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.

Continuous-variable cluster states offer a potentially promising method of implementing a quantum computer. This paper extends and further refines theoretical foundations and protocols for experimental implementation. We give a cluster-state implementation of the cubic phase gate through photon detection, which, together with homodyne detection, facilitates universal quantum computation. In addition, we characterize the offline squeezed resources required to generate an arbitrary graph state through passive linear optics. Most significantly, we prove that there are universal states for which the offline squeezing per mode does not increase with the size of the cluster. Simple representations of continuous-variable graph states are introduced to analyze graph state transformations under measurement and the existence of universal continuous-variable resource states

We derive necessary conditions in terms of the variances of position and momentum linear combinations for all kinds of separability of a multi-party multi-mode continuous-variable state. Their violations can be sufficient for genuine multipartite entanglement, provided the combinations contain both conjugate variables of all modes. Hence a complete state determination, for example by detecting the entire correlation matrix of a Gaussian state, is not needed.Comment: 13 pages, 3 figure

We show that one single-mode squeezed state distributed among N parties using linear optics suffices to produce a truly N -partite entangled state for any nonzero squeezing and arbitrarily many parties. From this N -partite entangled state, via quadrature measurements of N − 2 modes, bipartite entanglement between any two of the N parties can be 'distilled', which enables quantum teleportation with an experimentally determinable fidelity better than could be achieved in any classical scheme.PACS numbers: 03.65.Bz, 42.50.Dv Entanglement is seen as an essential ingredient in quantum communication and computation. For example, it enables quantum teleportation which was originally proposed for systems of discrete variables [1]. Later, quantum teleportation was also proposed for continuous variables [2,3]. The simplest teleportation schemes rely on bipartite entanglement, the entanglement of a pair of systems shared by two parties. For pure states, this kind of entanglement is well-understood and can be quantified [4]. Multipartite entanglement, the entanglement shared by more than two parties, is much more difficult to quantify [5]. Yet in the laboratory, the creation of tripartite discrete-variable entanglement, yielding so-called GHZ states [6], has been reported for single-photon polarization states [7] and using nuclear magnetic resonance [8].Continuous-variable quantum teleportation of arbitrary coherent states has been realized experimentally with bipartite entanglement built from two single-mode squeezed vacuum states combined at a beamsplitter [9]. In the absence of entanglement the best mean fidelity of the reconstructed coherent states is F = 1 2 [10]. Experimentally, F = 0.58 ± 0.02 was achieved. Though this limits our attention to the teleportation of a rather modest set of non-orthogonal states, the fidelity gives a clear experimental signal for the presence of entanglement.Now it is known that even one single-mode squeezed state incident on a beamsplitter yields a bipartite entangled state [11]. This result is in agreement with entropic measures of bipartite pure-state entanglement [12]. If one single-mode squeezed state were distributed among N parties using linear optics would we obtain a truly N -partite entangled state? We will show that we can answer this question using the fidelity criterion for teleporting unknown coherent states. In particular, we will see that one single-mode squeezed state is sufficient to allow quantum teleportation between any two of the N parties with the help of all other parties. The assistance by the other N − 2 parties only relies on local measurements and classical communication. Due to these N − 2 measurements, bipartite entangled states are 'distilled' from the initial N -partite entangled state.The 'position' and 'momentum' of a 1-D wavepacket (units-free withh = 1 2 as in Ref.[13]) are the electric quadrature amplitudes representing the quantum state of a single polarization of a single transverse mode of electromagnetic radiation. We define the action of an i...

We describe a quantum repeater protocol for long-distance quantum communication. In this scheme, entanglement is created between qubits at intermediate stations of the channel by using a weak dispersive light-matter interaction and distributing the outgoing bright coherent light pulses among the stations. Noisy entangled pairs of electronic spin are then prepared with high success probability via homodyne detection and postselection. The local gates for entanglement purification and swapping are deterministic and measurement-free, based upon the same coherent-light resources and weak interactions as for the initial entanglement distribution. Finally, the entanglement is stored in a nuclear-spin-based quantum memory. With our system, qubit-communication rates approaching 100 Hz over 1280 km with fidelities near 99% are possible for reasonable local gate errors.PACS numbers: 03.67. Hk, 03.67.Mn, 42.50.Pq In a quantum repeater, long-distance entanglement is created by distributing entangled states over sufficiently short segments of a channel such that the noisy entangled states in each segment can be purified and connected via entanglement swapping [1,2]. The resulting entanglement between the qubits at distant stations can then be used, for example, to teleport quantum information [3] or transmit secret classical information [4]. Existing approaches to quantum repeaters generate entanglement using postselection with single-photon detection [5,6,7]. In these schemes, high-fidelity entanglement is created and the subsequent entanglement purification is needed primarily to compensate the degrading effect of connecting the imperfect entangled pairs via swapping. However, due to their rather low success probabilities in the initial entanglement distribution, these protocols feature very low communication rates.More efficient schemes, compatible with existing classical optical communication networks, would involve bright multi-photon signals. In this Letter, we propose such a scheme that operates in a regime of modest initial fidelities, but creates entangled states at high speed. The high rate in the generation of entangled pairs is mainly due to the near-unit efficiencies for the homodyne detection of bright signals, as opposed to the low efficiencies of single-photon detectors. In our scheme, the resulting entangled pairs will be discrete atomic qubit states, but the probe system we use is a bright light pulse described and measured via a continuous phase observable; hence, our quantum repeater is "hybrid" not only because it employs matter signals and light probes (as in other schemes), but more distinctly, by utilizing both discrete and continuous quantum variables.In general, in order to realize universal quantum computation or, more relevant to us here, long-distance quantum communication, a nonlinear element is needed for the implementation. Optically, this nonlinear element may be introduced in at least two possible ways. The first method uses only linear transformations, but a measurement-induced nonlinear...

