Surface-Codes-Based Quantum Communication Networks

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Article Surface-Codes-Based Quantum Communication Networks

Ivan B. Djordjevic Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA; [email protected]; Tel.: +1-520-626-5119 Received: 8 August 2020; Accepted: 21 September 2020; Published: 22 September 2020

Abstract: In this paper, we propose the surface codes (SCs)-based multipartite quantum communication networks (QCNs). We describe an approach that enables us to simultaneously entangle multiple nodes in an arbitrary network topology based on the SCs. We also describe how to extend the transmission distance between arbitrary two nodes by using the SCs. The numerical results indicate that transmission distance between nodes can be extended to beyond 1000 km by employing simple syndrome decoding. Finally, we describe how to operate the proposed QCN by employing the software-deﬁned networking (SDN) concept.

Keywords: quantum key distribution (QKD); quantum communications networks (QCNs); quantum communications; entanglement; surface codes

1. Introduction Quantum information processing (QIP) opens up new avenues for reliable communications, high-precision sensing, and high-performance computing [1–20]. Entanglement represents a unique resource for QIP,which allows quantum computers to solve classically intractable problems [7], provides certifiable security [2] for data transmissions, and enables sensors to achieve measurement sensitivities beyond the classical limit [8]. The quantum communication is the key cornerstone to fully exploit the properties of entanglement. The modern classical communications tend to use heterogeneous networks capable of simultaneous data transmission between nodes connected via different types of channels, such as free-space optical (FSO) and fiber-optics links. Nodes in existing quantum communication networks (QCNs), however, have been limited to a single optical medium. Moreover, trusted node assumption [4] is required to operate the current QCNs. As a result, one compromised node in a QCN can undermine the security of the entire QCN. Several quantum key distribution (QKD) testbeds have been reported so far, such as the DARPA QKD network [5], Tokyo QKD network [6], and the secure communication based on quantum cryptography (SECOQC) network [7]. Unfortunately, these different QKD networks employ the dark fiber infrastructure. In this paper, we propose the multipartite heterogenous QCN employing the surface codes, which does not require the trusted node assumption. The research on multipartite entanglement is getting momentum with numerous experimental demonstrations, such as [8]. The surface codes, typically defined on a 2-D lattice, are closely related to the quantum topological codes on the boundary [1], introduced by Bravyi and Kitaev [11,12]. This class of codes is highly popular in quantum computing [13–15] because only local qubits are involved in stabilizers. In Litinski’s framework [14], the surface code for quantum computing is represented as a game, played on a board partitioned in a certain number of tiles. On each tile we can place a logical qubit, represented as a patch. The edges of qubits represent the logical Pauli operators [1]. The logical qubits correspond to the surface code (SC) patches. By placing the SC patches in nodes of a communication network, and connecting the neighbouring patches by d wavelength channels, corresponding to the distance of the underlying surface code, we

