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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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- 1.Long range failure-tolerant entanglement distribution Ying LiCentre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543Sean 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 andCentre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543Simon C. BenjaminarXiv:1209.4031v1 [quant-ph] 18 Sep 2012Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK andCentre 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 isbased on a chain of repeater stations, and exploits topological encoding to tolerate very high levelsof defects and errors. The repeater stations may employ probabilistic entanglement operations whichusually fail; ours is the rst protocol to explicitly allow for technologies of this kind. Given an errorrate between stations in excess of 10%, arbitrarily long range high delity entanglement distributionis possible even if the heralded failure rate within the stations is as high as 99%, providing thatunheralded 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 crucialCluster states are resources of measurement-based for quantum teleportation, quantum cryptography andquantum 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 andpaper, we propose a protocol of distributing entangle- the delity of the resulting quantum state decrease expo-ment by single-qubit measurements on a topologically nentially with distance. Therefore, one needs quantumprotected cluster (TPC) state [7] across the chain of re- repeaters to achieve long distance entanglement [1, 2]. Apeater stations. The TPC state must rst be grown good quantum repeater protocol should be fault-tolerantvia 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 noisemuch better performance than communications between in quantum communication channels, and of course errorsrepeaters (since the latter may be over distances of kilo- generated by operations within the repeaters. We assumemetres). We nd that the protocol is valid if the probabil- that the repeater stations may employ non-deterministicity 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 oftenby operations within repeaters are negligible. With errors fails but the failures are heralded. In addition there isless than the threshold, entanglement can be established of course a nite error rate even for the operations thatbetween two remote logical qubits encoded in two sepa- are deemed successful. Non-deterministic EOs will occurrated 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 quantumquantum computing [7]. Alternatively one can also de- registers together optically, i.e. qubits can be entangledcode each logical qubit to a physical qubit via single-qubit by joint measurements on single photons emitted frommeasurements. 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 failIn this protocol, the quality of the eventual entangle- both between and within repeater stations, we nd that ment between logical qubits is only limited by the numberof qubits in each repeater. Therefore, our protocol eec-

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 anTPC state (d) elongated shape and shown as a blue rectangular cuboid in Fig. 1(b). Once the TPC state is generated, measure-Zplugments 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 qubitdecoding(c)encoded in each plug. After all other quits are measured, and the outcomes are communicated to Alice and Bob, iithen 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 distributionshifting 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 quantumlattice of the original primal lattice. The TPC staterepeater stations, which are connected by optical quantumis stabilized by K(c) = ac Xa bc 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 Bellas X = asection Xa and Z = bline 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 subgure (e) and can be decoded as subg- 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 rstly reportedThe two stabilizers are K(ci ) = Z A Z B aci Xa andin Ref. [8], in which the Calderbank-Shor-Steane code K(cii ) = X A X B ac Xa , where A, B denote Alice andis employed. Subsequently 3D lattice-based distributioniihas 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 aface 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 eciency 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 rst 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 ecient even if EOs are far fromscheme 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 andshown 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 bres 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 we overcome the impact of high EO failure rates whenroot 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-structuregraph statetake 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 p1graph states can be generated by using parity projections 312 434 (PPs) [3]. A PP on roots of two individual trees canp2 fuse them into a double-size tree [Fig. 2(a)]. If all PPsDumbbell, 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. Snowake 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 snowake will ultimatelyfused into a snowake graph state as the following: fusing correspond to a specic qubit on the TPC state. Eacheach pair of trees into a bigger tree at rst; cutting two roots quarter of a snowake is used to establish a connectionby measurements in Z basis; fusing them into a snowakewith a neighboring snowake. We refer to the secondand cutting the unwanted qubit. (c) Two trees in dierentkind of building-block as a dumbbell. These are nonlocalquantum repeater stations are fused into a dumbbell graphbuilding 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 dierent stations. Oneof the photons (p2) will have travelled