One of our major goals is the development of a collection of techniques for computing the discrete Fourier transform. We shall find many such techniques, each with different advantages and each best-used in different circumstances. There are two basic strategies. One strategy is to change a one-dimensional Fourier transform into a two-dimensional Fourier transform of a form that is easier to compute. The second strategy is to change a one-dimensional Fourier transform into a small convolution, which is then computed by using the techniques described in Chapter 5. Good algorithms for computing the discrete Fourier transform will use either or both of these strategies to minimize the computational load. In Chapter 6, we shall describe how the fast Fourier transform algorithms are used to perform, in conjunction with the convolution theorem, the cyclic convolutions that are used to compute the long linear convolutions forming the output of a digital filter.

Throughout the chapter, we shall usually regard the complex field as the field of the computation, or perhaps the real field. However, most of the algorithms we study do not depend on the particular field over which the Fourier transform is defined. In such cases, the algorithms are valid in an arbitrary field. In some cases, the general idea behind an algorithm does not depend on the field over which the Fourier transform is defined, but some small detail of the algorithm may depend on the field.

In earlier chapters, we saw ways in which the Fourier transform can be broken into pieces and ways in which a convolution algorithm can be turned into an algorithm for the Fourier transform. It also is possible to break a multidimensional Fourier transform into pieces and to turn an algorithm for the multidimensional convolution into an algorithm for the multidimensional Fourier transform. The possibilities now are even richer than they were in the one-dimensional case. We shall discuss a variety of methods.

The algorithms for multidimensional Fourier transforms are studied in the easy way by studying only the algorithms for the two-dimensional Fourier transform as representative of the more general case. The discussion and the formulas for the two-dimensional Fourier transforms can be extended directly to a discussion of the multidimensional Fourier transforms.

Multidimensional Fourier transforms arise naturally from problems that are intrinsically multidimensional. They also arise artificially as a way of computing a one-dimensional Fourier transform. This chapter includes such methods for computing a one-dimensional Fourier transform, and so we will give practical methods for building large one-dimensional Fourier transform algorithms from the small one-dimensional Fourier transform algorithms that were studied in Chapter 3.

Small-radix Cooley–Tukey algorithms

A two-dimensional discrete Fourier transform can be computed by applying the Cooley–Tukey FFT first to each row and then to each column.

Just as one can define a one-dimensional convolution, so one can define a multidimensional convolution. Multidimensional linear convolutions and cyclic convolutions can be defined in any field of interest on multidimensional arrays of data and are useful in many ways. We have seen in earlier chapters that multidimensional arrays and multidimensional convolutions can be created artificially as part of an algorithm for processing a one-dimensional data vector. Multidimensional arrays also arise naturally in many signal-processing problems, especially in the processing of image data such as satellite reconnaissance photographs, medical imagery including X-ray images, seismic records, and electron micrographs.

This chapter will begin the study of fast algorithms for multidimensional convolutions by nesting fast algorithms for one-dimensional convolutions. Then we shall study ways to construct a fast algorithm for a one-dimensional cyclic convolution by temporarily mapping it into a multidimensional convolution, a procedure that is known as the Agarwal–Cooley convolution algorithm. The Agarwal–Cooley algorithm for one-dimensional cyclic convolution is a powerful adjunct to the convolution methods studied in Chapter 6, which become unwieldy for large blocklength. It gives a way to build algorithms for large one-dimensional cyclic convolutions by combining the small convolution algorithms. Then, in the latter half of the chapter, we shall study methods that are derived specifically to compute two-dimensional convolutions.

Nested convolution algorithms

A two-dimensional convolution is an operation on a pair of two-dimensional arrays.

Algorithms for computation are found everywhere, and efficient versions of these algorithms are highly valued by those who use them. We are mainly concerned with certain types of computation, primarily those related to signal processing, including the computations found in digital filters, discrete Fourier transforms, correlations, and spectral analysis. Our purpose is to present the advanced techniques for fast digital implementation of these computations. We are not concerned with the function of a digital filter or with how it should be designed to perform a certain task; our concern is only with the computational organization of its implementation. Nor are we concerned with why one should want to compute, for example, a discrete Fourier transform; our concern is only with how it can be computed efficiently. Surprisingly, there is an extensive body of theory dealing with this specialized topic – the topic of fast algorithms.

Introduction to fast algorithms

An algorithm, like most other engineering devices, can be described either by an input/output relationship or by a detailed explanation of its internal construction. When one applies the techniques of signal processing to a new problem one is concerned only with the input/output aspects of the algorithm. Given a signal, or a data record of some kind, one is concerned with what should be done to this data, that is, with what the output of the algorithm should be when such and such a data record is the input.

Qubits are not the only information carriers that can be used for quantum information processing. In this chapter, we will focus on quantum communication with ‘continuous quantum variables’, or continuous variables for short. In the context of quantum information processing we will also call continuous variables ‘qunats’. We have seen in Chapter 2 that a natural representation of a continuous variable is given by the position of a particle. The conjugate continuous variable is then the momentum of the particle. Unfortunately, the eigenstates of the position and momentum operators are not physical, and we have to construct suitable approximations to these states that can be created in the laboratory. Any practical information processing device must then take into account the deviation of the actual states from the ideal position and momentum eigenstates. Rather than the position and momentum of a particle, we will consider here the two position and momentum quadratures of an electromagnetic field mode. These operators obey the same commutation relations as the canonical position and momentum operators, but they are not the physical position and momentum of field excitations. Approximate eigenstates of the quadrature operators can be constructed in the form of squeezed coherent states. We define a quantum mechanical phase space for quadrature operators, similar to a classical phase space for position and momentum. Probability distributions in the classical phase space then become quasi-probability distributions over the quadrature phase space.We will develop one of these distributions, namely the Wigner function, and identify certain phase-space transformations of the Wigner function with linearoptical and squeezing operations.

