We introduce measurement-based quantum repeaters, where small-scale measurement-based quantum processors are used to perform entanglement purification and entanglement swapping in a long-range quantum communication protocol. In the scheme, pre-prepared entangled states stored at intermediate repeater stations are coupled with incoming photons by simple Bell-measurements, without the need of performing additional quantum gates or measurements. We show how to construct the required resource states, and how to minimize their size. We analyze the performance of the scheme under noise and imperfections, with focus on small-scale implementations involving entangled states of few qubits. We find measurement-based purification protocols with significantly improved noise thresholds. Furthermore we show that already resource states of small size suffice to significantly increase the maximal communication distance. We also discuss possible advantages of our scheme for different set-ups

Measurement-based quantum computation (MBQC) represents a powerful and flexible framework for quantum information processing, based on the notion of entangled quantum states as computational resources. The most prominent application is the one-way quantum computer, with the cluster state as its universal resource. Here we demonstrate the principles of MBQC using deterministically generated graph states of up to 7 qubits, in a system of trapped atomic ions. Firstly we implement a universal set of operations for quantum computing. Secondly we demonstrate a family of measurement-based quantum error correction codes, and show their improved performance as the code length is increased. We show that all our graph states violate a multipartite Bell inequality and are therefore capable of information processing tasks that cannot be described by a local hidden variable model. The methods presented can directly be scaled up to generate graph states of several tens of qubits.The circuit model of quantum computation is conceptually similar to a classical computer: a register of two-level systems in a simple initial product state are manipulated using unitary quantum logic gates [1]. MBQC [2] represents a conceptually and practically different approach: after preparing an entangled cluster state of qubits [3], computation proceeds by performing measurements and feedforward. Both approaches present different theoretical and practical challenges to realisation and warrant investigation in parallel.Recently, researchers have found novel applications for MBQC beyond universal QC, including e.g. blind quantum computation [4, 5] measurement-based entanglement purification [6] and quantum error correction [7], featuring very high thresholds. Owing to the two-stage process of MBQC -resource creation followed by its processing -resources states can be purified and manipulated beforehand. This offers a large degree of flexibility in optimizing and compressing schemes for quantum information processing. Schemes for correcting errors in universal MBQC have also been found with extremely high tolerence to errors, compared to those known for the circuit model of QC [7][8][9].Important experimental progress on MBQC has been made using entangled states of up to 8 photonic qubits [10][11][12]. Scaling up the non-deterministic methods used to generate entangled states in these works is very challenging, since their success probably reduces exponentially in photon number. Very recently there has been work on generating cluster states in continuous variables of light fields [13].In this work we present the first demonstration of MBQC using trapped ions. Furthermore, we make two experimental steps forward in the model of MBQC that are systemindependent: the deterministic generation of cluster states and the demonstration of quantum error correction (QEC). The paper is organised as follows; firstly MBQC is briefly reviewed and our approach to preparing cluster states is summarised; then a universal set of operations is presented using a 4 qubit...

We investigate measurement-based entanglement purification protocols (EPP) in the presence of local noise and imperfections. We derive a universal, protocol-independent threshold for the required quality of the local resource states, where we show that local noise per particle of up to 24% is tolerable. This corresponds to an increase of the noise threshold by almost an order of magnitude, based on the joint measurement-based implementation of sequential rounds of few-particle EPP. We generalize our results to multipartite EPP, where we encounter similarly high error thresholds.

We review and discuss the potential of using measurement-based elements in quantum communication schemes, where certain tasks are realized with the help of entangled resource states that are processed by measurements. We consider long-range quantum communication based on the transmission of encoded quantum states, where encoding, decoding and syndrome read-out are implemented using small-scale resource states. We also discuss entanglement-based schemes and consider measurement-based quantum repeaters. An important element in these schemes is entanglement purification, which can also be implemented in a measurement-based way. We analyze the influence of noise and imperfections in these schemes, and show that measurement-based implementation allows for very large error thresholds of the order of 10% noise per qubit and more. We show how to obtain optimal resource states for different tasks, and discuss first experimental realizations of measurement-based quantum error correction using trapped ions and photons.

