Long range failure-tolerant entanglement distribution

Ying LiCentre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543

Sean D. BarrettBlackett Laboratory and Institute for Mathematical Sciences,Imperial College London, London SW7 2PG, United Kingdom

Thomas M. StaceSchool 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. BenjaminDepartment 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 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 first protocol to explicitly allow for technologies of this kind. Given an errorrate between stations in excess of 10%, arbitrarily long range high fidelity 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 amongremote quantum computers is one of the fundamentaltasks of quantum information technologies. It is crucialfor quantum teleportation, quantum cryptography anddistributed quantum computing. Using direct transmis-sion, the success probability of transmitting a qubit andthe fidelity of the resulting quantum state decrease expo-nentially with distance. Therefore, one needs quantumrepeaters to achieve long distance entanglement [1, 2]. Agood quantum repeater protocol should be fault-tolerantand support a high communication rate. In this paper,we will propose a protocol to distribute entanglement be-tween two remote quantum computers. We consider noisein quantum communication channels, and of course errorsgenerated by operations within the repeaters. We assumethat the repeater stations may employ non-deterministicentanglement operations (EOs): that is, a means of en-tanglement, even within the a single repeater, that oftenfails but the failures are ‘heralded’. In addition there isof course a finite error rate even for the operations thatare deemed successful. Non-deterministic EOs will occurwithin individual repeater stations if, for example, theirinternal hardware is based on networking small quantumregisters together optically, i.e. qubits can be entangledby joint measurements on single photons emitted fromthese qubits rather than control of interactions [3, 4].Such an architecture may be much easier to implementin a scalable way than monolithic architectures e.g. largescale ion traps. Even with this assumption that EOs failboth between and within repeater stations, we find that

the rate of distributing entanglement decreases only log-arithmically with the communication distance.

Cluster states are resources of measurement-basedquantum computing [5], and long-range entanglementcan be established in noisy cluster states [6]. In thispaper, we propose a protocol of distributing entangle-ment by single-qubit measurements on a topologicallyprotected cluster (TPC) state [7] across the chain of re-peater stations. The TPC state must first be grownvia operations within repeaters together with quantumcommunication between pairs of neighboring repeaters.The operations within repeaters are expected to have amuch better performance than communications betweenrepeaters (since the latter may be over distances of kilo-metres). We find that the protocol is valid if the probabil-ity of an error occurring in the communication channel islower than a threshold, which is 15% when errors inducedby operations within repeaters are negligible. With errorsless than the threshold, entanglement can be establishedbetween two remote logical qubits encoded in two sepa-rated graph states, which may be used for further infor-mation processing via the topological measurement basedquantum computing [7]. Alternatively one can also de-code each logical qubit to a physical qubit via single-qubitmeasurements. Although we describe only the two-partyprotocol here, it should be straightforward to generalizefor distributing multi-party entanglement.

In this protocol, the quality of the eventual entangle-ment between logical qubits is only limited by the numberof qubits in each repeater. Therefore, our protocol effec-

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FIG. 1: The scheme of quantum entanglement distributionprotocol based on topologically protected cluster (TPC) state.(a) Alice and Bob can be entangled via a chain of quantumrepeater stations, which are connected by optical quantumcommunication channels. (b) Each station contains a ‘slice’of the TPC state. The TPC state contains two empty tubes(blue) without any qubit. (c) Once the TPC is complete, allqubits are measured in X basis except two parts of the TPCstate (green) in stations of Alice and Bob respectively; theseare called plugs and contain the eventual encoded shared Bellpair. (d) The elementary cell of the TPC state. Each logicalqubit is encoded as subfigure (e) and can be decoded as subfig-ure (f) (see text). (g) Two surfaces propagating correlationsbetween two logical qubits.

tively distills as well as distributes entanglement. Theidea of using an error correction code with protected log-ical qubits for remote entanglement was firstly reportedin Ref. [8], in which the Calderbank-Shor-Steane codeis employed. Subsequently 3D lattice-based distributionhas also been studied [9] and the extension to lower di-mensionality has been examined [10]. Recently, in aprotocol for quantum state transfer of a surface-code-encoded qubit, the efficiency of quantum communicationis greatly improved by removing the necessity of two-way communication [11]. Compared with these proto-cols, ours is the first to consider a probabilistic architec-ture within each repeater station, so that the entangle-ment distribution can be efficient even if EOs are far fromdeterministic.

