Entanglement generation in a quantum network at distance-independent rate
Ashlesha Patil, Mihir Pant, Dirk Englund, Don Towsley, Saikat Guha
EEntanglement generation in a quantum network at distance-independent rate
Ashlesha Patil, ∗ Mihir Pant, Dirk Englund, Don Towsley, and Saikat Guha College of Optical Sciences, University of Arizona,1630 East University Boulevard, Tucson, AZ 85721. Massachusetts Institute of Technology, 32 Vassar Street, Cambridge MA 02139 College of Information and Computer Sciences,University of Massachusetts, Amherst, MA 01002.
We develop a protocol for entanglement generation in the quantum internet that allows a repeaternode to use n -qubit Greenberger-Horne-Zeilinger (GHZ) projective measurements that can fuse n successfully-entangled links , i.e., two-qubit entangled Bell pairs shared across n network edges,incident at that node. Implementing n -fusion, for n ≥
3, is in principle no harder than 2-fusions(Bell-basis measurements) in solid-state qubit memories. If we allow even 3-fusions at the nodes,we find—by developing a connection to a modified version of the site-bond percolation problem—that despite lossy (hence probabilistic) link-level entanglement generation, and probabilistic successof the fusion measurements at nodes, one can generate entanglement between end parties Aliceand Bob at a rate that stays constant as the distance between them increases. We prove thatthis powerful network property is not possible to attain with any (non error-corrected) quantumnetworking protocol built with Bell measurements alone. We also design a two-party quantum keydistribution protocol that converts the entangled states shared between two nodes into a sharedsecret, at a key generation rate that is independent of the distance between the two parties.
I. INTRODUCTION
The Quantum Internet will provide the service of gen-erating shared entanglement of different kinds, betweendistant end-user pairs and groups, on demand, and athigh speeds. The entanglement generation rate betweentwo nodes decays linearly with the transmissivity η ofthe channel connecting them, which turns into an expo-nentially rate-vs.-distance decay over optical fiber, since η = e − αL for a length- L fiber [1]. The maximum attain-able rate is − log(1 − η ) ≈ . η , for η (cid:28)
1, ebits (pureBell states shared between two parties) per transmittedoptical mode [2].
Quantum repeaters need to be insertedalong the length of the optical channel in order to cir-cumvent this rate limit [3–5]. There is a wide variety ofrepeater and router protocols being researched, most ofwhich use Bell state measurements (BSMs) as a build-ing block. BSM is a two-qubit destructive measurementthat can fuse two entangled links (each entangled link be-ing a two-qubit Bell state shared across a network edge)incident at a node, into one entangled link over a two-hop path. For a linear chain of repeater nodes, whereeach repeater is equipped with quantum memories andemploys BSMs and switches, the entanglement rate out-performs what can be attained with a direct connectionconnecting the communicating end parties, but the ratestill decays exponentially with distance, i.e., R ∼ e − sαL ,with s <
1. Two communicating parties Alice and Bobsituated in a quantum network can take advantage ofmulti-path routing—with the same repeater-node capa-bilities as above but able to dynamically switch BSMapplications from one time slot to the next across lo- ∗ [email protected] cal qubit memories entangled with different neighbor-ing nodes, based on link-state knowledge of neighboringlinks—to attain an entanglement rate that exceeds whatis possible along a pre-determined linear repeater chainalong a single shortest path connecting Alice and Bob [6].However, the rate still decays exponentially with the dis-tance between the users. Assuming global link state in-formation is available, i.e., every node is instantly awareof the success-failure outcomes of entanglement attemptsacross all network edges in each time slot, further im-proves the multi-path rate advantage. However, evenwith such an unrealistic assumption, as long as singleBSM attempts succeed with probability less than 1, theend-to-end rate still decays exponentially with the dis-tance between the users [6].Various genres of quantum repeaters and associatederror-correction codes are under investigation [4]. For thepurposes of our paper, we will consider the following sim-ple model, and show a surprising result—that the end-to-end entanglement rate remains constant with increasingdistance when network nodes are able to measure morethan two qubits in a joint projective measurement. Ineach time slot, each network edge attempts to establishan entangled link: a Bell state of two qubits, each resid-ing in a quantum memory at nodes on either end of thelink. In every time slot, each link is established success-fully, i.i.d., with probability p proportional to the trans-missivity of the optical link. Subsequently, each node,based on local link-state