From quantum fusiliers to high-performance networks
W. J. Munro, K. A. Harrison, A. M. Stephens, S. J. Devitt, Kae Nemoto
aa r X i v : . [ qu a n t - ph ] O c t From quantum fusiliers to high-performance networks
W. J. Munro,
1, 2, ∗ K. A. Harrison, A. M. Stephens, S. J. Devitt, and Kae Nemoto National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom (Dated: August 29, 2018)Our objective was to design a quantum repeater capable of achieving one million entangled pairsper second over a distance of 1000km. We failed, but not by much. In this letter we will describethe series of developments that permitted us to approach our goal. We will describe a mechanismthat permits the creation of entanglement between two qubits, connected by fibre, with probabilityarbitrarily close to one and in constant time. This mechanism may be extended to ensure that theentanglement has high fidelity without compromising these properties. Finally, we describe how thismay be used to construct a quantum repeater that is capable of creating a linear quantum networkconnecting two distant qubits with high fidelity. The creation rate is shown to be a function of themaximum distance between two adjacent quantum repeaters.
PACS numbers:
INTRODUCTION
The twentieth century saw the discovery of quantummechanics, a set of principles describing physical realityat the atomic level of matter. These principles have beenused to develop much of today’s advanced technology in-cluding, for example, today’s microprocessors. Quantumphysics also allows a new paradigm for the processingof information — a field known as quantum informationprocessing [1, 2]. Over the last decade we have seen ahuge worldwide effort to develop and explore quantum-information based devices and technologies [3, 4]. Quan-tum key distribution (QKD) enabled devices are alreadycommercially available [5]. The next step after this islikely to be small scale processors, probably distributedin nature.Quantum repeaters are a natural candidate to consider[6]. Their role is to enable the creation of entangled statesbetween remote locations. Long-distance entanglementis achieved by placing a number of repeater nodes in-between two end points and creating entangled links be-tween the adjacent nodes. Once a node has links both tothe left and to the right, entanglement swapping withinthe nodes then allows longer-range entangled links to beformed. Once swapping operations have occurred at allthe intermediate nodes an end-to-end entangled link willhave been formed. These entangled pairs can then beused in QKD, quantum communication, or distributedquantum computation.The current goal of many research groups is to pro-duce a stream of entangled qubits over long distances,preferably with rates in the MHz range, There havebeen many proposals for how this could be achievedand a number of ”in-principle” demonstrations have beenperformed. Such proposals have generally focused onthe quantum components necessary to create entangledlinks between neighboring nodes, purification of theselinks, and swap operations to create longer-distance links [7, 8, 9, 10, 11, 12, 13]. The entangled links are generallycreated by entangling an optical signal (appendix 1) witha qubit and then transmitting that signal over a channelto the neighboring node. Here the signal entangles with aqubit within that node and then a measurement is madeon the quantum signal indicating successful generation ornot [8, 9]. The probability of successfully generating thelink scales at best as exp[ − L/L ], where L is the distancebetween repeater nodes and L the attenuation length ofthe fiber.The next step is to look at the overall design of the re-peater network, in terms of both the quantum and clas-sical components. The communication time for classicalmessages to be transmitted between nodes severely limitsthe performance of a repeater network. Messages gener-ally need to be sent between nodes in all of the three keyquantum stages of a repeater network: entanglement dis-tribution, purification, and swapping. In this letter wewill describe a pipe-lined architecture where one knowswhen the end-to-end entangled pairs are going to be avail-able. QUANTUM FUSILIERS AND FUSILANDS
The major issue affecting the performance for a quan-tum repeater is the probabilistic nature of the generationof entanglement between adjacent nodes and not know-ing when such a link is going to be available. This issuemeans that a confirmation signal needs to be sent backfrom the receiver to the transmitter side and so the gener-ation rate is ultimately limited by this round trip trans-mission time. With typical repeater nodes being sepa-rated by, say, 40km this would take on the order of 400 µ s. Now with the probability of success for entanglementgeneration being below 25%, quite a number of attemptsare going to be needed before we are ” guaranteed ” a link.A significant time delay results if the attempts are per-formed sequentially. One could parallelize the operationsbut with significantly more resources. One must be ableto do better!A simpler design does indeed exist which we depict inFig. (1). In this design each repeater node comprises twofundamental parts: a quantum fusillade containing mul-tiple fusiliers (transmitters) and quantum fusilands (re-ceivers). There are generally more fusiliers than fusilandsand for the moment we will consider a single fusiland.The creation of a constant-time entanglement link beginsby a classical pulse initiating all the fusiliers in that nodeto prepare individual quantum optical signals. These sig-nals then interact and become entangled with the qubitsin their respective fusilier cavities. The signals then prop-agate, temporally multiplexed together with the classicalheralding pulse, along the fiber to the fusiland in thenext repeater node. The classical pulse announces to thefusiland that a series of quantum signals are about to ar-rive and so the fusiland initializes the qubit into the ap-propriate state and then interacts with the first fusilier’squantum signal. The signal is then measured to deter-mine whether a successful entanglement-creation opera-tion has occurred. If not, the fusiland qubit is re-preparedfor the arrival of the second fusilier’s signal and the sameinteraction/measurement procedure is performed. Thiscontinues until a success is reported. A successful resulttriggers two operations: first, it stops any further signalsinteracting with the fusiland; and secondly, it dispatchesa classical message back to the fusiland informing it ofwhich fusilier was successful. The time taken from firingthe fusillade to receiving the classical message is essen-tially the round trip time between two adjacent repeatersand is a constant. With enough fusiliers we can ” guar-antee ” the entangled link exists. The failure probabil-ity is given by p f = (1 − p ) n , where n is the numberof fusiliers and p is the success probability of a singlefusilier/fusiland. With p = 0 .
