Long range failure-tolerant entanglement distribution                                                                     ...
2                                                                     is grown across quantum repeater stations via probab...
3                                                                                   we overcome the impact of high EO fail...
4                              Dumbbell                            stations are perfect (when heralded as successful); the...
5                0.12                                              0.15                                        FL=0.00    ...
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Long range failure tolerant entanglement distribution by ying li sean d barrett thomas m stace and simon c benjamin

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We introduce a protocol to distribute entanglement between remote parties. Our protocol is
based on a chain of repeater stations, and exploits topological encoding to tolerate very high levels
of defects and errors. The repeater stations may employ probabilistic entanglement operations which
usually fail; ours is the rst protocol to explicitly allow for technologies of this kind. Given an error
rate between stations in excess of 10%, arbitrarily long range high delity entanglement distribution
is possible even if the heralded failure rate within the stations is as high as 99%, providing that
unheralded errors are low (order 0:01%).
PACS numbers:
Introduction. Distributing an entangled state among
remote quantum computers is one of the fundamental
tasks of quantum information technologies. It is crucial
for quantum teleportation, quantum cryptography and
distributed quantum computing. Using direct transmis-
sion, the success probability of transmitting a qubit and
the delity of the resulting quantum state decrease expo-
nentially with distance. Therefore, one needs quantum
repeaters to achieve long distance entanglement [1, 2]. A
good quantum repeater protocol should be fault-tolerant
and support a high communication rate. In this paper,
we will propose a protocol to distribute entanglement be-
tween two remote quantum computers. We consider noise
in quantum communication channels, and of course errors
generated by operations within the repeaters. We assume
that the repeater stations may employ non-deterministic
entanglement operations (EOs): that is, a means of en-
tanglement, even within the a single repeater, that often
fails but the failures are `heralded'. In addition there is
of course a nite error rate even for the operations that
are deemed successful. Non-deterministic EOs will occur
within individual repeater stations if, for example, their
internal hardware is based on networking small quantum
registers together optically, i.e. qubits can be entangled
by joint measurements on single photons emitted from
these qubits rather than control of interactions [3, 4].
Such an architecture may be much easier to implement
in a scalable way than monolithic architectures e.g. large
scale ion traps

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Transcript of "Long range failure tolerant entanglement distribution by ying li sean d barrett thomas m stace and simon c benjamin"

  1. 1. Long range failure-tolerant entanglement distribution Ying Li Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 Sean D. Barrett Blackett Laboratory and Institute for Mathematical Sciences, Imperial College London, London SW7 2PG, United Kingdom Thomas M. Stace School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 Simon C. BenjaminarXiv:1209.4031v1 [quant-ph] 18 Sep 2012 Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 We introduce a protocol to distribute entanglement between remote parties. Our protocol is based on a chain of repeater stations, and exploits topological encoding to tolerate very high levels of defects and errors. The repeater stations may employ probabilistic entanglement operations which usually fail; ours is the first protocol to explicitly allow for technologies of this kind. Given an error rate between stations in excess of 10%, arbitrarily long range high fidelity entanglement distribution is possible even if the heralded failure rate within the stations is as high as 99%, providing that unheralded errors are low (order 0.01%). PACS numbers: Introduction. Distributing an entangled state among the rate of distributing entanglement decreases only log- remote quantum computers is one of the fundamental arithmically with the communication distance. tasks of quantum information technologies. It is crucial Cluster states are resources of measurement-based for quantum teleportation, quantum cryptography and quantum computing [5], and