information (i.e., which neigh-boring links succeeded in that time slot), and knowledgeof the location of the communicating parties Alice andBob, decides which pairs of successful links to fuse. Thetwo qubits that are fused with a BSM at a node are de-stroyed in the measurement process, while creating anentangled (Bell) state among the two qubits at the farends of the two links, thus creating a 2-hop entangled a r X i v : . [ qu a n t - ph ] M a y link traversing two network edges. A fusion attempt suc-ceeds with probability q . It was shown recently that witha simple distance-vector fusion rule, the achievable en-tanglement generation rate exceeds what is possible witha fusion schedule along a pre-determined single shortestpath connecting Alice and Bob [6]. Despite this rate ad-vantage from multipath entanglement routing, the ratedecays exponentially with the distance L between Aliceand Bob, for any value of p and q less than 1.In this paper, we develop a protocol that allows nodesto use n -qubit Greenberger-Horne-Zeilinger (GHZ) pro-jective measurements, i.e., n -fusions, that can fuse n suc-cessful links at a node. When n = 2, the nodes im-plement a two-qubit BSM. For n = 1, the nodes im-plement a single-qubit Pauli measurement. Implement-ing n -fusion, for n ≥ n -fusionattempts as q . We report a surprising and potentiallyhigh-impact result: if we allow even 3-fusions at the re-peater nodes, there is a non-trivial regime of ( p, q ) whereour protocol generates entanglement at a rate that staysconstant with L . We prove this is not possible with anyquantum network protocol that only uses Bell measure-ments (see Appendix C). Our protocol only uses local linkstate knowledge, but requires a single-round of classicalcommunications that adds to the latency of the protocol(however, does not affect the rate).Finally, we develop a quantum key distribution (QKD)protocol that allows a pair of users Alice and Bob, situ-ated in a network, to sift (two-party) secret keys start-ing from a pre-shared m + n -qubit Greenberger-Horne-Zeilinger (GHZ) state, m qubits of which are held byAlice and n by Bob. It is an extension of the BBM92protocol [9], a simplification of the E’91 protocol [10],which relies on shared Bell states and measurements byAlice and Bob in a matching pair of bases. Using ourabove described quantum network protocol that employs n -GHZ measurements at nodes, we thus have devised aQKD protocol over a quantum network whose secret-keygeneration rate is constant with increasing distance be-tween communicating parties, despite lossy channel seg-ments between nodes and probabilistic successes of the n -GHZ measurements at nodes.In Section II, we discuss the elementary multi-qubitprojective measurements used in our protocol. SectionIII describes the entanglement distribution protocol andits improved variations. We also map the problem ofdistributing entanglement over a quantum network toa mixed percolation problem studied in classical sta-tistical mechanics. We discuss the origin of distance-independence of the shared entanglement rate, along withnumerical calculations of the rate, in Section IV. We con-clude in Section VII by summarizing the results and dis-cussing open questions that can be studied as immediateextensions and applications of the proposed protocol. II. FUSING ENTANGLEMENT USINGGHZ-STATE PROJECTIONS
We use entanglement-swapping operations, namely,Bell State Measurements (BSMs) and n -qubit GHZ pro-jections at network nodes, for routing entanglement in aquantum network. An n -qubit GHZ projection is a vonNeumann projective measurement, that projects the n measured qubits into one of (the 2 n ) mutually-orthogonal n -qubit GHZ states, thereby producing a (random) n -bitclassical measurement result. The well-known BSM is a2-qubit GHZ projection. Entanglement swapping at aquantum (repeater) node extends the range of entangle-ment by fusing two Bell states shared across two adjacentedges of the network.We refer to n -qubit stabilizer states [11]with stabilizer generators of the form { ( − g X X . . . X n , ( − g Z Z , ( − g Z Z , . . . , ( − g n Z Z n } , g i ∈ { , } as n -GHZs, which includesthe case of n = 2 i.e., Bell states. X i and Z i aresingle-qubit Pauli operators for the i -th qubit. Weuse the (unconventional) notation of an n -star graphto represent an n -GHZ. This is not a star-topologycommonly-known graph state [12]. Furthermore, werefer to a projective measurement onto the n -GHZ basisas a ( n -qubit) fusion . The size of an n -qubit GHZ stateis n . An n -GHZ projection on a set of GHZ states ofsize m , m , ....m n results in a single GHZ state of size (cid:80) ni =1 m i − n , obtained by removing the qubits thatare fused from the original set of qubits and coalescingall the unmeasured qubits into a single GHZ state, asshown in Fig. 1. FIG. 1. 3-GHZ projection on three 3-GHZ states. Fusionsuccess creates a 6-GHZ state and failure performs X-basismeasurements on the fused qubits, resulting in three 2-qubitBell pairs.