Quantum Repeaters based on Cluster States. Alice andBob are entangled via a chain of quantum repeater sta-tions. Two nearby repeaters are connected by opticalquantum communication channels [Fig. 1(a)] – essen-tially a bundle of optical fibres that are used in parallel.To give an overview of the process: Firstly, a TPC state

is grown across quantum repeater stations via probabilis-tic EOs and quantum communications between nearbystations. The TPC state contains two parallel emptytubes, which terminate in stations of Alice and Bob.Each empty tube is a void in the TPC state, with anelongated shape and shown as a blue rectangular cuboidin Fig. 1(b). Once the TPC state is generated, measure-ments in the X basis are performed on all qubits excepttwo parts of the TPC state located in stations of Al-ice and Bob respectively [see Fig. 1(c)]. The two partswhich are to remain unmeasured are called plugs, andare connected with empty tubes. Two empty tubes andtwo plugs form a closed loop. There is one logical qubitencoded in each plug. After all other quits are measured,and the outcomes are communicated to Alice and Bob,then these two logical qubits are entangled as one of theBell states (determined by measurement outcomes).

The TPC state is a cluster state of qubits locatedon the a cubic lattice [7]. There is one qubit on eachface and edge of the elementary cell [Fig. 1(d)]. Byshifting the lattice, one can transfer qubits on faces toedges, and vice versa. The new lattice is called the duallattice of the original primal lattice. The TPC stateis stabilized by K(c) =

∏a∈cXa

∏b∈∂c Zb, where c is

an arbitrary primal (dual) surface and ∂c is the primal(dual) chain as the boundary of c. Qubits in the set c(∂c) are located on faces (edges) composing the surface(chain) c (∂c). The logical qubit is encoded in a plugas X =

∏a∈sectionXa and Z =

∏b∈line Zb, where X

and Z are Pauli operators of the logical qubit. Here,section is a dual surface across the plug, and line is aprimal chain on the surface of the plug and connectingtwo 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-plug loop, and (ii) cii is a closed dual surface envelop-ing one empty tube and crossing two plugs [Fig. 1(g)].The two stabilizers are K(ci) = ZAZB

∏a∈ci Xa and

K(cii) = XAXB

∏a∈c′ii

Xa, where A,B denote Alice and

Bob respectively, and c′ii denotes the part of the sur-face cii outside two plugs. After measurements in theX basis, one can replace Xa with measurement out-comes. Then, we get two new stabilizers ZAZB = ±1and XAXB = ±1, i.e. the two logical qubits are stabi-lized as one of Bell states. Here, the two signs depend onmeasurement outcomes.

Besides two-party entanglement, we note that ourscheme can be directly generalized to multi-party entan-glement, e.g. three-party and four-party entanglement asshown in Ref. [12].

Noise in quantum communication channels and imper-fections in operations will give rise to phase errors onthe TPC state. In order to eliminate errors from the Bellstate of two logical qubits, we monitor errors on the TPCstate by parity check operators K(cc), where cc are min-

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p2 3 4 3 4

Snowflake, local resource

Dumbbell, nonlocal resource

Tree-structure graph state

root

(a)

(b) (c)