long-range entanglement distributed quantum computing. Using direct transmis- can be established in noisy cluster states [6]. In this sion, the success probability of transmitting a qubit and paper, we propose a protocol of distributing entangle- the fidelity of the resulting quantum state decrease expo- ment by single-qubit measurements on a topologically nentially with distance. Therefore, one needs quantum protected cluster (TPC) state [7] across the chain of re- repeaters to achieve long distance entanglement [1, 2]. A peater stations. The TPC state must first be grown good quantum repeater protocol should be fault-tolerant via operations within repeaters together with quantum and support a high communication rate. In this paper, communication between pairs of neighboring repeaters. we will propose a protocol to distribute entanglement be- The operations within repeaters are expected to have a tween two remote quantum computers. We consider noise much better performance than communications between in quantum communication channels, and of course errors repeaters (since the latter may be over distances of kilo- generated by operations within the repeaters. We assume metres). We find that the protocol is valid if the probabil- that the repeater stations may employ non-deterministic ity of an error occurring in the communication channel is entanglement operations (EOs): that is, a means of en- lower than a threshold, which is 15% when errors induced tanglement, even within the a single repeater, that often by operations within repeaters are negligible. With errors fails but the failures are ‘heralded’. In addition there is less than the threshold, entanglement can be established of course a finite error rate even for the operations that between two remote logical qubits encoded in two sepa- are deemed successful. Non-deterministic EOs will occur rated graph states, which may be used for further infor- within individual repeater stations if, for example, their mation processing via the topological measurement based internal hardware is based on networking small quantum quantum computing [7]. Alternatively one can also de- registers together optically, i.e. qubits can be entangled code each logical qubit to a physical qubit via single-qubit by joint measurements on single photons emitted from measurements. Although we describe only the two-party these qubits rather than control of interactions [3, 4]. protocol here, it should be straightforward to generalize Such an architecture may be much easier to implement for distributing multi-party entanglement. in a scalable way than monolithic architectures e.g. large scale ion traps. Even with this assumption that EOs fail In this protocol, the quality of the eventual entangle- both between and within repeater stations, we find that ment between logical qubits is only limited by the number of qubits in each repeater. Therefore, our protocol effec-
  2. 2. 2 is grown across quantum repeater stations via probabilis- Bob tic EOs and quantum communications between nearby stations. The TPC state contains two parallel empty Alice (a) tubes, which terminate in stations of Alice and Bob. Each empty tube is a void in the TPC state, with an TPC  state (d) elongated shape and shown as a blue rectangular cuboid in Fig. 1(b). Once the TPC state is generated, measure- Z plug ments in the X basis are performed on all qubits except (b) two parts of the TPC state located in stations of Al- ice and Bob respectively [see Fig. 1(c)]. The two parts which are to remain unmeasured are called plugs, and X (e) are connected with empty tubes. Two empty tubes and two plugs form a closed loop. There is one logical qubit decoding (c) encoded in each plug. After all other quits are measured, and the outcomes are communicated to Alice and Bob, ii then these two logical qubits are entangled as one of the Bell states (determined by measurement outcomes). The TPC state is a cluster state of qubits located i (g) (f) on the a cubic lattice [7]. There is one qubit on each face and edge of the elementary cell [Fig. 1(d)]. ByFIG. 