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 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.
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.
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.
We describe a generalization of the cluster-state model of quantum computation to continuous-variable systems, along with a proposal for an optical implementation using squeezed-light sources, linear optics, and homodyne detection. For universal quantum computation, a nonlinear element is required. This can be satisfied by adding to the toolbox any single-mode non-Gaussian measurement, while the initial cluster state itself remains Gaussian. Homodyne detection alone suffices to perform an arbitrary multimode Gaussian transformation via the cluster state. We also propose an experiment to demonstrate cluster-based error reduction when implementing Gaussian operations.
One-way quantum computation is a very promising candidate to fulfill the capabilities of quantum information processing. Here we demonstrate an important set of unitary operations for continuous variables using a linear cluster state of four entangled optical modes. These operations are performed in a fully measurement-controlled and completely unconditional fashion. We implement three different levels of squeezing operations and a Fourier transformation, all of which are accessible by selecting the correct quadrature measurement angles of the homodyne detections. Though not sufficient, these linear transformations are necessary for universal quantum computation.
In the previous chapter we developed some of the basic aspects of quantum information processing with single photons as qubits. Apart from noting the obvious benefit of using light for quantum communication, we identified some difficulties in manipulating quantum information that is encoded in photons. In particular, it is difficult to construct two-qubit gates for photonic qubits. In this chapter we will have to face this difficulty head-on in our discussion of quantum computation with single photons and linear-optical elements. The possibility of a quantum computer based on single photons, linear optics, and photon counting was demonstrated in a landmark paper by Knill, Laflamme, and Milburn in 2001. Their protocol is commonly referred to as the ‘ klm protocol’. In subsequent years, the klm protocol has been dramatically slimmed down in terms of complexity and the necessary resources to create the universal set of quantum gates. For pedagogical reasons we give the streamlined version here, and we briefly describe the original klm protocol in Appendix 2. We will make extensive use of the results in Section 1.4 about mode transformations, and Section 2.3 about cluster-state quantum computing. We start this chapter, however, with a description of linear-optical networks, and how they fail as deterministic quantum computers. In Section 6.2 we discuss the principle of post-selection, and how this can be used to create probabilistic gates with deterministic feed-forward control.
With the exception of references made to optical elements and detection techniques when discussing Gaussian operations and POVMs, and in their applications, the theory developed in this book has been largely presented as an abstract mathematical construction. This chapter intends to liaise our theoretical pictures to the wide world of controllable, coherent quantum systems that share the continuous variable description. Emphasis will be on the emergence of the canonical commutation relations in different contexts and on the introduction of the relevant Hamiltonian descriptions with the understanding that, once the basic formal analogy is established, the approaches and results of the previous chapters will be in principle applicable to the system at hand. Note, as a caveat, that some of the Hamiltonians listed in this chapter are not quadratic, notably those involving optical nonlinearities, superconducting systems and cold atoms, so that the Gaussian theory would not apply to them. In fact, we shall take the opportunity to briefly discuss the more exhaustive control, beyond the Gaussian regime, allowed by nonlinear Hamiltonians. Upon the description of the corresponding degrees of freedom, we will also venture to sketch two well-known proposals for the realisation of hardware for quantum computation, based respectively on optical modes and trapped ions, where quantum continuous variables play a significant role. The aim of this chapter is hence not providing a thorough and exhaustive discussion of each continuous variable system, but rather to grant the reader a degree of awareness on the possibilities offered by the diverse experimental platforms to which the theory applies. Thus, the conceptual elements will typically overshadow the (often very important when applications are concerned!) technical aspects, and glaring simplifications will be adopted. Given the central role played by the optical embodiment of quantum continuous variables, an exception to this cursory treatment has been made for the quantisation of the electromagnetic field, carried out below in detail from the classical theory. This will also present the opportunity to provide a full derivation of the input-output formalism and the associated quantum Langevin equations, which to our knowledge were never previously dealt with at a pedagogical level in the time domain, and to discuss a bag of quantum optical tools, such as the rotating wave approximation, which, although featuring in a number of introductory quantum optics textbooks, find such wide application to deserve a place in any attempt at a self-contained treatment.