FIG. 2: Resource graph states, i.e. building-blocks, for grow-ing the topologically protected cluster state. Each red dottedline denotes a parity projection (PP). (a) Tree graph statescan be grown by PPs on roots of trees. (b) Four trees can befused into a ‘snowflake’ graph state as the following: fusingeach pair of trees into a bigger tree at first; cutting two rootsby measurements in Z basis; fusing them into a snowflakeand cutting the unwanted qubit. (c) Two trees in differentquantum repeater stations are fused into a dumbbell graphstate by a Bell measurement on photon-p1 and the photon-p2, each associated with a qubit in a different stations. Oneof the photons (p2) will have travelled between stations. TheBell measurement is followed by a measurement in the Y ba-sis on the qubit-1 and a measurement in the X basis on thequbit-2, in order to get the desired dumbbell graph state.

imum closed surfaces. Usually, minimum closed surfacesare surfaces of elementary cubes. However, some qubitson the TPC state may be missing. The parity check op-erator of an elementary cube with missing qubits can notbe used to detect errors. Then, one has to use productsof parity check operators connected by missing qubitsto form a new set of parity check operators [23]. Par-ity check operators reveal the endpoints of error chains,where an error chain (ring) is a sequence of phase er-rors. If the number of phase errors on the surface cc isodd, the existence of errors can be identified by K(cc),which is called an error syndrome. Errors are not activelycorrected, rather parities of

∏a∈ci Xa and

∏a∈c′ii

Xa, are

modified by knowledge of the total number of error chainscrossing surfaces ci and c′ii respectively. After the errorcorrection, only error rings encircling the tube-plug loop,error chains connecting two empty tubes and error chainsconnecting the loop with the boundary of the TPC state[12], may contribute an error on logical qubits. If noiseand imperfections are less than a threshold, the probabil-ity of an error on logical qubits decreases exponentiallywith the minimum length of these error rings and errorchains [7].

Cluster State Growth.- In order to grow the TPC stateacross quantum repeater stations, some ‘building-block’graph states should first be prepared within each repeaterdevice. It is through the use of these building-blocks that

we overcome the impact of high EO failure rates whenwe create the large scale TPC state. The structure ofthese elementary graph states can be a star [13], a line[14], a cross [15], or a tree [16, 17]. In this paper, wetake the tree structure as an example, and the schemecan be adapted to other structures. The tree structureaccumulates fewer errors than other structures when thesuccess probability of EOs is low [17, 18]. Tree-structuregraph states can be generated by using parity projections(PPs) [3]. A PP on roots of two individual trees canfuse them into a double-size tree [Fig. 2(a)]. If all PPsare successful, after n steps, one can grow a tree with2n qubits from separated qubits, where the integer n iscalled the generation of the tree.

Trees are fused into two kinds of building-block graphstates. Snowflake graph states are prepared by fusingfour trees [Fig. 2(b)]. Each snowflake will ultimatelycorrespond to a specific qubit on the TPC state. Eachquarter of a snowflake is used to establish a connectionwith a neighboring snowflake. We refer to the secondkind of building-block as a dumbbell. These are nonlocalbuilding blocks connecting two nearby quantum repeaterstations [Fig. 2(c)]. A dumbbell is formed by two treeslocated in different stations. For example, suppose thatthe basic qubits are optically active atoms: then in orderto prepare a dumbbell, we cause each root qubit emita 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-tational basis and the photonic qubit can be encoded inpolarization, frequency [19] or time-bin [20]. One pho-ton is transmitted from one station to another. After aBell measurement on two photons and single-qubit mea-surements on roots, we obtain the dumbbell graph state[12].

Making a building-block graph state requires all oper-ations to be successful, whose probability may be quitesmall. Therefore, building-block graph states are pro-duced with a post selection strategy: if an operation isheralded as failed, the corresponding graph state is aban-doned with the qubits reinitialized.