Depending upon the choice of quantum memory andprocessor hardware at the quantum repeater node, fusionoperations may be probabilistic. We model the result of afailed fusion attempt as performing an X -basis measure-ment on all qubits that were used as part of the fusion,as shown in Fig. 1. Measuring a qubit of an n -GHZ statein the Pauli- X basis results in a ( n − FIG. 2. Measuring a qubit in X-basis removes it from the n -GHZ state. III. THE PROTOCOL
In this paper, we study two kinds of quantum networks:a two-dimensional square-grid, and a configuration-model random graph with a given node-degree distri-bution [13]. First, let us consider a square-grid graph.Each node is a quantum repeater (blue circles in Fig. 3a)with four quantum memories (black dots in Fig. 3a) as-sociated with each neighboring edge. Each repeater iseither a “consumer” of entanglement i.e., Alice and Bob,or a “helper” i.e., they help to establish entanglementbetween the consumer nodes. In the first time step, eachnetwork edge attempts to establish an entangled link:a Bell state of two qubits, each residing in a quantummemory at nodes on either end of the link. Each link isestablished successfully, i.i.d., with probability p , whichis proportional to the transmissivity of the respective op-tical link [1, 2]. The repeater nodes have only local link-state knowledge, i.e., a repeater knows the success-failureoutcomes at each time slot of its own link generation at-tempts (across its neighboring edges). Each repeater isalso aware whether it is a consumer or a helper node,knows the overall network topology, and the locationof the consumer nodes (if it is a helper node). In thenext time step, all helper nodes that have more than onesuccessfully-created link, attempt fusions on the qubitsheld in their respective quantum memories tied to a sub-set of those successful links. The fusion success proba-bility is taken to be q . A successful fusion at a repeatercreates a Bell pair or a GHZ state shared between a sub-set of its neighbours. If a helper node has only one linksuccess in a time slot, it performs an X-basis measure-ment on the corresponding locally-held qubit, which un-entangles and dissociates that qubit from any others inthe network.We consider three protocols for the square-grid net-work which differ in the operations available at repeaternodes, and allow for different entanglement generationrates: (1) the 4-GHZ protocol, (2) the 3-GHZ protocoland (3) the 3-GHZ brickwork protocolIn the 4-GHZ protocol, a repeater performs a fusion onall locally-held qubits successfully entanlged with neigh-boring nodes at each time step. Hence, the largest mea-surement in such a protocol is a 4-GHZ measurement,which is done when all 4 links are successfully created. Ina time step when only 3 or 2 links are successful, a 3-GHZmeasurement or a 2-GHZ (i.e., Bell) measurement is per-formed. If only one link is successful, the corresponding qubit is measured in the X-basis. In the 3-GHZ proto-col, the maximum size of the GHZ projection allowed islimited to 3, which may be imposed due to hardware con-straints. If the number of successful neighboring links ofa helper node is less than or equal to three, the repeaterperforms a fusion between the corresponding qubits, i.e.,behaves the same as the 4-GHZ protocol. However, iffour neighboring links are successful, the repeater ran-domly chooses three qubits and performs a fusion onthem. It performs an X measurement on the fourth re-maining qubit if this happens. Every helper node sendsits local link state knowledge, fusion success outcomes,and X-basis measurement outcomes to the consumersAlice and Bob using a classical communication overlaychannel. This classical communication time determinesthe overall latency of the entanglement generation proto-col, but the entanglement rate is determined by the rateat which each entangled link is attempted across eachnetwork edge.It is important to note that all Bell state measure-ments, GHZ projections and Pauli X-basis measurementsacross the entire network are performed during the sametime step. This is allowed because all of these operationsand measurements commute with one another. At theend of this step, the consumers obtain (potentially morethan one) shared n -GHZ state(s) with a probability thatdepends on the network topology, p , q , and which of thetwo protocols described above is used.We discuss the rules for the Brickwork protocol in sec-tion V B, which instead of being fully randomized asabove, imposes some additional structure on which fu-sions to attempt, and can outperform the 3-GHZ proto-col in certain regimes.We also study the n -GHZ protocol for a random graphnetwork, with an arbitrary node degree distribution p k .Here, p k is the probability that a randomly chosen nodehas degree k . In other words, it is the probability thata randomly chosen quantum repeater node has k edges.In an n -GHZ protocol, repeaters are allowed to performup to n -GHZ projections for fusions. For the n -GHZprotocol over a random network, if a degree k helpernode has l successful links in a time slot such that l ≤ n ,it performs an l -GHZ fusion. If l > n , it performs an n -GHZ fusion on the n qubits corresponding to n randomlychosen links (of the l ). The remaining steps are sameas the 3- and 4-GHZ protocols described above for thesquare grid.Immediately after the time slot when all helper nodesperform their measurements and