1: The scheme of quantum entanglement distribution shifting the lattice, one can transfer qubits on faces toprotocol based on topologically protected cluster (TPC) state. edges, and vice versa. The new lattice is called the dual(a) Alice and Bob can be entangled via a chain of quantum lattice of the original primal lattice. The TPC staterepeater stations, which are connected by optical quantum is stabilized by K(c) = a∈c Xa b∈∂c Zb , where c iscommunication channels. (b) Each station contains a ‘slice’ an arbitrary primal (dual) surface and ∂c is the primalof the TPC state. The TPC state contains two empty tubes(blue) without any qubit. (c) Once the TPC is complete, all (dual) chain as the boundary of c. Qubits in the set cqubits are measured in X basis except two parts of the TPC (∂c) are located on faces (edges) composing the surfacestate (green) in stations of Alice and Bob respectively; these (chain) c (∂c). The logical qubit is encoded in a plugare called plugs and contain the eventual encoded shared Bell as X = a∈section Xa and Z = b∈line Zb , where Xpair. (d) The elementary cell of the TPC state. Each logical and Z are Pauli operators of the logical qubit. Here,qubit is encoded as subfigure (e) and can be decoded as subfig- section is a dual surface across the plug, and line is aure (f) (see text). (g) Two surfaces propagating correlationsbetween two logical qubits. primal chain on the surface of the plug and connecting two empty tubes [Fig. 1(e)]. We consider two stabiliz- ers according to the following surfaces: (i) ci is a pri- mal surface whose boundary is enclosed by the tube-tively distills as well as distributes entanglement. The plug loop, and (ii) cii is a closed dual surface envelop-idea of using an error correction code with protected log- ing one empty tube and crossing two plugs [Fig. 1(g)].ical qubits for remote entanglement was firstly reported The two stabilizers are K(ci ) = Z A Z B a∈ci Xa andin Ref. [8], in which the Calderbank-Shor-Steane code K(cii ) = X A X B a∈c Xa , where A, B denote Alice andis employed. Subsequently 3D lattice-based distribution iihas also been studied [9] and the extension to lower di- Bob respectively, and cii denotes the part of the sur-mensionality has been examined [10]. Recently, in a face cii outside two plugs. After measurements in theprotocol for quantum state transfer of a surface-code- X basis, one can replace Xa with measurement out-encoded qubit, the efficiency of quantum communication comes. Then, we get two new stabilizers Z A Z B = ±1is greatly improved by removing the necessity of two- and X A X B = ±1, i.e. the two logical qubits are stabi-way communication [11]. Compared with these proto- lized as one of Bell states. Here, the two signs depend oncols, ours is the first to consider a probabilistic architec- measurement outcomes.ture within each repeater station, so that the entangle- Besides two-party entanglement, we note that ourment distribution can be efficient even if EOs are far from scheme can be directly generalized to multi-party entan-deterministic. glement, e.g. three-party and four-party entanglement as Quantum Repeaters based on Cluster States. Alice and shown in Ref. [12].Bob are entangled via a chain of quantum repeater sta- Noise in quantum communication channels and imper-tions. Two nearby repeaters are connected by optical fections in operations will give rise to phase errors onquantum communication channels [Fig. 1(a)] – essen- the TPC state. In order to eliminate errors from the Belltially a bundle of optical fibres that are used in parallel. state of two logical qubits, we monitor errors on the TPCTo give an overview of the process: Firstly, a TPC state state by parity check operators K(cc ), where cc are min-
  3. 3. 3 we overcome the impact of high EO failure rates when root we create the large scale TPC state. The structure of these elementary graph states can be a star [13], a line [14], a cross [15], or a tree [16, 17]. In this paper, we Tree-structure graph state take the tree structure as an example, and the scheme (a) can be adapted to other structures. The tree structure accumulates fewer errors than other structures when the Snowflake, local resource success probability of EOs is low [17, 18]. Tree-structure p1 graph states can be generated by using parity projections 3 1 2 4 3 4 (PPs) [3]. A PP on roots of two individual trees can p2 fuse them into a double-size tree [Fig. 2(a)]. If all PPs Dumbbell, nonlocal resource are successful, after n steps, one can grow a tree with (c) (b) 2n qubits from separated qubits, where the integer n is called the generation of the tree.FIG. 2: Resource graph states, i.e. building-blocks, for grow- Trees are fused into two kinds of building-block graphing the topologically protected cluster state. Each red dotted states. Snowflake graph states are prepared by fusingline denotes a parity projection (PP). (a) Tree graph statescan be grown by PPs on roots of trees. (b) Four trees can be four trees [Fig. 2(b)]. Each snowflake will ultimatelyfused into a ‘snowflake’ graph state as the following: fusing correspond to a specific qubit on the TPC state. Eacheach pair of trees into a bigger tree at first; cutting two roots quarter of a snowflake is used to establish a connectionby measurements in Z basis; fusing them into a snowflake with a neighboring snowflake. We refer to the secondand cutting the unwanted qubit. (c) Two