Once a sufficient number of each resource (snowflakesand dumbbells) have been generated, we can assemblethem to create a suitable TPC state. Snowflakes are as-sembled by PPs on leaves, which are qubits on the edge ofa snowflake (Fig. 3). Two snowflakes in the same quan-tum repeater station can be connected directly, while twosnowflakes in different stations are connected by bridg-ing them with a dumbbell shared by these two stations.The number of leaf qubits on each quarter of a snowflakeis 2n−1. Therefore, the failure probability of connectingtwo snowflakes in the same station is FL = f2

n−1

, andthe failure probability of connecting two snowflakes indifferent stations is FNL ' 2FL, where f is the basic fail-ure probability of EOs. After establishing connectionsbetween snowflakes, all qubits except those at the center

4

Dumbbell

3 4 5 6

7

5 6

7

TPC state

FIG. 3: The strategy of assembling resource graph statesinto the full topologically protected cluster (TPC) state whichspans all quantum repeater stations. (a) Snowflakes withinthe same station are connected directly to each other by par-ity projections (red dotted lines) on leaves. Two snowflakesin different stations can be connected via a dumbbell whichincorporates the required nonlocal connection (dash line). (b)After extraneous quits are removed, ultimatelty the qubits atthe heart of each snowflake survive as nodes of the TPC state.

of each snowflakes are removed by appropriate single-qubit measurements, so that the surviving qubits formthe TPC state. Here, the measurement pattern for re-moving qubits can be found in Ref. [12]. Since somesnowflakes have failed to connect, this implies some miss-ing connections on the TPC state. We presently describesimulations establishing that when connections are rarelymissing, i.e. FL < 5%, then the cluster state is well con-nected: it is easy to find surfaces propagating correlationsbetween two logical qubits, indeed this is guaranteed inthe scaling limit (as expected from percolation theory)[21, 23].

As a footnote to this section we note that the ‘building-block’ strategy is not always necessary. If the failureprobability of EOs is low enough f < 5%, one may gen-erate the TPC state directly, for example, by using con-trol phase gates [7], where control-phase gates on twoqubits located in different quantum repeater stations canbe simulated by consuming entanglement prepared viaquantum communication [24]. However here we are in-terested in the general case where the failure probabilitymay be very high.

Noise, Imperfections and Error Correction.- Bothnoise in quantum communication channels and imper-fections in operations can give rise to errors on the TPCstate. We assume communication noise is depolarized,and described by the superoperator E = (1 − ε)[1p2] +ε([Xp2] + [Yp2] + [Zp2])/3 [see Fig. 2(c)]. We call qubitswith nonlocal connections ‘joint qubits’ [gray circles inFig. 3]. Errors induced by communication noise maymake phase errors on corresponding joint qubits (qubits5 and 6) with a probability 2ε/3 for each of them [12].Consider first the case that internal operations within

stations are perfect (when heralded as successful); thenonly joint qubits have errors, and these imperfect qubitsexist in specific non-adjacent layers of the TPC state.Then 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-nected lattice [26]. Moreover a near-perfectly connectedlattice would indeed be achievable since, given error freeEOs within repeaters, one could always grow sufficientlybig tree structures to make FL as low as desired. There-fore, with perfect operations, the condition of getting acorrect correlation between two logical qubits faithfullyis 2ε/3 . 10%, i.e. the error threshold of communicationnoise is εt ' 15%.

With imperfect operations, all qubits on the TPC statemay affected by phase errors. If the distribution of phaseerrors is uniform, i.e. all qubits may have a phase errorwith the same probability, the threshold of phase errorsis about 3% for perfectly connected TPC state [25]. How-ever, in our case, the TPC state grown by probabilisticEOs is unlikely to be perfectly connected and there aremore errors on joint qubits than others. Our strategyis to treat missing connections by transforming them toqubit loss, by means of deleting the qubits with missingconnections using measurements in the Z basis. Then,the loss probability of joint qubits is 5FL, and the lossprobability of other qubits is 4FL. We determine errorthresholds for general cases numerically as shown in Fig.4(a), using the method developed in Ref. [22, 23].