sends, via unicast com-munications, the requisite classical communication to theconsumer nodes, the network edges re-attempt entangle-ment generation in the next time step, and the helpernodes again make their measurements based on the pro-tocol described above using local link state information,until the end of the protocol’s duration. The lengthof each time step determines the rate of the protocol,whereas the classical communication time determines thelatency. Consumers hold on to all their qubits for thetime required to receive the classical information regard-ing the results of measurements made during a specifictime slot from every helper node in the network. Theyuse the local link state knowledge from the helpers to de-termine which one of their qubits (from the correspond-ing time slot) are part of a shared entangled state heldbetween Alice and Bob. In each time slot, Alice and Bobgenerate 0, 1, 2, 3 or 4 shared GHZ states. Each of thoseshared GHZ states could have more than 2 qubits. Forexample, Alice and Bob could generate one 3-qubit GHZstate two of whose qubits are held by Alice and one byBob, and one 2-qubit GHZ (i.e., Bell) state one qubit ofwhich is with Alice and the other with Bob.At this point, Alice and Bob can use their sharedentangled state for a quantum information processingprotocol, e.g., QKD, entanglement enhanced sensing, ordistributed quantum computing implemented by a tele-ported gate. If the protocol requires a particular n -GHZstate as a resource, it is always possible for Alice and Bobto correct the state by applying local unitary operations,or for some protocols such as QKD by correcting theoutcome of the protocol during classical post-processingusing the measurement results received from the helpers. IV. ENTANGLEMENT RATES
We calculate the shared entanglement generation ratefor the square-grid topology of the quantum network un-der three different fusion rules (Fig. 3) as a functionof link and fusion success probabilities ( p, q ) and the dis-tance between the consumers. We define rate as expectednumber of n -GHZs (including Bell pairs) shared betweenthe consumers per cycle. We can think of the quantumnetwork shown in Fig. 3(a) as a graph G ( V, E ) suchthat each quantum memory is a vertex v ∈ V , and eachlink e ∈ E is created with probability p (a successfullycreated link Bell pair). Fusion operations are then exe-cuted at vertices with at least two neighbors creating anew graph G (cid:48) ( V (cid:48) , E (cid:48) ). In this graph, v ∈ V (cid:48) is a quantummemory that has undergone a successful fusion operation. G (cid:48) ( V (cid:48) , E (cid:48) ) has additional edges that represent the edgescreated due to successful fusions between vertices. In ad-dition, consumers Alice and Bob have four vertices each.They share an entangled state at the end of the fusionstage, if they belong to the same connected componentof graph G (cid:48) . The number of n -GHZ states shared be-tween Alice and Bob equals the number of disconnectedsub-graphs of G (cid:48) containing at least one vertex each fromboth Alice and Bob. Hence, the maximum value the ratecan take would be 4 n -GHZ/cycle. In the following sec-tions, we compute and compare the shared entanglementgeneration rates for the different protocols over square-grid and random networks. We refer to the protocol inwhich repeaters can perform up to n -qubit GHZ projec-tions as n -GHZ protocol. (a)(b)(c) FIG. 3. The schematic of the quantum network during var-ious stages of the random 3-GHZ protocol. (a) The quan-tum network after link generation. The successfully generatedlinks are shown using black solid lines. The green trianglesand rectangles denote the successful fusions. Their red coun-terparts represent the failed fusion attempts. (b) The threeGHZ states (green, orange, and magenta lines) generated afterperforming fusions at the repeaters. The quantum memoriesmarked in red perform X-basis measurements on the qubitsheld in them. (c) The GHZ state shared between Alice andBob (orange, and magenta lines).
A. Perfect repeaters
We first study the case where repeaters always success-fully perform fusions, i.e., q = 1. In the n -GHZ protocolover a certain network topology, calculating the probabil-ity that the consumers are a part of the same connectedcomponent of G (cid:48) ( V (cid:48) , E (cid:48) ) translates to a bond percola-tion problem on the underlying network topology [14].The link generation probability p is equivalent to thebond occupation probability in the percolation problem.Percolation is a phase transition phenomenon such thatwhen p < p c (sub-critical regime), where p c is a thresh-old that depends on the lattice geometry, the probabilitythat two randomly chosen sites are connected decays ex-ponentially with distance between the two sites. On theother hand, if p > p c (super-critical regime), this prob-ability remains constant with the distance. This impliesthe probability that the consumers belong to the sameconnected component doesn’t vary with the distance be-tween the consumers in the super-critical regime. Thisresult forms the basis of our protocols to achieve distance-independent shared entanglement generation rates. Thecorrelation length changes from a finite value in the sub-critical regime to infinity in the super-critical regime.The phase transition in the correlation length leads toa sharp transition in the rate, similar to a percolationplot, at p = 0 .