trees in different kind of building-block as a dumbbell. These are nonlocalquantum repeater stations are fused into a dumbbell graph building blocks connecting two nearby quantum repeaterstate by a Bell measurement on photon-p1 and the photon- stations [Fig. 2(c)]. A dumbbell is formed by two treesp2, each associated with a qubit in a different stations. Oneof the photons (p2) will have travelled between stations. The located in different stations. For example, suppose thatBell measurement is followed by a measurement in the Y ba- the basic qubits are optically active atoms: then in ordersis on the qubit-1 and a measurement in the X basis on the to prepare a dumbbell, we cause each root qubit emitqubit-2, in order to get the desired dumbbell graph state. a single photon as |η j → |η j |η pj , where j = 1, 2 de- notes a root qubit, ‘pj’ denotes the corresponding pho- tonic qubit, η = 0, 1 is the label the state in the compu-imum closed surfaces. Usually, minimum closed surfaces tational basis and the photonic qubit can be encoded inare surfaces of elementary cubes. However, some qubits polarization, frequency [19] or time-bin [20]. One pho-on the TPC state may be missing. The parity check op- ton is transmitted from one station to another. After aerator of an elementary cube with missing qubits can not Bell measurement on two photons and single-qubit mea-be used to detect errors. Then, one has to use products surements on roots, we obtain the dumbbell graph stateof parity check operators connected by missing qubits [12].to form a new set of parity check operators [23]. Par- Making a building-block graph state requires all oper-ity check operators reveal the endpoints of error chains, ations to be successful, whose probability may be quitewhere an error chain (ring) is a sequence of phase er- small. Therefore, building-block graph states are pro-rors. If the number of phase errors on the surface cc is duced with a post selection strategy: if an operation isodd, the existence of errors can be identified by K(cc ), heralded as failed, the corresponding graph state is aban-which is called an error syndrome. Errors are not actively doned with the qubits reinitialized.corrected, rather parities of a∈ci Xa and a∈c Xa , are Once a sufficient number of each resource (snowflakes iimodified by knowledge of the total number of error chains and dumbbells) have been generated, we can assemblecrossing surfaces ci and cii respectively. After the error them to create a suitable TPC state. Snowflakes are as-correction, only error rings encircling the tube-plug loop, sembled by PPs on leaves, which are qubits on the edge oferror chains connecting two empty tubes and error chains a snowflake (Fig. 3). Two snowflakes in the same quan-connecting the loop with the boundary of the TPC state tum repeater station can be connected directly, while two[12], may contribute an error on logical qubits. If noise snowflakes in different stations are connected by bridg-and imperfections are less than a threshold, the probabil- ing them with a dumbbell shared by these two stations.ity of an error on logical qubits decreases exponentially The number of leaf qubits on each quarter of a snowflakewith the minimum length of these error rings and error is 2n−1 . Therefore, the failure probability of connecting n−1chains [7]. two snowflakes in the same station is FL = f 2 , and Cluster State Growth.- In order to grow the TPC state the failure probability of connecting two snowflakes inacross quantum repeater stations, some ‘building-block’ different stations is FNL 2FL , where f is the basic fail-graph states should first be prepared within each repeater ure probability of EOs. After establishing connectionsdevice. It is through the use of these building-blocks that between snowflakes, all qubits except those at the center
  4. 4. 4 Dumbbell stations are perfect (when heralded as successful); then only joint qubits have errors, and these imperfect qubits exist in specific non-adjacent layers of the TPC state. 5 3 4 6 Then error correction can be performed independently on each such layer. The error threshold of a two dimen- sional layer is about 10% in the limit of a perfectly con- 5 6 nected lattice [26]. Moreover a near-perfectly connected lattice would indeed be achievable since, given error free EOs within repeaters, one could always grow sufficiently 7 big tree structures to make FL as low as desired. There- 7 fore, with perfect operations, the condition of getting a correct correlation between two logical qubits faithfully TPC state is 2 /3 10%, i.e. the error threshold of communication noise is t 15%.FIG. 3: The strategy of assembling resource graph statesinto the full topologically protected cluster (TPC) state