Entropy 2020, 22, 1059; doi:10.3390/e22091059 www.mdpi.com/journal/entropy Entropy 2020, 22, x FOR PEER REVIEW 2 of 10 underlying surface code, we can create the quantum communication network. The SC patches placed in intermediate nodes can be operated as the SC-based quantum repeaters, thus extending significantly the transmission distance. When the patch edges in the tiles of neighbouring network nodes are different, we can perform the product measurements to entangle them. For instance, the product Z⊗Z between adjacent nodes’ patches can be simultaneously measured to introduce the entanglement between two adjacent quantum nodes. Namely, we start with the state |++ = 0.5(|00 + |11 + |01 + |10) and perform the measurement on Z⊗Z operator. If the result of the measurement Entropy 2020, 22, 1059 2 of 10 is +1, the qubits end up in state 2−1/2(|00 + |11); otherwise, they end up in state 2−1/2 (|01 + |10). In either case, the qubits are maximally entangled. This indicates that the proposed SC-based QCN is highlycan create flexible the quantumand have communication numerous applications, network. Theincluding: SC patches (i) to placed teleport in intermediatequantum states nodes between can be anyoperated two nodes as the in SC-based the network, quantum (ii) repeaters,to develop thus the extendinginformation significantly infrastructure the transmissionwith unprecedented distance. securityWhen the level, patch (iii) edges to enable in the distributed tiles of neighbouring quantum computing, network nodes and (iv) are to di ffenableerent, ultra-high we can perform precision the forproduct quantum measurements sensing applications. to entangle To them. operate For such instance, a quantum the product network,Z Zwebetween propose adjacent to employ nodes’ the ⊗ software-definedpatches can be simultaneously networking (SDN) measured concepts. to introduce the entanglement between two adjacent quantum nodes.The Namely, paper is we organized start with theas follows. state |++ In= Section0.5(|00 2,+ |we11 +introduce |01 + |10 the) and surface perform codes the and measurement describe i i i i i brieflyon Z Ztheoperator. Litinski’s If formalism the result ofneeded the measurement in incoming issections.+1, the In qubits Section end 3, up we in describe state 2 1the/2(| 00proposed+ |11 ); ⊗ − i i SC-basedotherwise, QCN they concept. end up inIn stateSection 2 1 /4,2 (we|01 describe+ |10 ). Inour either approach case, theto extend qubits the are transmission maximally entangled. distance − i i betweenThis indicates QCN nodes. that the In proposed Section 5, SC-based we provide QCN illustrative is highly numerical flexible and results. have In numerous Section 6, applications,we describe howincluding: to operate (i) to the teleport proposed quantum SC-based states QCN between by utiliz anying two the nodes SDN inconcepts. the network, Finally, (ii) in to Section develop 7, we the provideinformation some infrastructure important concluding with unprecedented remarks. security level, (iii) to enable distributed quantum computing, and (iv) to enable ultra-high precision for quantum sensing applications. To operate such a 2.quantum Surface network,Codes for we Quantum propose Networking to employ the and software-defined Distributed Computing networking (SDN) concepts. TheThe surface paper iscode organized belongs as to follows. the class In of Section topological2, we codes introduce [1] and the it surface is defined codes on anda 2-D describe lattice, withbriefly one the illustrativeLitinski’s formalismexample provided needed inin incomingFigure 1, with sections. qubits In being Section clearly3, we describeindicated the in Figure proposed 1a. TheSC-based stabilizers QCN of concept. plaquette In Sectiontype can4, be we defined describe as our provided approach in toFigure extend 1b. the Each transmission plaquette stabilizer distance denotedbetween by QCN X (Z nodes.) is composed In Section of5 ,Pauli we provide X (Z)-operators illustrative on numerical qubits located results. in Inthe Section intersection6, we describeof edges ofhow corresponding to operate the plaquette. proposed As SC-based an illustra QCNtion, by the utilizing plaquette the stabilizer SDN concepts. denoted Finally, by X related in Section to qubits7, we 1provide and 2 will some be important X1×2. The concludingplaquette stabilizer remarks. denoted by Z, related to qubits 5, 6, 8, and 9, will be Z5Z6Z8Z9. To simplify the notation, we can use the representation provided in Figure 1c, where the 2. Surface Codes for Quantum Networking and Distributed Computing shaded plaquettes correspond to all-X containing operators’ stabilizers, while the white plaquettes correspondThe surface to all-Z code containing belongs to op theerators’ class of stabilizers. topological The codes stabilizers [1] and itrequire is defined only on local a 2-D qubits’ lattice, interaction,with one illustrative which is not example true for provided other classes in Figure of quantum1, with qubits error correction being clearly codes. indicated in Figure1a. The stabilizersThe weight-2 of plaquettestabilizers type are allocated can be defined around as perimeter, provided inwhile Figure weight-41b. Each stabilizers plaquette are stabilizer located indenoted the interior. by X ( ZThe) is logical composed operators of Pauli forX this(Z)-operators code are run on qubits over both located sides in theof the intersection lattice, as of shown edges in of corresponding plaquette. As an illustration,== the plaquette stabilizer denoted by X related to qubits 1 and Figure 1d, and can be represented as XXXXZZZZ369, 123. The codeword length is determined as 2 will be X1 2. The plaquette stabilizer denoted by Z, related to qubits 5, 6, 8, and 9, will be Z5Z6Z8Z9. the product ×of side lengths, expressed in number of qubits, and, for the surface code from Figure 1, weTo simplifyhave that the n = notation, Lx·Lz = 3·3 we = 9. can On use the the other representation hand, the number provided of information in Figure1c, qubits where is the k = shaded 1. The minimumplaquettes distance correspond of this to all- codeX containing is determin operators’ed as the stabilizers, minimum whileside length, the white that plaquettes is d = min( correspondLx,Lz) = 3, indicatingto all-Z containing that this operators’code can correct stabilizers. a single The qubit stabilizers error.require only local qubits’ interaction, which is not true for other classes of quantum error correction codes.

= Z ZZZ123 X X 1 2 3 X 1 2 1 2 3 Z X Z 4 5 6 4 5 6 = X XXX369 Z X Z 7 8 9 Z5Z6Z8Z9 7 8 9 X

Z X

(a) (b) (c) (d)

Figure 1. Illustration of a surface code: (a) the qubits are located in the lattice positions, (b) all-X and all-Z plaquette operators, (c) popular representation of surface codes in which stabilizers are clearly indicated, and (d) logical operators.