5, which is the bond percolation threshold p c of the square lattice as shown in Fig. 4(a). Simi-larly, the 3-GHZ protocol over the square-grid network,in which the repeaters can’t perform 4-GHZ projections,the protocol becomes a different bond percolation prob-lem on the square lattice. For this problem, the bondpercolation threshold p c ≈ .
53 (Fig. 4(b)). For both ofthese fusion rules at the repeaters, when p > p c , the ratedoesn’t decay exponentially with the distance betweenthe consumers, but remains constant instead. B. Imperfect repeaters
Depending on the quantum hardware used at the re-peaters, fusion operations can be probabilistic [15]. Inthis paper, if a repeater fails to perform fusion, it is equiv-alent to performing X-basis measurements on the qubitsinvolved in the fusion. Calculating the probability thata pair of users end up with shared entanglement whenboth link generation and fusions are probabilistic, now,becomes a site-bond percolation problem [16] over theunderlying network topology lattice (e.g. the square lat-tice). Site-bond percolation is the generalized version of apercolation problem in which sites and bonds are presentwith probabilities q and p respectively. The boundarybetween the super- and sub- critical regimes becomes acurve in the ( p, q ) plane. For our protocol, the fusionsuccess probability at each repeater translates to the siteoccupation probability q . Here, we assume that all fu-sion operations succeed with the same probability q . Weanalytically calculate the site-bond region for an n -GHZ (a)(b) FIG. 4. Entanglement rate over the square grid networkassuming q = 1 for, (a) the 4-GHZ protocol (b) the ran-dom 3-GHZ protocol. We see that (a) above the threshold p > p c , the entanglement rate becomes independent of thedistance between communicating parties while it scales withthe Manhattan distance when p < p c ; and (b) the threshold p c is higher (0 .
53 versus 0 .
5) for the 3-GHZ protocol. Thethreshold p c for the 4-GHZ protocol is the standard bond-percolation threshold of the 2D square lattice. protocol over a random graph in Appendix A. Fig. 5(a)shows the site-bond region for the lattices formed afterthe fusion step in 3- and 4-GHZ projection protocols ona square-grid network, simulated using the Newman-Ziffmethod [17] and 3-GHZ protocol on a constant degree-4random graph network using the analytical formula. Thesite-bond curve gives the percolation thresholds ( p c , q c )of the underlying lattice. The probability that the twoconsumers are connected is distance-independent when p > p c and q > q c . Thus, the link generation and fu-sion success probabilities need to lie above the site-bondcurve to achieve distance-independent rate. To demon- (a)(b) FIG. 5. (a) Site-bond percolation regions for the percolationproblems corresponding to the 4-GHZ protocol over square-grid network, 3-GHZ protocol over square-grid and constantdegree-4 random graph networks. The curves represent thecritical regime of percolation. p and q need to lie above thecurves for distance independent rate. (b) Rate vs distance forpoints in three different regions of the site-bond curve markedin (a) for the square-grid network. The dashed and solid linescorrespond to 4- and random 3-GHZ protocols respectively. strate this, we plot the rate as a function of distance forthree pairs of ( p, q ) that lie in three different regions ofthe site bond curves of the 4- and 3- GHZ protocols inFig. 5(b). V. IMPROVED n -GHZ PROTOCOL We observe a curious turnaround in the site-bondcurves for the 3-GHZ and some n -GHZ protocols oversquare-grid and random networks, respectively. For the3-GHZ protocol, when two neighbouring repeaters havefour link successes each and they are limited to doing3-GHZ projections, they might choose to sacrifice differ-ent edges as the repeaters don’t communicate with each FIG. 6. Rate vs link success probability for the 3-GHZ pro-tocol over the square-grid network other to decide which links to choose to perform fusionon. This effect is even more pronounced when q < p regime, the rate starts decaying with p when q < n -GHZ protocol.In the following sections, we discuss three strategies toimprove the turnaround. A. Thinning the network
Let p ∗ be the link generation probability at which theturnaround occurs. The adversarial behaviour of the pro-tocol is observed only beyond p ∗ . We can get rid of theturnaround by randomly removing links in the high p regime. We modify the protocol such that when p > p ∗ ,each link is deleted with probability ( p − p ∗ ) /p . Thismakes the effective link generation probability p ∗ when p > p ∗ as shown in FIG. 5(a). B. The Brickwork network