which With imperfect operations, all qubits on the TPC statespans all quantum repeater stations. (a) Snowflakes within may affected by phase errors. If the distribution of phasethe same station are connected directly to each other by par- errors is uniform, i.e. all qubits may have a phase errority projections (red dotted lines) on leaves. Two snowflakes with the same probability, the threshold of phase errorsin different stations can be connected via a dumbbell which is about 3% for perfectly connected TPC state [25]. How-incorporates the required nonlocal connection (dash line). (b) ever, in our case, the TPC state grown by probabilisticAfter extraneous quits are removed, ultimatelty the qubits atthe heart of each snowflake survive as nodes of the TPC state. EOs is unlikely to be perfectly connected and there are more errors on joint qubits than others. Our strategy is to treat missing connections by transforming them toof each snowflakes are removed by appropriate single- qubit loss, by means of deleting the qubits with missingqubit measurements, so that the surviving qubits form connections using measurements in the Z basis. Then,the TPC state. Here, the measurement pattern for re- the loss probability of joint qubits is 5FL , and the lossmoving qubits can be found in Ref. [12]. Since some probability of other qubits is 4FL . We determine errorsnowflakes have failed to connect, this implies some miss- thresholds for general cases numerically as shown in Fig.ing connections on the TPC state. We presently describe 4(a), using the method developed in Ref. [22, 23].simulations establishing that when connections are rarely The error rate of imperfect operations must be lowermissing, i.e. FL < 5%, then the cluster state is well con- than the threshold of fault-tolerant quantum comput-nected: it is easy to find surfaces propagating correlations ing (FTQC). The threshold of FTQC on the TPC statebetween two logical qubits, indeed this is guaranteed in with non-deterministic EOs (deterministic control-phasethe scaling limit (as expected from percolation theory) gates) is about 2×10−4 [18] (5×10−3 [7]). By optimizing[21, 23]. the size of trees, (a bigger tree can reduce missing con- As a footnote to this section we note that the ‘building- nections but generate more errors), we have obtained theblock’ strategy is not always necessary. If the failure thresholds of tolerable communication noise in the pres-probability of EOs is low enough f < 5%, one may gen- ence of finite error rates for internal EOs, see Fig. 4(b). Iferate the TPC state directly, for example, by using con- the error rate of operations is 10−4 , the threshold of com-trol phase gates [7], where control-phase gates on two munication noise is about 11% when the success probabil-qubits located in different quantum repeater stations can ity of entangling operations is 1%. In contrast, by usingbe simulated by consuming entanglement prepared via control-phase gates to generate the TPC state directly,quantum communication [24]. However here we are in- the threshold of communication noise is still above 10%terested in the general case where the failure probability even if the error rate of operation is 2 × 10−3 , but themay be very high. success probability must be higher than 98%. Noise, Imperfections and Error Correction.- Both Full decoding.- A logical qubit can be decoded into anoise in quantum communication channels and imper- physical qubit by measurements on the correspondingfections in operations can give rise to errors on the TPC plug, leaving just one qubit unmeasured. The residualstate. We assume communication noise is depolarized, qubit carries the quantum state of the logical qubit. Forand described by the superoperator E = (1 − )[1p2 ] + decoding, two (blue) pyramids inside the plug, whose ([Xp2 ] + [Yp2 ] + [Zp2 ])/3 [see Fig. 2(c)]. We call qubits apexes hold the residual qubit (red circle) and bases con-with nonlocal connections ‘joint qubits’ [gray circles in nect tubes, are measured in the Z basis, while otherFig. 3]. Errors induced by communication noise may qubits are measured in the X basis [see Fig. 1(f)]. Themake phase errors on corresponding joint qubits (qubits residual qubit can acquire an error if there is an error5 and 6) with a probability 2 /3 for each of them [12]. chain connecting two pyramids. Therefore, the proba-Consider first the case that internal operations within bility of an error on the residual qubit is p + O(p3 ) [7],