The weight-2 stabilizers are allocated around perimeter, while weight-4 stabilizers are located in the interior. The logical operators for this code are run over both sides of the lattice, as shown in Figure1d, and can be represented as X = X3X6X9, Z = Z1Z2Z3. The codeword length is determined as the product of side lengths, expressed in number of qubits, and, for the surface code from Figure1, we have that n = Lx Lz = 3 3 = 9. On the other hand, the number of information qubits is k = 1. The × × Entropy 2020, 22, x FOR PEER REVIEW 3 of 10 Entropy 2020, 22, x FOR PEER REVIEW 3 of 10 Figure 1. Illustration of a surface code: (a) the qubits are located in the lattice positions, (b) all-X and Entropy 2020, 22, 1059 3 of 10 Figureall-Z plaquette 1. Illustration operators, of a surface (c) popular code: representation (a) the qubits areof surfacelocated codesin the inlattice which positions, stabilizers (b )are all-X clearly and all-Zindicated, plaquette and operators,(d) logical operators.(c) popular representation of surface codes in which stabilizers are clearly indicated, and (d) logical operators. minimum distance of this code is determined as the minimum side length, that is d = min(Lx,Lz) = 3, Let us now specify the rules of the game, that is the operations that can be applied to the patches indicating that this code can correct a single qubit error. (qubits),Let us which now specifycan be thecategorized rules of the as game [14]:, that (i) initializationis the operations, (ii) qubitthat canmeasurements be applied, toand the (iii) patches patch Let us now specify the rules of the game, that is the operations that can be applied to the patches (qubits),deformations which. Compared can be categorized to computing as only[14]: limited(i) initialization number, of(ii) operations qubit measurements are required, and in (iii)quantum patch (qubits), which can be categorized as [14]: (i) initialization, (ii) qubit measurements, and (iii) patch deformationsnetworking.. ComparedWith each ofto thesecomputing operations, only limitedwe asso numberciate the of cost, operations expressed are in required terms of in time-steps, quantum deformations. Compared to computing only limited number of operations are required in quantum networking.with each time-step With each (t.s.) of correspondingthese operations, to ∼wed code asso ciatecycles the (related cost, expressed to the measuring in terms all of stabilizerstime-steps, d networking. With each of these operations,∼ we associate the cost, expressed in terms of time-steps, withtimes), each with time-step d being (t.s.) the distancecorresponding of underlying to d code surface cycles code (related per tile.to the One-qubit measuring patches, all stabilizers shown ind times),with each with time-step d being (t.s.)the distance corresponding of underlying to ~d codesurface cycles code (related per tile. to One-qubit the measuring patches, all stabilizersshown in Figure 2 as |q1 and |q2, can be initialized to |0 or |+ states, while two-qubit patches, shown in d times), with d being the distance of underlying surface code per tile. One-qubit patches, shown Figure 2 as |q1 and |q2, can be initialized to |0 or |+ states, while two-qubit patches, shown in Figure 2 as |q3, to |00 or |++ states, with associated cost being 0 t.s. (The logic |+-state indicates in Figure2 as |q1 and |q2 , can be initialized to |0 or |+ states, while two-qubit patches, shown in Figurethat all 2 physical as |q3, toqubitsi |00 areori |++initialized states, into with |+ associated-state.)i In costprinciple,i being one-qubit0 t.s. (The patches logic |+ can-state be initializedindicates Figure2 as |q3 , to |00 or |++ states, with associated cost being 0 t.s. (The logic |+ -state indicates thatto the all arbitrary physicali qubitsstates,i aresuch initialized asi the magic into state |+-state.) |m = |0In principle, + exp(jπ/4)|1 one-qubit; however, patches an canundetectedi be initialized Pauli that all physical qubits are initialized into |+ -state.) In principle, one-qubit patches can be initialized toerror the [1]arbitrary can spoil states, the initialized such as the state. magic The state single-patchi |m = |0 measurements + exp(jπ/4)|1 can; however, be performed an undetected in X or Z bases,Pauli to the arbitrary states, such as the magic state |m = |0 + exp(jπ/4)|1 ; however, an undetected Pauli errorand after [1] can the spoil measurement the initialized the state.corresponding The single-patch patchei measurementsis get removed cani from be performedthe board, inthus X orfreeing Z bases, up error [1] can spoil the initialized state. The single-patch measurements can be performed in X or Z bases, andthe occupiedafter the measurementtiles for future the use. corresponding The cost associat patcheed withs get single-patch removed from measurements the board, thus is 0 t.s.freeing For two- up and after the measurement the corresponding patches get removed from the board, thus freeing up thepatch occupied measurements tiles for, when future the use. edges The costin neighboring associated with tiles single-patch are different, measurements we can perform is 0 t.s.the Forproduct two- the occupied tiles for future use. The cost associated with single-patch measurements is 0 t.s. For patchmeasurements. measurements As, anwhen illustration, the edges the in neighboringproduct Z⊗Z tiles between are different, adjacent wepatches can perform can be measuredthe product as two-patch measurements, when the edges in neighboring⊗ tiles are diﬀerent, we can perform the product measurements.illustrated in Figure As an 3 (left).illustration, the product Z Z between adjacent patches can be measured as measurements. As an illustration, the product Z Z between adjacent patches can be measured as illustrated in Figure 3 (left). ⊗ illustrated in Figure3 (left).

(a) (b) (a) (b) Figure 2. Illustration of one-qubit and two-qubits patches: (a) notation and (b) actual physical Figure 2. Illustration of one-qubit and two-qubits patches: (a) notation and (b) actual Figureimplementation. 2. Illustration of one-qubit and two-qubits patches: (a) notation and (b) actual physical implementation.physical implementation. Z⊗Z measurement Z⊗Z measurement

Lattice Latticesurgery surgery

X X X X | | Z X+ Z Z X+ Z Z X|+ Z Z X|+ Z | Z | Z Z +X Z +X Z |+X Z Z |+X Z X X X X

FigureFigure 3.3. Illustrating thethe latticelattice surgerysurgery procedureprocedure forfor thethe measurementmeasurement onon thethe productproduct ZZ⊗Z. Figure 3. Illustrating the lattice surgery procedure for the measurement on the product Z⊗Z. InIn surfacesurface codes, codes, this this corresponds corresponds to the tolattice the surgerylattice [14surgery,15], in [14,15], which wein changewhich thewe conﬁguration change the asconfiguration shownIn surface in Figure as codes, shown3 (right) thisin Figure bycorresponds introducing 3 (right) byto the introducingthe patches lattice withsurgery the dark-redpatches [14,15], with edges; in dark-red which after that, edges;we wechange after measure that,the configurationthe stabilizers as for shownd cycles in Figure to get the3 (right) outcome by introducing of measurements, the patches and splitwith again.dark-red The edges; cost associatedafter that, with the lattice surgery is 1 t.s. This represents the way to introduce the entanglement between two