We provide a unified graphical calculus for all Gaussian pure states, including graph transformation rules for all local and semi-local Gaussian unitary operations, as well as local quadrature measurements. We then use this graphical calculus to analyze continuous-variable (CV) cluster states, the essential resource for oneway quantum computing with CV systems. Current graphical approaches to CV cluster states are only valid in the unphysical limit of infinite squeezing, and the associated graph transformation rules only apply when the initial and final states are of this form. Our formalism applies to all Gaussian pure states and subsumes these rules in a natural way. In addition, the term "CV graph state" currently has several inequivalent definitions in use. Using this formalism we provide a single unifying definition that encompasses all of them. We provide many examples of how the formalism may be used in the context of CV cluster states: defining the "closest" CV cluster state to a given Gaussian pure state and quantifying the error in the approximation due to finite squeezing; analyzing the optimality of certain methods of generating CV cluster states; drawing connections between this new graphical formalism and bosonic Hamiltonians with Gaussian ground states, including those useful for CV one-way quantum computing; and deriving a graphical measure of bipartite entanglement for certain classes of CV cluster states. We mention other possible applications of this formalism and conclude with a brief note on fault tolerance in CV one-way quantum computing.

The linear optical creation of Gaussian cluster states, a potential resource for universal quantum computation, is investigated. We show that for any Gaussian cluster state, the canonical generation scheme in terms of QND-type interactions, can be entirely replaced by off-line squeezers and beam splitters. Moreover, we find that, in terms of squeezing resources, the canonical states are rather wasteful and we propose a systematic way to create cheaper states. As an application, we consider Gaussian cluster computation in multiple-rail encoding. This encoding may reduce errors due to finite squeezing, even when the extra rails are achieved through off-line squeezing and linear optics.PACS numbers: 03.67. Lx, 42.50.Dv, 42.25.Hz Introduction.-The cluster-state model for quantum computation [1] is a conceptually interesting alternative to the more conventional circuit model [2]. Once a suitable multi-party entangled cluster state has been prepared, universal quantum gates can be effected through the cluster via only single-party projective measurements and feedforward. Though originally based upon qubits, the cluster-state model can be also applied to other discrete-variable systems (qudits) as well as to continuous quantum variables [3].Linear optics represents one of the most practical approaches to the realization of quantum information protocols, both for discrete-variable (DV) [4] and continuous-variable (CV) implementations [5]. In the DV case, efficient entangling gates cannot be achieved with single photons and linear optics. Nonetheless, probabilistic gates can be applied off-line to an entangled multiphoton state that eventually serves as a resource for the on-line computation [6]. A similar approach uses DV photonic cluster states, leading to a significant reduction in the resource consumption [7,8]. However, the generation of the optical cluster states remains highly probabilistic in this case.Although up to six-qubit single-photon cluster states have been created via postselection using nonlinear and linear optics [9, 10], a possible deterministic, unconditional realization of optical cluster states would be based on continuous variables. Here, the resources are squeezed states of light and the Gaussian cluster states may be created via quadratic quantum nondemolition (QND) interactions [11]. These interactions, however, cannot be realized through beam splitters alone. Additional "on-line" squeezers are needed for every single link of the cluster state, again rendering the mechanism for cluster generation rather inefficient with current technology. Moreover, the squeezing of the resource states will always be finite, inevitably resulting in errors in the cluster computation. In this paper, we will address both issues: the avoidance of on-line squeezing in cluster-state generation and the reduction of finite-squeezing induced errors in cluster-state computation.

The development of quantum information processing has traditionally followed two separate and not immediately connected lines of study. The main line has focused on the implementation of quantum bit (qubit) based protocols whereas the other line has been devoted to implementations based on high-dimensional Gaussian states (such as coherent and squeezed states). The separation has been driven by the experimental difficulty in interconnecting the standard technologies of the two lines. However, in recent years, there has been a significant experimental progress in refining and connecting the technologies of the two fields which has resulted in the development and experimental realization of numerous new hybrid protocols. In this Review, we summarize these recent efforts on hybridizing the two types of schemes based on discrete and continuous variables.