The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

Linear optics with photon counting is a prominent candidate for practical quantum computing. The protocol by Knill, Laflamme, and Milburn ͓2001, Nature ͑London͒ 409, 46͔ explicitly demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projective measurements is possible. Subsequently, several improvements on this protocol have started to bridge the gap between theoretical scalability and practical implementation. The original theory and its improvements are reviewed, and a few examples of experimental two-qubit gates are given. The use of realistic components, the errors they induce in the computation, and how these errors can be corrected is discussed.

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. Introduction.-One-way quantum computation [1] provides the ability to perform universal quantum computation (QC) using only single-qubit projective measurements, given a specially prepared and highly entangled cluster state. This is in contrast to the traditional circuit model, where unitary evolution and coherent control of individual qubits are required [2]. Apart from its conceptual importance, the cluster-state approach can also lead to practical advantages. For example, the resources required for QC using linear optics [3] can be significantly reduced by first creating photonic cluster states via nondeterministic gates [4 -6]. Recently, a four-qubit cluster state has been demonstrated optically in the single-photon regime [7].While qubits are typically used in QC, Lloyd and Braunstein [8] proposed the use of continuous variables for QC and proved that only a finite set of continuousvariable (CV) gates are needed for universal QC. In the CV approach, the continuous degree of freedom may be used directly or lower-dimensional systems may be encoded within the modes, such as in the Gottesman-KitaevPreskill (GKP) proposal [9], which encodes one qubit into each mode. This allows, for instance, for the application of standard qubit protocols to CV systems. The optical modes of the electromagnetic field provide an experimental test bed for these ideas [10].In this Letter, we describe a model of universal QC using CV cluster states. We also propose an optical implementation of our scheme where squeezed-light sources serve as the nodes of the cluster. The main advantage of this approach is that not only can computations with the cluster be performed deterministically, but also the preparation of the cluster state, including connecting the nodes, can be done unconditionally. This is in contrast to the discrete-variable linear-optics schemes [4,6,11], where cluster states are created probabilistically. Therefore, the CV approach appears to be particularly suited for further experimental demonstration of the general principles of cluster-state QC.

The promise of tremendous computational power, coupled with the development of robust error-correcting schemes, has fuelled extensive efforts to build a quantum computer. The requirements for realizing such a device are confounding: scalable quantum bits (two-level quantum systems, or qubits) that can be well isolated from the environment, but also initialized, measured and made to undergo controllable interactions to implement a universal set of quantum logic gates. The usual set consists of single qubit rotations and a controlled-NOT (CNOT) gate, which flips the state of a target qubit conditional on the control qubit being in the state 1. Here we report an unambiguous experimental demonstration and comprehensive characterization of quantum CNOT operation in an optical system. We produce all four entangled Bell states as a function of only the input qubits' logical values, for a single operating condition of the gate. The gate is probabilistic (the qubits are destroyed upon failure), but with the addition of linear optical quantum non-demolition measurements, it is equivalent to the CNOT gate required for scalable all-optical quantum computation.