Entropy 2020, 22, x FOR PEER REVIEW 4 of 10 Entropy 2020, 22, 1059 4 of 10 we measure the stabilizers for d cycles to get the outcome of measurements, and split again. The cost associated with the lattice surgery is 1 t.s. This represents the way to introduce the entanglement adjacent patches. Namely, we start with the state |++ = 0.5(|00 + |11 + |01 + |10 ) and perform between two adjacent patches. Namely, we start with thei state |++i = 0.5(|00i  + |11i  + |01i  + |10) and the measurement on Z Z operator. If the result of the measurement is +1, the qubits end up in state perform1/2 the measurement⊗ on Z⊗Z operator. If the result1/2 of the measurement is +1, the qubits end up 2− (|00 + |11 ); otherwise, they end up in state 2− (|01 + |10 ). In either case, the qubits are in statei 2−1/2(|00i  + |11); otherwise, they end up in state 2−1/2 (|01i  + |10i ). In either case, the qubits are maximally entangled. We can apply the similar procedure to the X Z product operator. Of course, it is maximally entangled. We can apply the similar procedure to the ⊗X⊗Z product operator. Of course, it also possible to measure the product operator involving the Y operator, which is really not needed in is also possible to measure the product operator involving the Y operator, which is really not needed our proposed QCNs. What is even more interesting is that it is possible to measure the product for in our proposed QCNs. What is even more interesting is that it is possible to measure the product for more than two encoded Pauli operators through multi-patch measurements [14]. more than two encoded Pauli operators through multi-patch measurements [14]. 3. Proposed Surface-Codes-Based Quantum Communications Networks 3. Proposed Surface-Codes-Based Quantum Communications Networks To enable the next generation of quantum communication networking, we propose to employ To enable the next generation of quantum communication networking, we propose to employ the surface codes so that the logical qubits are located at different nodes in the network. The logical the surface codes so that the logical qubits are located at different nodes in the network. The logical qubits are represented by the patches introduced in the previous section. For simplicity, we assume qubits are represented by the patches introduced in the previous section. For simplicity, we assume that the surface code is defined on a d d grid. The neighboring nodes are connected by employing d that the surface code is defined on a d× × d grid. The neighboring nodes are connected by employing d wavelengths, as illustrated in Figure4. Some of the optical links could be FSO links. wavelengths, as illustrated in Figure 4. Some of the optical links could be FSO links. By performing the Z Z measurement, as described in the previous section, the logical qubits By performing the Z⊗⊗Z measurement, as described in the previous section, the logical qubits create the Einstein–Podolsky–Rosen (EPR) pair. The results of the measurements have been passed to create the Einstein–Podolsky–Rosen (EPR) pair. The results of the measurements have been passed the SDN controller, which will know the exact EPR pair being created. To create the desired QCN, to the SDN controller, which will know the exact EPR pair being created. To create the desired QCN, thethe corresponding corresponding product product measurementsmeasurements needneed toto bebe simultaneouslysimultaneously performed. As As an an illustration, illustration, letlet us us consider consider the the ring ring network network composedcomposed ofof fourfour nodes,nodes, as shown in Figure 55 (left), with each node beingbeing equipped equipped with with the the surface surface code code patch patch representing representing the corresponding the corresponding logical qubits.logical Inqubits. principle, In oneprinciple, surface one patch surface can be patch split can between be split multiple between nodes, mult but,iple tonodes, facilitate but, explanations,to facilitate explanations, we assume thatwe each node contains a single SC patch. By performing the simultaneous product Z Z measurements assume that each node contains a single SC patch. By performing the simultaneous⊗ product Z⊗Z q q q q q q q q betweenmeasurements logical qubitsbetween1 andlogical2, qubits2 and q31, and3 and q2, q42, and4 and q3, q13, weand can q4, entangleq4 and q1 the, we nodes can entangle 1–4 and thusthe createnodes the 1–4 ring and QCN. thus create On the the other ring hand, QCN for. On the the four-node other hand, mesh for network the four-node shown mesh in Figure network5 (right), shown by performing the simultaneous product Z Z measurements between⊗ logical qubits q1 and q2, q2 and q3, in Figure 5 (right), by performing the⊗ simultaneous product Z Z measurements between logical q and q , q and q as well as the simultaneous X X measurements between⊗ q and q , q and q , we 3qubits q41 and4 q2, q12 and q3, q3 and q4, q4 and q1 as well⊗ as the simultaneous X X measurements1 3 2 between4 canq1 and entangle q3, q2 theand four q4, we qubits can into entangle the mesh the configuration.four qubits into By the providing mesh configuration. the results of theBy providing measurements the toresults the SDN of the control measurements plane, the to exact the SDN maximum control entangledplane, the stateexact betweenmaximum nodes entangled in the state QCN between will be known.nodes in Clearly, the QCN this will approach be known. allows Clearly, us to this entangle approach the allows logical us qubits to entangle in an arbitrarythe logical network. qubits in Thean trappedarbitrary ions-based network. The technology trapped representsions-based a technology perfect candidate represents for a practical perfect candidate implementation for practical of the proposedimplementation QCN. By of equipping the proposed every QCN. node By in equippin the proposedg every QCN node by multiplein the proposed qubit patches, QCN by in multiple principle, wequbit can simultaneouslypatches, in principle, perform we quantum can simultaneously networking andperform quantum quantum distributed networking computing. and quantum Instead of wavelength-divisiondistributed computing. multiplexing Instead of (WDM),wavelength-division the multicore multiplexing fiber can also (WDM), be used the to multicore connect the fiber logical can qubitsalso be [16 used]. The to proposedconnect the QCN logical does qubits not require[16]. The the proposed trusted nodeQCN assumption,does not require but itthe is assumedtrusted node that Eveassumption, does not havebut it access is assumed to SDN that controller. Eve does not have access to SDN controller.