The random selection of the links to fuse degrades therate when repeaters can fail. To overcome this issue, wepropose a deterministic link selection rule that doesn’tlet neighbouring repeaters make conflicting fusions. Con-sider the square-grid topology of the quantum repeaters.This network has two types of links - red and black. Bothred and black links have the same success probability p .Links are arranged such that the black links form a brick-work lattice. Each repeater has a maximum three blacklinks and one red link. In the fusion step of the protocol,a repeater uses the red link only if it has two or fewerblack links as shown in Fig. 7(b). This protocol is equiv-alent to percolation over a brickwork lattice with an extraoptional bond at each site. Hence, we observe in FIG. 8that the repeater success probability threshold is equalto the site percolation threshold of the brickwork lattice.And the link success probability is higher than the bondpercolation threshold of the brickwork lattice due to theadditional bond. This fixed selection rule gets rid of theadversarial nature of the previous protocol without hav-ing neighbouring repeaters communicate with each other.Fig. 7(c) shows the rate vs. link success probability ( p )curve doesn’t decay when the repeaters fail to performfusions ( q < n -GHZprotocol over a random network to make the protocolpartially deterministic, each node can have maximum n black edges and the rest are red edges. If the totalnumber of edges at a node is less that n , all of themare black. Each repeater (node) uses the red links forfusion only if it has less than n black links. We com-pare the site-bond regions for the 3-GHZ brickwork pro-tocol for various network topology with mean degree ≈ n as shown in Fig. 11.The analytical expression for the site-bond region of thisbrickwork-like model for random graphs is derived in Ap-pendix B. C. Dividing the network
As discussed earlier, the entanglement generation rateis proportional to the number of disconnected sub-graphs of the graph generated after fusion ( G (cid:48) ( V, E ))that are shared between the consumers. In the high( p, q ) regime, for the square-grid network, due to 3-/4-qubit projections, as the overall connectivity of G (cid:48) ( V, E )improves, its disconnected sub-graphs start merging to-gether. Hence, this framework fails to achieve the maxi-mum rate possible for the underlying network topology,(4 GHZ states/cycle in our case). When p and q bothequal one, we end up with only one GHZ state sharedbetween the consumers. This issue can be overcome bydividing the network into four disconnected sub-graphssuch that exactly one quantum memory from each con-sumer resides in one sub-graph. The sub-graphs are neverallowed to merge into each other by permanently erasingthe edges joining them. (a)(b)(c) FIG. 7. The brickwork protocol over the square-grid network- (a) The dotted lines show the red and black link generationattempt. The dotted black lines form a brickwork lattice.(b) Red links are used for fusion only if there are less than 3black links present. (c) Comparison between the 3-GHZ pro-tocol and the brickwork protocol for q = 0 .
85 and Manhattandistance = 85 units. The rate for the brickwork protocol doesnot degrade when p is high and q < FIG. 8. The site-bond region for the 3-GHZ brickwork pro-tocol for the square-grid, Poisson-degree distributed randomgraph with mean degree 4, and constant degree-4 randomgraph network topologies
VI. QUANTUM KEY DISTRIBUTION
In this section, we briefly discuss a Quantum Key gen-eration protocol to share a secret key between a pair ofusers using n -GHZ states. This protocol is an extensionof the BBM’92 quantum key distribution protocol [9].The protocol consists of the following steps - • Step 1:
Alice and Bob start with multiple m + n ≥ m -and n qubits of the GHZ state. | ψ AB (cid:105) = | (cid:105) ⊗ mA | (cid:105) ⊗ nB + | (cid:105) ⊗ mA | (cid:105) ⊗ nB √ m and n can vary across the collection ofshared GHZ states Alice and Bob possess. • Step 2:
They independently and randomly choosebetween the computational basis (0/1-basis) andthe Hadamard basis (+/- -basis) for measurement.Each user measures all their qubits of the GHZstate using (their) randomly-chosen measurement.Alice and Bob get m − and n - bit results, respec-tively, after performing the measurements. • Step 3:
They use a classical channel to inform eachother the basis they have used to measure theirrespective qubits. The measurements instanceswhere Alice and Bob used the same basis are usedfor key generation. This step is similar to theBBM92 protocol. • Step 4:
If both of them used the computationalbasis in a given round of the protocol, they get bitstring of either all 0’s or all 1’s. In this case, that bit becomes the key. When Alice and Bob both use theHadamard basis, they get measurement outcomebits strings a a . . . a m and b b . . . b n , respectively,such that ( a + a + · · · + a m ) mod 2 = ( b + b + · · · + b n ) mod 2. Here, the key would be the parityof their respective bit strings.We leave the security proof for this protocol as an openquestion. But we believe that it can be done as an ex-tension of the security proof for BBM’92. AliceBasis Output bits Key+/- 1010 00/1 0 0+/- 1101 -+/- 100 10/1 1111 -0/1 1 10/1 00 - BobBasis Output bits Key+/- 0 00/1 0 00/1 11 -+/- 010 1+/- 01 -0/1 111 1+/- 110 -TABLE I. Quantum key generation from shared GHZ-statesusing the protocol described in VI. When Alice and Bob bothuse the 0/1 basis, the secret key bit is the bit repeated inthe output bit-string. When both of them use the +/- basis,the secret key bit is the parity of their respective output bit-strings.