25, 16 fusiliers are neededfor p f < . n fusiliers and m fusilands then theprobability that all m links have not been establishedis p f ( m ) = P mj =1 (cid:0) nj − (cid:1) p j − (1 − p ) n − j +1 . For p f =0 .
01 and p = 0 .
25, the numbers of fusiliers/fusilandsneeded are (n=16,m=1), (n=24,m=2), (n=70,m=10),(n=485,m=100) which in the asymptotic limit of large m goes to (n=m/p, m). This clearly shows the advantage ofhaving multiple fusilands in terms of resource efficiency.With multiple entanglement links available betweenadjacent nodes, there are various possibilities for howthese can be used. The simplest is just to use them optical switch classicalpulseprobe pulses channelclassical channelquantum fusillade quantum qusilandquantum fusilandLocal Repeater Node cavityqubitquantum fusilier FIG. 1: Schematic representation of a quantum repeater nodeand its link to its next neighbor. The basic repeater is com-posed of two fundamental components: a quantum fusilladecontaining multiple fusiliers (transmission cavities each with aqubit within them) and a quantum fusiland (receiving cavitywith a qubit within it and a signal detector). in parallel to improve the overall network performance,however as our entangled links may not be perfect weneed to be able to purify them. Normal purification pro-tocols are problematic since they are probabilistic andrequire two-way communication to determine if we havesucceeded or failed [14, 15]. Upon failure our entangledlinks are destroyed and we must start the link generationagain. This is a major performance issue but it can besolved by using quantum error correction [16].The particular error-correction code to be used willdepend on the errors induced in the entanglement gen-eration process and on the failure rate of the quantumgates at each node. If we assume perfect local gates andthat the predominant channel error (excluding loss) is abit-flip error, then our entangled link can be representedby ρ [ F ] = F | gg + ee ih gg + ee | + 1 − F | ge + eg ih eg + ge | , (1)where F measures the fidelity (quality) of the entangledpairs one is trying to create. In this case, to create an en-tangled pair with fidelity F ′ > F we make use of a three-qubit repetition code, which corrects a single bit-flip er-ror as follows: Given three copies of ρ ( F ) we performnon-destructive parity measurements on the first and sec-ond and then on the second and third fusiliers, recordingthe results p and p . The second and third fusiliersare then measured out in the X basis. On the fusilandside identical parity measurements are performed with re-sults, say, r and r and then the second/third fusilandsare measured out in the X basis. The resulting entangledstate is ρ [ F ′ = F + 3 F (1 − F )] up to a bit-flip correc-tion determined by p , p , r , and r and a phase-flipcorrection determined by the results of the four X mea-surements. These corrections simply update the Pauliframe and need only be communicated to one end of thenetwork. This means we do not need to wait and so thefusiliers and fusilands can be further processed.This simple protocol is quite effective at increasing thefidelity of the remaining pair relative to that of the initialpairs; for instance if we started with F = 0 .
95 we wouldhave F ′ ≥ .