WDM MUX WDM SMF DEMUX WDM WDM

FigureFigure 4. 4. SimpliﬁedSimplified description of of connecting connecting two two logical logical qubits qubits from from two two neighboring neighboring nodes nodes by byd dwavelengths.wavelengths.

Entropy 2020, 22, x FOR PEER REVIEW 5 of 10

| | q2 q2

| Entropy 2020, 22, 1059 | | 5 of 10 | q1 q3 q1 q3 Entropy 2020, 22, x FOR PEER REVIEW 5 of 10

| | q2 q2 | | q4 q4

| | |  |  Figure 5.q1 Illustrative four-node ring (left) andq3 four-nodeq1 mesh (right) quantum communicationq3 networks.

4. Extending the Distance between | the Nodes in the Proposed SC-Based | QCN q4 q4

To extend the transmission distance between neighboring nodes in QCN, we propose to use the Figure 5.5. IllustrativeIllustrative four-node four-node ring ring (left ()left and) four-nodeand four-node mesh (rightmesh) quantum(right) quantum communication communicationnetworks. quantum errornetworks. correction (QEC)-based repeaters. So far, QEC-based repeaters are based on two- dimensional4. Extending QE-based the Distancerepeaters, between such theas the Nodes du inal-containing the Proposed Calderbank–Shor–Steane SC-Based QCN (CSS)-codes- based repeaters4. ExtendingTo extend [17] the theand Distance transmission surface-codes-based between distance the Nodes between inrepeaters the neighboring Proposed [18]. nodes SC-Based Unfortunately, in QCN, QCN we propose dual-containing to use the CSS codes arequantum essentiallyTo extend error correction the girth-4 transmission (QEC)-based quantum distance repeaters low-densit between. So far, ne QEC-basedyighboring parity-check repeaters nodes in are(LDPC)QCN, based we on proposecodes two-dimensional withto use thepoor error correctionquantumQE-based performance error repeaters, correction [19]. such On(QEC)-based asthe the other dual-containing repeaters hand,. Sothe far, surface Calderbank–Shor–Steane QEC-based codes repeaters proposed are (CSS)-codes-based in based [18] onintroduce two- large dimensionalrepeaters [17 QE-based] and surface-codes-based repeaters, such as repeatersthe dual-containing [18]. Unfortunately, Calderbank–Shor–Steane dual-containing (CSS)-codes- CSS codes latency and are not compatible with the QCN proposed in the previous section. Here, we propose a basedare essentially repeaters girth-4 [17] and quantum surface-codes-based low-density parity-check repeaters [18]. (LDPC) Unfortunately, codes with poordual-containing error correction CSS differentcodesperformance approach are essentially to [19 interpret]. On thegirth-4 other an intermediatequantum hand, the low-densit surface no codesdey asparity-check proposedan SC patch in (LDPC) [18 and] introduce applycodes largewiththe patch latencypoor errordeformation and approachcorrectionare due not to compatible Litinskiperformance withand [19]. thethus On QCN extend the proposed other the hand, logi in thethecal previous surfacequbit tocodes section. two proposed spatially Here, wein separated[18] propose introduce a dipatches,ff largeerent which is illustratedlatencyapproach in and Figure to are interpret not 6. compatible In an this intermediate example, with the node QCNthree as anproposed wavelengths SC patch in and the applyprevious are needed the patchsection. to deformation Here, interact we propose approachremote a patches. Once thedifferentdue logical to Litinski approach qubit and to thusis interpret extended extend an the intermediate logicalto the qubit intermediate no tode two as spatially an SC node, patch separated andwe apply patches,further the whichpatch perform deformation is illustrated product X⊗Z approachin Figure6 due. In thisto Litinski example, and three thus wavelengths extend the logi arecal needed qubit to to interact two spatially remote separated patches. Once patches, the logicalwhich measurements to entangle the logical qubits q1 and q2. This approach is applicable to several isqubit illustrated is extended in Figure to the 6. intermediate In this example, node, wethree further wavelengths perform productare neededX Ztomeasurements interact remote to entanglepatches. ⊗ intermediateOncethe logical nodes,the logical qubits thus qqubit1 andoffering isq2 .extended This the approach potential to the is intermediate applicable to significantly to node, several weextend intermediate further the perform distance nodes, product thus between off eringX⊗Z any two desired nodesmeasurementsthe potential in the to QCN.to significantly entangle the extend logical the distancequbits q between1 and q2 any. This two approach desired nodes is applicable in the QCN. to several intermediate nodes, thus offering the potential to significantly extend the distance between any two desired nodes in the QCN.