VII. CONCLUSION
We have designed a quantum-network-based entangle-ment generation protocol, which affords a rate that isindependent of the distance between the users. The pro-tocol only uses local link state information, and has theaforesaid property of distance-independent entanglementrate in a certain region of the link-level entanglementsuccess probability p (which is proportional to the link’soptical transmissivity, and hence range) and an individ-ual repeater’s success probability q (in performing an n -GHZ projective measurement). This ( p, q ) region thatachieves distance-independent rate is the site-bond re-gion of a modified mixed percolation problem, defined onthe underlying network such that the bond and site occu-pation probabilities are given by the link generation andrepeater success probabilities, respectively. Our protocolrequires only certain local Clifford operations, Pauli mea-surements, and classical communications. We performmulti-qubit projections at each node of the 2D networkmaking it a multi-path routing protocol. It outperformsthe multi-path routing protocol that only uses Bell statemeasurements (BSMs) [6]. All BSM based entanglementprotocols exhibit rates that decay with distance eventhose that use non-local-link state knowledge. To studyour protocol for complex quantum networks, we analyt-ically derived the site-bond region for a configuration-model random network with an arbitrary node degreedistribution. This shows an excellent match with the FIG. 9. Schematic representation of the sum rule for the con-nected component of vertices reached by following a randomlychosen link. numerically-evaluated site-bond region of our modifiedmixed percolation problem using the Newman-Ziff al-gorithm. We also discussed a two-party quantum keydistribution protocol that can be implemented using theshared entangled state obtained from the entanglementgeneration protocol.A few other questions that can be solved as an ex-tension of this protocol are - (1) generating shared en-tanglement between multiple consumer pairs simultane-ously (2) The repeater failure model we have assumedhere is very simple. One can study more realistic modelsrepeater failure due to unsuccessful fusions, photon loss,etc.
ACKNOWLEDGMENTS
AP and SG acknowledge the National Science Foun-dation (NSF) EFRI-ACQUIRE program, grant numberECCS-1640959. DT’s work was supported in part by theNSF under grant CNS-1617437.
Appendix A: Site-bond region for n -GHZ protocolover configuration graph Consider a configuration graph with node degree dis-tribution given by the generating function G ( x ) = ∞ (cid:88) d =0 p d x d (A1)where p d is the probability that a randomly chosen nodehas degree d . The average node degree is z = (cid:80) ∞ d =0 dp d .The generating function for the excess degree distributionis G ( x ) = G (cid:48) ( x ) z = (cid:80) ∞ d =0 dp d x d − z = ∞ (cid:88) d =0 e d x d (A2)In the percolation problem for the n -GHZ protocol,we are allowed to perform up to n -qubit GHZ projec-tion at each node (repeater). We start with a random graph with node degree distribution given by (A1). Inthis graph, each edge is occupied with probability p , thelink generation probability. We call the edges that areoccupied “links”. In this section, we derive the site-bondregion for a configuration-graph random network by gen-eralizing the formalism in [13].Let H ( x ) be the generating function for the distri-bution of the size of the component that is reached bychoosing a random link and counting all of the nodesthat can be reached through one of its end points. Fig. 9shows the schematic of the sum rule for H ( x ), the con-nected component (square) reached by following a ran-domly chosen link (black lines). We denote nodes bycircles and unoccupied edges by grey lines. The distri-bution of connected component consists of a node at theend of the link we started with and clusters attached(squares) to the node via links (if any). We refer to thenode reached by following the link as vertex. The sizeof the component is zero, if the fusion fails at the vertexwith probability (1 − q ). Excess edges are the edges ofa node other than the one used to reach the node. Thisdefinition can be extended to excess links as well. Be-cause of the n -GHZ fusion rule, a link always leads toa connected component as long as the number of excessedges at its vertex ( k ) is less than or equal to n − k > n −