The error rate of imperfect operations must be lowerthan the threshold of fault-tolerant quantum comput-ing (FTQC). The threshold of FTQC on the TPC statewith non-deterministic EOs (deterministic control-phasegates) is about 2×10−4 [18] (5×10−3 [7]). By optimizingthe size of trees, (a bigger tree can reduce missing con-nections but generate more errors), we have obtained thethresholds of tolerable communication noise in the pres-ence of finite error rates for internal EOs, see Fig. 4(b). Ifthe error rate of operations is 10−4, the threshold of com-munication noise is about 11% when the success probabil-ity of entangling operations is 1%. In contrast, by usingcontrol-phase gates to generate the TPC state directly,the threshold of communication noise is still above 10%even if the error rate of operation is 2 × 10−3, but thesuccess probability must be higher than 98%.

Full decoding.- A logical qubit can be decoded into aphysical qubit by measurements on the correspondingplug, leaving just one qubit unmeasured. The residualqubit carries the quantum state of the logical qubit. Fordecoding, two (blue) pyramids inside the plug, whoseapexes hold the residual qubit (red circle) and bases con-nect tubes, are measured in the Z basis, while otherqubits are measured in the X basis [see Fig. 1(f)]. Theresidual qubit can acquire an error if there is an errorchain connecting two pyramids. Therefore, the proba-bility of an error on the residual qubit is p + O(p3) [7],

5

0.0 0.2 0.4 0.6 0.80.00

0.04

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0.12thresholds  of  p

J

p/pJ

 FL=0.00  FL=0.01  FL=0.04

0.90 0.95 1.000.05

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10-­3 10-­2 10-­1 1000.00

0.05

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thresholds  of  ¦

f(a) (b)

FIG. 4: Thresholds of error correction on the topologicallyprotected cluster (TPC) state. (a) Thresholds of phase errorson joint qubits, which is dependent on the ratio between theerror probability on joint qubits (pJ) and the error probabilityon other qubits (p). (b) Thresholds of communication noise εwith operational error rate 10−4 (solid line), evaluated fromthe linear interpolation of data in subfigure (a). By usingcontrol-phase gates to generate the TPC state directly, theerror rate can be much higher (2 × 10−3) but only a failureprobability (f) lower than 4% is tolerable (dash line). Herewe have assumed that memory errors happen at a lower ratethan operational errors. Memory errors at 10% of the opera-tional error rate can lower the threshold, but not dramatically(dotted line).

where p is the probability of phase errors on the residualqubit, which is usually lower than 3%.

Performance.- The probability of errors on two en-tangled logical qubits decreases exponentially with theminimum length of error rings and error chains [7]. Wedesign the TPC state as follows: the perimeters of twoempty tubes, the distance between empty tubes, and thedistance between each empty tube and the boundary,are each proportional to the same length scale L. Thelength of the TPC state, i.e., the number of quantumrepeater stations, can increase the probability of errorrings and error chains linearly [6]. Therefore, the over-all probability of errors on two entangled logical qubitsis εE ∝ Ne−κL, where N is the number of stations, κis a constant depending on p, pJ and FL. To achievea given quality of entanglement, we need a TPCS withL = O(log(N/εE)/κ). The number of photonic qubitstransferred between two nearby stations is proportionalto L2. Therefore the overall entanglement distributionrate of our scheme is RN = O(log−2(N/εE)/κ).

In conclusion, we have described an advanced proto-col for distributing entanglement through the use of re-peater stations which together generate a topologicallyprotected cluster state. We find that the approach is re-markably robust to errors, while the resource cost withineach repeater scales only logarithmically with the totaldistance over which entanglement is to be shared.

While preparing this document we became aware ofa manuscript describing closely related research: Ash-ley Stephens, Jingjing Huang, Kae Nemoto and WilliamJ. Munro, “Fault-tolerant quantum communication withrare-earth elements and superconducting circuits”.

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