Intermediate patch SMF Intermediate SMF SMF patch |q’ SMF |q  | | 2 q’ q2

| | q1q1

Logical qubit Logical qubit with extended edge with extended edge Figure 6. Extending the distance between two nodes in a quantum communication network (QCN) Figure 6. Extending the distance between two nodes in a quantum communication network (QCN) by Figure 6.by Extending creating the thelogical distance qubit spanning between two two spatially nodes separated in a quantum surface code communication (SC) patches. network (QCN) creating the logical qubit spanning two spatially separated surface code (SC) patches. by creating the logical qubit spanning two spatially separated surface code (SC) patches. 5. Illustrative Illustrative Numerical Numerical Results Results 5. IllustrativeAlthough Numerical the the channel Results loss dominates the perfor performancemance of of quantum repeaters, there there will be quantum errors associated with each stage, whic whichh can be represented by using the quantum channel Althoughmodel provided the channel in Figure loss 7, wheredominates X and Z the quantum perfor errorsmance occur of wi quantumth the same repeaters, probability pthere. The will be quantumcorresponding errors associated Kraus represen with eachtation stage, [1] is givenwhic by:h can be represented by using the quantum channel model provided in Figure 7, whereρξρ==−+ X() and Z ( quantum ) ρ errors ρ +occur ρ with the same probability p. The f 12ppXXpZZ . (1) corresponding Kraus representation [1] is given by: ρξρ==−+() ( ) ρ ρ + ρ f 12ppXXpZZ . (1)

Entropy 2020, 22, 1059 6 of 10 model provided in Figure7, where X and Z quantum errors occur with the same probability p. The corresponding Kraus representation [1] is given by:

ρ f = ξ(ρ) = (1 2p)ρ + pX ρX + pZ ρZ. (1) Entropy 2020, 22, x FOR PEER REVIEW − 6 of 10

s=1−2p s=1−2p |ψ |ψ ρ ρ

p p

X|ψ Xρ X p p

Z|ψ Zρ Z

(a) (b)

FigureFigure 7.7. Quantum channel model underunder study:study: ( a) PauliPauli operatoroperator descriptiondescription andand ((bb)) densitydensity operatoroperator description.description.

LetLet usus consider consider the the BB84 BB84 protocol protocol by by employing employing the the approach approach introduced introduced in previous in previous section. section. The correspondingThe corresponding secret-key secret-key rate afterrate afterN sections N sections willbe: will be:

N () () =−{ () } N −()(Z)ZX − (X()) SKRSKR =1max1,0,[1 PP E(E)] TT max ()1 h h22q qNeNfeh f hq q , 0 ,(2) (2) − − 2 N − 2 N ()XZ () where fe denotes the error correction inefficiency (fe ≥ 1),(X )qqNN(Z) denotes the quantum bit-error where fe denotes the error correction inefficiency (fe 1), q q denotes the quantum bit-error rate ≥ N N (QBER)rate (QBER) in the inX -basisthe X-basis (Z-basis) (Z-basis) after N afterstages, N stages,T represents T represents the single the linksingle transmissivity, link transmissivity, and h (x and) is 2 (Z) ()Z 2 ()=− ()()() − − − () theh (x binary) is the entropy binary entropy function functionh2(x) = hx22x log ( xxlog) (1 xx) 1log x log(1 2 1x). x The. The term termh2 qhq2 Nrepresentsrepresents the − 2 − − 2 − N amount of information Eve was able to learn during the raw key transmission, which can be removed the amount of information Eve was able to learn during the raw key transmission, which can be (X) () from the final key during the privacy amplification phase. The term feh2 q representsX the amount removed from the final key during the privacy amplification phase. The Nterm fheN2 () q represents of information revealed to Eve during the information reconciliation stage. The dark counts, device the amount of information revealed to Eve during the information reconciliation stage. The dark imperfections, and errors introduced by Eve are all contributed to the Eve and included in transition counts, device imperfections, and errors introduced by Eve are all contributed to the Eve and probability p. The QBER after N stages can be estimated by: included in transition probability p. The QBER after N stages can be estimated by: N 1 −s N qN = 1− s , s = 1 2p. (3) qsp==−2 ,12.− (3) N 2 The probability of the syndrome decoding error is bounded by [1]: The probability of the syndrome decoding error is bounded by [1]: 2 Xd 2 ! d2 d2 j d2 j P(E) d (1 s)j s 2− , s = 1 2p, (4) ()≤ ≤−=−j (− ) dj− − P Essspj = (d 1)/2 +11,12, (4) b=−− c + jd()1/2 1j So, [1 P(E)]T represents the success probability for the single stage. The total success probability − can beSo, estimated [1−P(E)]T byrepresents {[1 P( Ethe)]T success}N and probability is illustrated forin the Figure single8 stage.by setting The total the successX (Z) qubitprobability error can be estimated by2 {[1−P(E)]T }N and is illustrated in Figure 8 by setting the X (Z) qubit error probability to p = 10− and transmissivity to T = 1, for different d d surface codes. probability to p = 10−2 and transmissivity to T = 1, for different d ×× d surface codes.