between stations. Thelocated in dierent 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 | j | j | pj , where j = 1, 2 de- notes a root qubit, pj denotes the corresponding pho- tonic qubit, = 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 productssurements 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 identied 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 aci Xa and ac Xa , areOnce a sucient number of each resource (snowakesiimodied 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. Snowakes 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 snowake (Fig. 3). Two snowakes 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 noisesnowakes in dierent 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 exponentiallyThe number of leaf qubits on each quarter of a snowakewith the minimum length of these error rings and error is 2n1 . Therefore, the failure probability of connectingn1chains [7].two snowakes 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 snowakes inacross quantum repeater stations, some building-blockdierent stations is FNL 2FL , where f is the basic fail-graph states should rst be prepared within each repeaterure probability of EOs. After establishing connectionsdevice. It is through the use of these building-blocks thatbetween snowakes, all qubits except those at the center 4. 4Dumbbellstations are perfect (when heralded as successful); thenonly joint qubits have errors, and these imperfect qubitsexist in specic non-adjacent layers of the TPC state.53 4 6Then error correction can be performed independentlyon each such layer. The error threshold of a two dimen-sional layer is about 10% in the limit of a perfectly con- 5 6nected lattice [26]. Moreover a near-perfectly connectedlattice would indeed be achievable since, given error freeEOs within repeaters, one could always grow suciently 7big tree structures to make FL as low as desired. There-7fore, with perfect operations, the condition of getting acorrect correlation between two logical qubits faithfully TPC stateis 2 /3 10%, i.e. the error threshold of communicationnoise is t 15%.FIG. 3: The strategy of assembling resource graph statesinto the full topologically protected cluster (TPC) state whichWith imperfect operations, all qubits on the TPC statespans all quantum repeater stations. (a) Snowakes within may aected 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 snowakes with the same probability, the threshold of phase errorsin dierent stations can be connected via a dumbbell whichis 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 snowake survive as nodes of the TPC state.EOs is unlikely to be perfectly connected and there aremore errors on joint qubits than others. Our strategyis to treat missing connections by transforming them toof each snowakes 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 errorsnowakes 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 rarelyThe 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 nd surfaces propagating correlations ing (FTQC). The threshold of FTQC on the TPC statebetween two logical qubits, indeed this is guaranteed inwith non-deterministic EOs (deterministic control-phasethe scaling limit (as expected from percolation theory) gates) is about 2104 [18] (5103 [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 nite 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 104 , the threshold of com-trol phase gates [7], where control-phase gates on twomunication noise is about 11% when the success probabil-qubits located in dierent quantum repeater stations canity 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 probabilityeven if the error rate of operation is 2 103 , but themay be very high. success probability must be higher than 98%. Noise, Imperfections and Error Correction.- BothFull 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 mayqubits 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 rst the case that internal operations withinbility of an error on the residual qubit is p + O(p3 ) [7], 5. 50.120.15 FL=0.00While preparing this document we became aware of FL=0.010.15 a manuscript describing closely related research: Ash-thresholds of pJ thresholds of 0.080.10 FL=0.040.10 ley Stephens, Jingjing Huang, Kae Nemoto and William 0.05 0.90 0.95 1.00 J. Munro, Fault-tolerant quantum communication with0.040.05rare-earth elements and superconducting circuits.0.000.00 -3 0.0 0.2 0.40.60.8 10-2 10 -110 100 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 104 (solid line), evaluated from5188 (2001); R. Raussendorf, D. E. Browne, and H. J.the linear interpolation of data in subgure (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 (2 103 ) 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 rate321, 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 dramaticallyHarrington, 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. TheRev. 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 qubitsKohn, Jr., P. Maunz, L.-M. Duan, and C. Monroe, J. Opt. Soc. Am. B 24, 300 (2007).is E N eL , where N is the number of stations, [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 )/). The number of photonic qubits[22] T. M. Stace, S. D. Barrett, A. C. Doherty, Phys. Rev.transferred between two nearby stations is proportionalLett. 102, 200501 (2009); T. M. Stace, S. D. Barrett,to L2 . Therefore the overall entanglement distributionPhys. Rev. A 81, 022317 (2010).rate of our scheme is RN = O(log2 (N/ E )/).[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 topologicallyPhys. B 697, 462 (2004).protected cluster state. We nd that the approach is re-[26] C. Wang, J. Harrington, and J. Preskill, Annals ofmarkably robust to errors, while the resource cost withinPhysics 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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