99. Importantly, the non-determinism inher-ent in purification-based schemes is not present in thisscheme, allowing for pipe-lining of the overall repeaternetwork. To extend entanglement beyond neighboringnodes we perform swap operations (achieved by paritygates) between local fusiliers and fusilands when the lo-cal fusilier is entangled to the right and the fusiland tothe left. This removes those local fusiliers and fusilandsand creates a longer range link. After the swap operationthe quality of the new link is likely to have degraded andso more error correction may be required.In the case of a general channel error and faulty localgates, to achieve fault tolerance whilst keeping with thespirit our design we can simply replace physical qubitswith logical qubits encoded with a concatenated codesuch as the Bacon-Shor code [17]. Then error correctionis performed at the same time as entanglement swappingwithout any need for additional protocols [18]. This con-trasts with other recent schemes based on planar codesand cluster states [19, 20, 21]. Since logical Bell pairsare required to perform error correction, one promisingapproach is to produce many logical Bell pairs at eachnode, rejecting pairs when errors are detected, so that ahigh-quality pair is always available when required [22].We expect that this will yield a scheme which has a highthreshold ( > and gate errors whilst re-taining the deterministic nature of the protocol. As withall error-correction schemes, the maximum error rate thatis tolerable will depend ultimately on the number of en-tangled links we have available, the number of qubits ateach node, and our target fidelity. However, we are con-fident that our method for establishing entanglement be-tween repeater nodes gives us the flexibility to tailor errorcorrection to communication tasks to ensure high fidelityentanglement with a practical amount of resources. A QUASI-ASYNCHRONOUS DESIGN
With all the quantum components available we nowneed to consider appropriate strategies for putting thisnetwork together and how it will operate. This will needto involve both the quantum resources and the classi-cal communication resources. The two logical choices forhow such a network could operate are basically eithera synchronous or asynchronous scheme. A synchronousscheme requires all the individual repeater nodes to havea shared clock which in certain circumstances could bechallenging. An asynchronous design does not requirethis and so it is the design we will focus on here. Wedepict such a scheme in Fig. (2) and note that an advan-tage of this design is that the distance between adjacentnodes need not be the same (some could be at 10km say,others at 40km).
TimeLHS RHS successful fusilierquantum fusilandclassical messageentanglement swappingclassical heralding pulse
FIG. 2: Schematic representation of a quasi–asynchronous re-peater network. Entanglement generation is initiated on theleft-hand side (LHS) where the system clock is located. In thisdesign the classical heralding pulse from the left-hand mostnode propagates to the furthermost right-hand node (RHS)initiating all the fusiliers as it propagates. Swap operationswithin local nodes occur when the local fusiliers and fusilandshave links to their neighbors. The left-hand node can start itsnext entanglement generation cycle after the round trip timefor a entanglement generation between neighboring nodes.The classical heralding pulse on this next round picks up thePauli frame information as it propagates through the networkand makes it available to the right end node as it arrives.
The quasi-asynchronous design begins with the clock inthe left hand network node initiating the classical herald-ing pulse that is going to propagate along the whole net-work from left to right. As it goes it will initiate thefusiliers to fire the signals to the fusiland in the adjacentnode and thus we will see the fusiliers firing in temporalprogression from the left hand side of the network to theright hand side. Each of the adjacent nodes reports bya classical message which fusilands were successful andwhen that node has a link both to the left and the rightthe entanglement swap operation is performed, creating alonger distance link and freeing the fusiliers and fusilandsin that node for future operations. The results of the par-ity measurements and swap operations are then availableat that local node. We propagate this information on theheralding pulse for the next round of long-range entangle-ment generation. It is important that the next heraldingpulse arrives at the repeater nodes after the swappingoperations have been performed as the herald will pickthis information up. It also means we know exactly whenthe entanglement link is ready to use and so we have anefficient pipe-lined design.
A BUTTERFLY DESIGN
As the entanglement generation is effectively flowingfrom left to right, the left-hand fusilier and the right-handfusiland become entangled at quite different times. ForQKD-like applications this is not an issue. For compu-tational applications this could be an issue, but a simplesolution is to split the network into two halves. The ac-tual location of the split depends on the topology of thenetwork, but is chosen to maximize throughput and tobalance the availability of the left and right qubits. Eachside is going to see a generalized parity for its half of thenetwork. The two halves can be simply connected by en-tanglement swapping and this information propagated toeither the left- or right-hand end with the next heraldingpulse. It also means these resources in the ”central” nodeare freed relatively quickly and consequently we do notneed exceptionally long lived qubits anywhere in the re-peater network. This should significantly lessen the tech-nological challenge inherent in distributed quantum in-formation processing as the quantum resources now haveto be good on time scales associated with the round triptime between adjacent nodes and not the propagationtime over the whole network.