Entropy 2020, 22, x FOR PEER REVIEW 7 of 10

100 s, tot s, P

10-1 Entropy 2020, 22, 1059 7 of 10 Entropy 2020, 22, x FOR PEER REVIEW 7 of 10

100

10-2 s, tot s,

P -2 Total success probability, probability, success Total p=10 : d=3 10-1 d=7 d=9

10-3 0 50 100 150 200 250 300

-2 Number of stages, N 10 -2 Total success probability, probability, success Total p=10 : Figure 8. Total success probability defined as (1−P(E))N, where N is the number of stages, when the d d=3 × d surface code is used, and syndrome d=7 decoding is applied. d=9 The numerical results10 for-3 secret-key rate (SKR) for different transmissivities T (assuming that fe = 1) vs. the number of stages0 N are summarized 50 100 in Figures 150 9 200 and 10. 250 The channel 300 transmittance in Number of stages, N Figure 9 is set to T = 0.95, while in Figure 10 it is set to T = 0.85. The qubit error transition probability Figure 8. Total success probability defined as (1 P(E))N, where N is the number of stages, when the p is used Figureas a parameter.8. Total success In probability both figures, defined the as (17−− P×( E7)) Nsurface, where Ncode is the isnumber used. of Given stages, whenthat the d effective d d surface code is used, and syndrome decoding is applied. transmission× ×d surface distance code of is used,the fiber and syndrome is given decodingby [20]: is applied. The numerical results for secret-key rate (SKR)−αL for different transmissivities T (assuming that The numerical results for secret-key rate1 (SKR)−e for different transmissivities T (assuming that fe f = 1) vs. the number of stages N are summarized=≈ in Figuresα9 and 10. The channel transmittance in =e 1) vs. the number of stages N are summarizedLeff in Figures1/ 9 , and 10. The channel transmittance in (5) Figure9 is set to T = 0.95, while in Figure 10 it isα set to T = 0.85. The qubit error transition probability Figure 9 is set to T = 0.95, while in Figure 10 it is set to T = 0.85. The qubit error transition probability p is used as a parameter. In both figures, the 7 7 surface code is used. Given that the effective Forp is ultra-lowused as a parameter.loss fiber introducedIn both figures, in [21]the 7with ×× 7 surfaceattenuation code iscoefficient used. Given α =that 0.1419 the effectivedB/km, we transmission distance of the fiber is given by [20]: obtaintransmission that Leff = 30.606 distance km. of Thethe fiber total is transmission given by [20]: length can be now estimated by: 1 e−ααLL LLNLT==1−−e − ln1/ .α , (5) Lefftot=≈α eff ≈1/α , (5) (6) eff α

For ultra-low loss 10fiber0 introduced in [21] with attenuation coefficient α = 0.1419 dB/km, we obtain that Leff = 30.606 km. The total transmission length can be now estimated by: d=7, T=0.95: -3 -1 = 10 LNLTtot eff ln . p=10 (6) -4 SKR p=5x10 p=10-4 -2 10 100

d=7, T=0.95: -3 -1 10-3 10 p=10 -4 SKR p=5x10 p=10-4 10-2 10-4

-3 Normalized secret-key rate, rate, secret-key Normalized 10 10-5

10-4 10-6

Normalized secret-key rate, rate, secret-key Normalized 0 50 100 150 200 250 -5 10 Number of stages, N

Figure 9. Normalized secret-key rate (SKR) vs. the number of stages N assuming that the 7 7 surface Figure 9. Normalized secret-key10-6 rate (SKR) vs. the number of stages N assuming that the× 7 × 7 surface code is used, and single link0 channel transmittance50 100 is T = 0.95. 150 200 250 code is used, and single link channel transmittance is T = 0.95. Number of stages, N

Figure 9. Normalized secret-key rate (SKR) vs. the number of stages N assuming that the 7 × 7 surface code is used, and single link channel transmittance is T = 0.95.

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100 -1 10 d=7, T=0.85: -2 10 p=10-3 -3 10 -4 SKR p=5x10 -4 10 p=10-4 10-5 10-6 10-7 -8 10 10-9 10-10 10-11

Normalized secret-key rate, rate, secret-key Normalized 10-12 10-13 10-14 10-15 0 40 80 120 160 200 Number of stages, N