  5. 5. 5 0.12 0.15  FL=0.00 While preparing this document we became aware of  FL=0.01 0.15 a manuscript describing closely related research: Ash-thresholds  of  pJ thresholds  of   0.08 0.10  FL=0.04 0.10 ley Stephens, Jingjing Huang, Kae Nemoto and William   0.05 0.90 0.95 1.00 J. Munro, “Fault-tolerant quantum communication with 0.04 0.05 rare-earth elements and superconducting circuits”.   0.00 0.00 -­3 0.0 0.2 0.4 0.6 0.8 10 -­2 10 -­1 10 10 0 p/pJ (a) f (b) [1] H.-J. Briegel et al., Phys. Rev. Lett. 81, 5932 (1998).FIG. 4: Thresholds of error correction on the topologically [2] L.-M. Duan et al., Nature 414, 413 (2001).protected cluster (TPC) state. (a) Thresholds of phase errors [3] S. C. Benjamin, B. W. Lovett, and J. M. Smith, Laser &on joint qubits, which is dependent on the ratio between the Photonics Reviews, 3, 556 (2009).error probability on joint qubits (pJ ) and the error probability [4] D. L. Moehring et al., J. Opt. Soc. Am. B 24, 300 (2007).on other qubits (p). (b) Thresholds of communication noise [5] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86,with operational error rate 10−4 (solid line), evaluated from 5188 (2001); R. Raussendorf, D. E. Browne, and H. J.the linear interpolation of data in subfigure (a). By using Briegel, Phys. Rev. A 68, 022312 (2003).control-phase gates to generate the TPC state directly, the [6] Robert Raussendorf, Sergey Bravyi, and Jim Harrington,error rate can be much higher (2 × 10−3 ) but only a failure Phys. Rev. A 71, 062313 (2005).probability (f ) lower than 4% is tolerable (dash line). Here [7] R. Raussendorf, J. Harrington, and K. Goyal, Ann. Phys.we have assumed that memory errors happen at a lower rate 321, 2242 (2006); R. Raussendorf and J. Harrington,than operational errors. Memory errors at 10% of the opera- Phys. Rev. Lett. 98, 190504 (2007); R. Raussendorf, J.tional error rate can lower the threshold, but not dramatically Harrington, and K. Goyal, N. J. Phys. 9, 199 (2007).(dotted line). [8] Liang Jiang et al., Phys. Rev. A. 79, 032325 (2009). [9] S. Perseguers, Phys. Rev. A 81, 012310 (2010). [10] A. Grudka et al arXiv:1202.1016 [quant-ph]. [11] A. G. Fowler et al., Phys. Rev. Lett. 104, 180503 (2010).where p is the probability of phase errors on the residual [12] Supplementary material, http://qunat.org/papers/topComqubit, which is usually lower than 3%. [13] M. Nielsen, Phys. Rev. Lett. 95, 080503 (2005). Performance.- The probability of errors on two en- [14] S. D. Barrett and P. Kok, Phys. Rev. A 71, 060310tangled logical qubits decreases exponentially with the (2005); S. C. Benjamin, Phys. Rev. A 72, 056302 (2005).minimum length of error rings and error chains [7]. We [15] L.-M. Duan and R. Raussendorf, Phys. Rev. Lett. 95,design the TPC state as follows: the perimeters of two 080503 (2005). [16] T. P. Bodiya and L.-M. Duan, Phys. Rev. Lett. 97,empty tubes, the distance between empty tubes, and the 143601 (2006).distance between each empty tube and the boundary, [17] Y. Matsuzaki, S. C. Benjamin, and J. Fitzsimons, Phys.are each proportional to the same length scale L. The Rev. Lett. 104, 050501 (2010).length of the TPC state, i.e., the number of quantum [18] Y. Li, S. D. Barrett, T. M. Stace, and S. C. Benjamin,repeater stations, can increase the probability of error Phys. Rev. Lett. 105, 250502 (2010).rings and error chains linearly [6]. Therefore, the over- [19] D. L. Moehring, M. J. Madsen, K. C. Younge, R. N.all probability of errors on two entangled logical qubits Kohn, Jr., P. Maunz, L.-M. Duan, and C. Monroe, J. Opt. Soc. Am. B 24, 300 (2007).is E ∝ N e−κL , where N is the number of stations, κ [20] S. D. Barrett and P. Kok, Phys. Rev. A 71, 060310is a constant depending on p, pJ and FL . To achieve (2005).a given quality of entanglement, we need a TPCS with [21] C. D. Lorenz and R. M. Zi , Phys. Rev. E 57, 230 (1998).L = O(log(N/ E )/κ). The number of photonic qubits [22] T. M. Stace, S. D. Barrett, A. C. Doherty, Phys. Rev.transferred between two nearby stations is proportional Lett. 102, 200501 (2009); T. M. Stace, S. D. Barrett,to L2 . Therefore the overall entanglement distribution Phys. Rev. A 81, 022317 (2010).rate of our scheme is RN = O(log−2 (N/ E )/κ). [23] S. D. Barrett, T. M. Stace, Phys. Rev. Lett. 105, 200502 (2010). [24] J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Plenio, In conclusion, we have described an advanced proto- Phys. Rev. A 62, 052317 (2000).col for distributing entanglement through the use of re- [25] T. Ohno, G. Arakawa, I. Ichinose and T. Matsui, Nucl.peater stations which together generate a topologically Phys. B 697, 462 (2004).protected cluster state. We find that the approach is re- [26] C. Wang, J. Harrington, and J. Preskill, Annals ofmarkably robust to errors, while the resource cost within Physics 303, 31 (2003); F. Merz and J. T. Chalker, Phys.each repeater scales only logarithmically with the total Rev. B 65, 054425 (2002).distance over which entanglement is to be shared.

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