We present a hybrid scheme for quantum computation that combines the modular structure of elementary building blocks used in the circuit model with the advantages of a measurement-based approach to quantum computation. We show how to construct optimal resource states of minimal size to implement elementary building blocks for encoded quantum computation in a measurement-based way, including states for error correction and encoded gates. The performance of the scheme is determined by the quality of the resource states, where within the considered error model a threshold of the order of 10% local noise per particle for fault-tolerant quantum computation and quantum communication.

The endeavor to develop quantum networks gave rise to a rapidly developing field with far-reaching applications such as secure communication and the realization of distributed computing tasks. This ultimately calls for the creation of flexible multiuser structures that allow for quantum communication between arbitrary pairs of parties in the network and facilitate also multiuser applications. To address this challenge, we propose a two-dimensional quantum repeater architecture to establish long-distance entanglement shared between multiple communication partners in the presence of channel noise and imperfect local control operations. The scheme is based on the creation of self-similar multiqubit entanglement structures at growing scale, where variants of entanglement swapping and multiparty entanglement purification are combined to create high-fidelity entangled states. We show how such networks can be implemented using trapped ions in cavities.

We introduce an alternative type of quantum repeater for long-range quantum communication with improved scaling with the distance. We show that by employing hashing, a deterministic entanglement distillation protocol with one-way communication, one obtains a scalable scheme that allows one to reach arbitrary distances, with constant overhead in resources per repeater station, and ultrahigh rates. In practical terms, we show that, also with moderate resources of a few hundred qubits at each repeater station, one can reach intercontinental distances. At the same time, a measurement-based implementation allows one to tolerate high loss but also operational and memory errors of the order of several percent per qubit. This opens the way for long-distance communication of big quantum data.

We report on the experimental violation of multipartite Bell inequalities by entangled states of trapped ions. First we consider resource states for measurement-based quantum computation of between 3 and 7 ions and show that all strongly violate a Bell-type inequality for graph states, where the criterion for violation is a sufficiently high fidelity. Second we analyze GHZ states of up to 14 ions generated in a previous experiment using stronger Mermin-Klyshko inequalities, and show that in this case the violation of local realism increases exponentially with system size. These experiments represent a violation of multipartite Bell-type inequalities of deterministically prepared entangled states. In addition, the detection loophole is closed. Introduction -How strong can physical correlations be? Bell inequalities set a bound on the possible strength of non-local correlations that could be explained by a theory based on some fundamental assumptions known as "local realism". Quantum mechanics predicts the existence of states which violate Bell's inequality, rendering a description of these states by a local hidden variable (LHV) model impossible. While first discovered for bipartite systems in a two-measurement setting [1], Bell inequalities have been extended to multi-measurement settings and multipartite systems, leading to a more profound violation for larger systems of different kinds [2][3][4][5][6].In particular, it was shown that all graph states violate local realism, where the possible violation increases exponentially with the number of qubits for certain types of states [4][5][6]. Graph states [7,8] are a large class of multiqubit states that include a number of interesting, highly entangled states, such as the 2D cluster states [9] or the GHZ states. They serve as resources for various tasks in quantum information processing, including measurement-based quantum computation (MBQC) [10,11] or quantum error correction [12]. The results of [4,5] (see also [13]) provide an interesting connection between the usability of states for quantum information processing and the possibility to describe them by a LHV model.Here we experimentally demonstrate the violation of multi-partite Bell-type inequalities for graph states generated with trapped ions. First we consider a range of graph states that find application in MBQC and observe strong violations in all cases. Second, for a different class of graph states, we investigate the scaling of the multipartite Bell violation with system size and confirm an exponential increase: that is the quantum correlations in these systems become exponentially stronger than allowed by any LHV model.To be more precise, in the first part of our work we consider graph states that allow one to perform singlequbit and two-qubit gates in MBQC, as well as resource states for measurement-based quantum error correction [14]. That is, we demonstrate that not only the codewords of quantum error correction codes violate local re-