Figure 10. Normalized SKR vs. number of stages N assuming that 7 7 surface code is used, and Figure 10. Normalized SKR vs. number of stages N assuming that 7× × 7 surface code is used, and single link channel transmittance is T = 0.85. single link channel transmittance is T = 0.85. For ultra-low loss fiber introduced in [21] with attenuation coefficient α = 0.1419 dB/km, we obtain For T = 0.95, by setting the qubit error probability to p = 10−4, we can see from Figure 9 that the that Leff = 30.606 km. The total transmission length can be now estimated by: achievable total transmission distance for normalized SKR of 10−6 is Ltot = 252 × 30.606 × |ln0.95| = 395.61 km. On the other hand, for T = 0.85, by setting the qubit error probability to p = 10−4, we can Ltot = NLeff ln T . (6) see from Figure 10 that the achievable total transmission| distance| for normalized SKR of 10−15 (typical 4 for discreteFor T = variable0.95, by QKD setting schemes the qubit [2]) is error Ltot = probability208 × 30.606 to × |ln0.85|p = 10− =, we1034.61 can km, see and from this Figure results9 6 thatis comparable the achievable to the total recently transmission proposed distance hybrid forQKD-postquantum normalized SKR cryptography of 10− is Ltot scheme= 252 [22,23].30.606 By × ×4 |ln0.95employing| = 395.61 higher km. complexity On the other quan hand,tum for sum-productT = 0.85, by algorithm setting the [1] qubit in each error stage, probability instead to ofp = simple10− , 15 wesyndrome can see fromdecoding, Figure the 10 thattotal thetransmission achievable dist totalance transmission well beyond distance 1000 forkm normalizedcan be achieved. SKR of Typical 10− (typicalQKD transmission for discrete variabledistances QKD are sign schemesificantly [2]) shorter, is Ltot = even208 when30.606 the most|ln0.85 advanced| = 1034.61 twin-field km, and QKD this × × resultsschemes is comparable are used [24]. to the recently proposed hybrid QKD-postquantum cryptography scheme [22,23]. By employing higher complexity quantum sum-product algorithm [1] in each stage, instead of simple syndrome6. Operating decoding, the Proposed the total QCN transmission by SDN Control distance well beyond 1000 km can be achieved. Typical QKDThe transmission SDN has distancesbeen introduced are significantly to separate shorter, the control even when plane the and most data advanced plane, manage twin-field network QKD schemesservices arethrough used [24abstraction]. of higher-level functionality, and implement new applications and algorithms efficiently [25,26]. It has already been studied to enable the coexistence of classical and 6. Operating the Proposed QCN by SDN Control quantum communication channels [27]. To enhance the security of the software-defined optical networks,The SDN authors has beenin [28] introduced proposed a to four-layer separate thearchitecture control plane composed and data of: application, plane, manage control, network QKD, servicesand data through layers. abstractionThe SDN-based of higher-level QCN architecture functionality, compatible and implement with the proposed new applications QCN should and algorithmscontain three effi layersciently only—namely, [25,26]. It has alreadyapplication been layer, studied control to enable layer, theand coexistence QCN layer. of Users classical will andsend quantumtheir requests communication from the application channels layer [27]. with To enhancethe help of the northbound security of interface the software-defined to the SDN controller. optical networks,The SDN authorscontroller in [will28] proposed allocate the a four-layer QCN resources architecture with composedthe help of of: its application, global map control, through QKD, the andsouthbound data layers. interface. The SDN-based The QCN QCN layer architecture can be comp compatibleosed of DWDM with the links proposed and QCN QCN nodes. should Each contain QCN threenode layersshould only—namely, contain quantum application transceivers, layer, integrated control layer, on the and same QCN chip, layer. together Users with will a sendd × d theirarray requestsof physical from qubits. the application Any two nodes layer within QCN the can help comm of northboundunicate through interface either to the a dedicated SDN controller. SMF link The or SDNby d controllerwavelength will channels. allocate theTo enable QCN resources so, we could with employ the help our of itsrecently global proposed map through bidirectional the southbound optical interface.space switch The [29], QCN to reconfigure layer can be the composed QCN. Other of DWDM alternative links optical and QCNswitches nodes. can be Each used QCN as well. node In shouldaddition contain to conventional quantum modules, transceivers, the application integrated layer on the should same also chip, have together modules with to aprovided d array security of × physicalmanagement qubits. services. Any two On nodes the other in QCN hand, can the communicate control layer, through in addition either to a controlling dedicated SMFthe QCN link orlayer, by dshouldwavelength provide channels. allocation To of enable resources so, we as couldwell as employ provide our services recently for proposed multiple bidirectionalapplications. opticalTo deal spacewith switchtime-varying [29], to channel reconfigure conditions the QCN. over Other he alternativeterogeneous optical links, switcheswe can canadapt be usedthe aschannel well. configuration based on both application requirements and link conditions.

Entropy 2020, 22, 1059 9 of 10

In addition to conventional modules, the application layer should also have modules to provide security management services. On the other hand, the control layer, in addition to controlling the QCN layer, should provide allocation of resources as well as provide services for multiple applications. To deal with time-varying channel conditions over heterogeneous links, we can adapt the channel conﬁguration based on both application requirements and link conditions.

7. Concluding Remarks To enable the next generation of quantum communication networks, we have proposed to employ the surface-codes-based patches as quantum nodes. We have described how to simultaneously entangle multiple quantum nodes in any quantum network topology by employing the SCs. We have also described how to extend the transmission distance between any two quantum nodes to beyond 1000 km. Finally, we have described how to operate the proposed QCN by employing the SDN concept. The trapped ion technology is an excellent candidate to be used as an enabling technology to implement SC-based QCNs. One important issue will be to implement a portable, rack-mounted ion-trap-based quantum interface, and some progress has already been made by researchers from Duke University in collaboration with ColdQuanta, Inc [30]. To improve the eﬃciency of the proposed QCNs, the high-dimensional SCs should be employed. By employing high-dimensional-based quantum error correction, we can achieve error correction capability comparable to 2D but with signiﬁcantly shorter codeword lengths as discussed in [31]. An alternative approach to the proposed QCN will be a recently introduced cluster-state-based QCN [32].

Funding: This research received no external funding. Conﬂicts of Interest: The author declares no conﬂict of interest.

References

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