application of unitary operators to qubit states. QC algorithms are designed by diagramming quantum circuits, casting those circuits into sequences of operators applied to qubit states, and converting the operators into matrix forms. These are the tools of QC. Several algorithms that demonstrate a range of computational tasks from simple arithmetic through quantum Fourier analysis complete the chapter.
The authors present briefly the basic notions of the quantum formalism: pure and mixed states; quantum observables; quantum probability; Born’s rule, superposition, and ‘state collapse’; the projection postulate (von Neumann–Lüders postulate); Dirac’s ket and bra-vector notations; elements of quantum information theory and quantum logic; Schrödinger’s and von Neumann’s equations; unitary dynamics; and positive operator valued measures. They also define the basic mathematical notions related to the quantum formalism: Hilbert space; scalar product; norm; Hermitian operator (matrix); projector; and unitary operator and adjoint operator. This chapter may be useful for newcomers to the field, but for those readers who have preliminary knowledge about quantum mechanics (QM) they can proceed directly to advanced chapters. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.