DISCUSSION
We have so far presented a highly optimized design fora quantum repeater and its associated use in a networkwhere the key element is the construction of a constant-time, near-deterministic, high fidelity entanglement linkgenerator between neighboring repeater nodes. This timeis of the order of the round trip time between adjacentnodes, approximately 0.4ms for a 40km link (0.1ms for a10km link) and so allows a maximum rate/fusiland of2500 (10000) entangled pairs between adjacent nodes.With more fusilands per repeater node one can approacha MHz rate. By utilizing oen way error correction thenear-deterministic nature can be maintained without anysignificant time cost. Finally by utilizing a butterfly net-work design the end nodes becomes entangled at roughlythe same time with the classical generalized parity re-sults arriving one cycle (round trip) later. This allowsa highly efficient and pipe-lined architecture. While wehave considered a linear design the network topology canbe easily generalized..
Acknowledgments : We would like to thank Clare Hors-man and Tim Spiller for valuable discussions. This workwas supported in part by MEXT and NICT in Japan andthe EU project HIP. ∗ Electronic address: [email protected][1] M. A. Nielsen and I. L Chuang, Quantum Computationand Quantum Information. Cambridge; New York: Cam-bridge University Press (2000).[2] T. P. Spiller, W. J. Munro, S. D. Barrett, and P.Kok,Contemporary Physics , 407 (2005).[3] J. P. Dowling and G. J. Milburn, QuantumTechnology: The Second Quantum Revolution,arXiv:quant-ph/0206091 (2000). [4] T P Spiller and W J Munro, J. Phys.: Condens. Matter , 1 (2006).[5] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev.Mod. Phys. , 145 (2002).[6] H.-J. Briegel, W. D¨ur, J. I. Cirac, and P. Zoller, Phys.Rev. Lett. , 5932 (1998).[7] N. Sangouard, C. Simon, H. de Riedmatten, and N.Gisin, Quantum repeaters based on atomic ensemblesand linear optics, arXiv:0906.2699v2 (2009) and refer-ences within.[8] P. van Loock, T. D. Ladd, K. Sanaka, F. Yamaguchi,Kae Nemoto, W. J. Munro, and Y. Yamamoto, Phys.Rev. Lett. , 240501 (2006).[9] W. J. Munro, R. Van Meter, Sebastien G. R. Louis, andKae Nemoto, Phys. Rev. Lett. , 040502 (2008).[10] L. Childress, J. M. Taylor, A. S. Sorensen, and M. D.Lukin, Phys. Rev. Lett. , 070504 (2006).[11] S. J. Enk, J. I. Cirac, and P. Zoller, Science , 205(1998).[12] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Na-ture , 413 (2001).[13] B. Zhao, Z.-B. Chen, Y.-A. Chen, J. Schmiedmayer, andJ.-W. Pan, Phys. Rev. Lett. 98, 240502 (2007).[14] W. D¨ur and H. J. Briegel, Rep. Prog. Phys. , 1381(2007).[15] J. Pan, C. Simon, C. Brukner, and A. Zeilinger, Nature , 1067 (2001).[16] S. J. Devitt, K. Nemoto, and W. J. Munro, The id-iots guide to Quantum Error Correction, arXiv:0905.2794(2009).[17] D. Bacon, Phys. Rev. A , 012340 (2006).[18] L. Jiang, J. M. Taylor, Kae Nemoto, W. J. Munro, R.Van Meter, and M. D. Lukin, Phys. Rev A , 032325(2009).[19] S. Perseguers, L. Jiang, N. Schuch, F. Verstraete, M.D.Lukin, J.I. Cirac, and K.G.H. Vollbrecht, Phys. Rev. A , 062324 (2008).[20] S. Perseguers, Fidelity threshold for long-range entangle-ment in quantum networks, arXiv:0910.1459v1 (2009).[21] A. G. Fowler, D. S. Wang, T. D. Ladd, R. Van Meter,and L. C. L. Hollenberg, Fast, fault-tolerant quantumcommunication, arXiv:0910.XXXXv1 (2009).[22] E. Knill, Nature , 39 (2005). APPENDIX - ENTANGLEMENT LINKS
One of the core elements necessary in any repeaterdesign is the creation of entanglement between nearestneighbor links. This entanglement will be created be-tween two electronic spins placed in cavities at neigh-boring repeater stations with nuclear spins available forquantum storage. The electronic and nuclear-spin sys-tems may be achieved, for example, by single electronstrapped in quantum dots, by neutral donor impuritiesin semiconductors or NV diamond centers. For a suffi-cient interaction between the electron and the light field,the system should be placed in a cavity resonant withthe light. The mechanism for the entanglement betweennodes generally fall into two categories. • The heralded creation of very high fidelity entan-gled links utilising single photon or weak coher-ent sources generally with a low probability of suc-cess. The qubit-light field can operate in a numberof regimes including on-resonance and dispersive.Moderate to strong coupling regimes are generallyrequired. ••