The field of quantum information processing has reached a level of maturity, and spans such a wide variety of topics, that it merits further specialization. In this book, we consider quantum information processing with optical systems, including quantum communication, quantum computation, and quantum metrology. Optical systems are the obvious choice for quantum communication, since photons are excellent carriers of quantum information due to their relatively slow decoherence. Indeed, many aspects of quantum communication have been demonstrated to the extent that commercial products are now available. The importance of optical systems for quantum communication leads us to ask whether we can construct integrated systems for communication and computation in which all processing takes place in optical systems. Recent developments indicate that while full-scale quantum computing is still extremely challenging, optical systems are one of the most promising approaches to a fully functional quantum computer.

This book is aimed at beginning graduate students who are starting their research career in optical quantum information processing, and it can be used as a textbook for an advanced master's course. The reader is assumed to have a background knowledge in classical electrodynamics and quantum mechanics at the level of an undergraduate physics course. The nature of the topic requires familiarity with quantized fields, and since this is not always a core topic in undergraduate physics, we derive the quantum mechanical formulation of the free electromagnetic field from first principles. Similarly, we aim to present the topics in quantum information theory in a self-contained manner.

In the previous chapter we have discussed some important quantum information processing techniques based on continuous variables, or ‘qunats’. In this chapter, we describe methods for extending these techniques to quantum computation with optical qunats. We will first show in Section 9.1 how the optical qunat states are initialized using single-mode squeezing, and we give optical implementations of some of the most important single-qunat gates, such as the Heisenberg-Weyl operators and the Fourier transform. In Section 9.2, we study the theory of two-mode Gaussian operations, and we construct a general optical circuit that can implement any two-mode Gaussian operation. In Section 9.3, we introduce the stabilizer formalism for qunats, and use it to derive a Gottesman-Knill theorem. This leads to the general requirement of nonlinear gates for qunats, and their optical implementation will be discussed in Section 9.4. Having thus established the optical implementation of universal quantum computing with continuous variables, we extend this to the one-way model in Section 9.5, where we introduce cluster states for continuous variables. Finally, in Section 9.6, we discuss various quantum error-correction techniques for qunat quantum computing.

Single-mode optical qunat gates

We have already seen in Chapter 2 that the single qunat gates are fundamentally different from qubit gates. In the context of optical implementations, this difference arises due to the complementary nature of qubits and qunats: qubits that are represented by single photons can be considered particles, whereas qunats are encoded in the wave properties of the optical field.

In Chapter 7 we discussed the interaction between atoms and photons. We found that the coupling between a single atom and a single photon is rather weak, unless the atom and the photon are contained within a small cavity. The only way to significantly affect a free atom with light is to use a state of light with many photons, and in particular we showed how to use classical laser fields to prepare particular atomic states. However, there is another way of enhancing the interaction in a useful way. We can use many atoms rather than a single one. In this chapter, we will show that the interaction between a single photon and an ensemble of atoms can be greatly increased through the phenomenon of collective enhancement. We will show that this gives an atomic ensemble a relatively large susceptibility, and that multilevel atoms can be used to generate large optical nonlinearities. These offer the possibility of slowing, and even stopping, a beam of light; a phenomenon that goes hand in hand with electromagnetically induced transparency. We will discuss how a quantum state of light can be stored in and retrieved from an ensemble of atoms, and how the states of different atomic ensembles can be entangled with one another. We will show that collective states of atoms can even be used as a single qubit or several qubits, and finally, we will show that atomic ensembles can be use to mediate a photon–photon interaction.

In general, all forms of information processing can be considered in a quantum mechanical context, including for example communication, channel capacities, and the quantum limits to information extraction. In this chapter we introduce the basic features of quantum information processing. We first present qubits as abstractions of two-level addressable quantum systems, before generalizing this concept to higher-dimensional qudit systems, and continuous variables. We introduce the stabilizer formalism towards the beginning of our discussion as a convenient way to track quantum information. In Section 2.2 we derive the no-cloning theorem, which leads us to cryptography, teleportation, and quantum repeaters. Section 2.3 describes quantum computing with qubits. We start with a brief description of the circuit model, and use it to define cluster states and the one-way model. We study the properties of cluster states and discuss error correction. In the last section we introduce quantum computing over continuous variables.

Quantum information

Classically, information is carried by ‘bits’, which are physical systems that have two (macroscopic) states denoted by 0 and 1. These bits are described classically, which immediately raises the question of how we should extend this information-theoretic description to the quantum mechanical case. This will lead to the quantum unit of information, the quantum bit or ‘qubit’.

2.1.1 Classical and quantum bits

We can define the qubit as a two-dimensional quantum system, which means that the state space of the system is a two-dimensional complex Hilbert space ℋ.