1, it leadsto two possible scenarios - (1) when the excess links atthe vertex l ≤ n −
1. In this case, the link connects toa connected component if the fusion succeeds. (2) when l > n −
1, the size of the component is non-zero if the ver-tex chooses the link we started with as one of the links forfusion. This happens with probability n/ ( l + 1). Whenthe link is excluded from the fusion, the size of the con-nected component becomes zero. As we are following alink and not a node, we are interested in the distributionof excess links at the vertex. The probability that a nodewith k excess edges has l excess links, given each edge isoccupied with probability p is - P ( l | k ) = (cid:18) kl (cid:19) p l (1 − p ) k − l (A3)Assuming the fusion success probability is q , we writedown the sum rule for H ( x ) - H ( x ) = 1 − q + qx n − (cid:88) k =0 e k k (cid:88) l =0 P ( l | k )[ H ( x )] l + qx ∞ (cid:88) k = n e k (cid:34) n − (cid:88) l =0 P ( l | k )[ H ( x )] l + k (cid:88) l = n P ( l | k ) nl + 1 [ H ( x )] n − (cid:35) + q ∞ (cid:88) k = n e k k (cid:88) l = n P ( l | k ) l + 1 − nl + 1 (A4)The generating function for the distribution of the size0of the component that a random node belongs is H ( x ) = 1 − q + qx ∞ (cid:88) k =0 p k k (cid:88) l =0 P ( l | k )[ H ( x )] l (A5)and the mean component size is (cid:104) s (cid:105) = H (cid:48) (1) = q ∞ (cid:88) k =0 p k k (cid:88) l =0 P ( l | k )(1 + lH (cid:48) (1)) (A6) H (cid:48) (1) diverges when q ( p ) ≥ n − (cid:88) k =0 e k k (cid:88) l =0 lP ( l | k ) + ∞ (cid:88) k = n e k (cid:34) n − (cid:88) l =0 lP ( l | k )+ k (cid:88) l = n P ( l | k ) n ( n − l + 1 (cid:35) (A7)This marks the phase transition for percolation and (A7)gives the site-bond curve. FIG. 10. Analytically calculated (using (A7)) and simulatedsite-bond region for the 10-GHZ protocol over a configurationgraph network with Poisson degree distribution with mean λ = 50. Appendix B: Brickwork-like model for configurationgraph
In the site-bond curve for the n -GHZ protocol over aconfiguration graph network, after a certain value of p ,the turnaround point, q starts increasing with p as shownin Fig. 10. This happens due to the adversarial nature ofthe protocol explained in V. In this section, we calculatethe site-bond region for the brickwork-like strategy forconfiguration graphs to improve the entanglement gener-ation rate beyond the turnaround point. For the n -GHZ protocol over a configuration graph net-work whose degree distribution is given by (A1), to makethe protocol deterministic, we divide the edges into twocategories - black and red. Each node can have maximum n black edges and the rest are red edges. If the total num-ber of edges at a node is less that n , all of them are black.Each repeater (node) uses the red links for fusion only ifit has less than n black links. Let H ( x ) , H ( x ) be thedistribution of the sizes of components that are reachedby following black and red links, respectively. Let l and l be respectively the number of black and red excesslinks at a node such that l = l + l . H ( x ) = 1 − q + qx n − (cid:88) k =0 e k k (cid:88) l =0 P ( l | k )[ H ( x )] l + qx ∞ (cid:88) k = n e k n − (cid:88) l =0 (cid:34) n − − l (cid:88) l =0 p l + l (1 − p ) k − l − l (cid:18) n − l (cid:19) × (cid:18) k − n + 1 l (cid:19) [ H ( x )] l [ H ( x )] l + k − n +1 (cid:88) l = n − l p l + l (1 − p ) k − l − l (cid:18) n − l (cid:19)(cid:18) k − n + 1 l (cid:19) × [ H ( x )] l [ H ( x )] n − − l (cid:35) (B1) H ( x ) = 1 − q + qx ∞ (cid:88) k = n e k n − (cid:88) l =0 (cid:34) n − − l (cid:88) l =0 p l + l (1 − p ) k − l − l × (cid:18) nl (cid:19)(cid:18) k − nl (cid:19) [ H ( x )] l [ H ( x )] l + k − n (cid:88) l = n − l p l + l (1 − p ) k − l − l (cid:18) nl (cid:19)(cid:18) k − nl (cid:19) [ H ( x )] l × [ H ( x )] n − − l n − l l + 1 (cid:35) + q ∞ (cid:88) k = n e k n (cid:88) l =0 k − n (cid:88) l = n − l p l + l (1 − p ) k − l − l (cid:18) nl (cid:19)(cid:18) k − nl (cid:19) l + l − nl + 1 (B2)The distribution of sizes of components to which a ran-1domly chosen node belongs is given by - H ( x ) = 1 − q + qx n (cid:88) k =0 p k k (cid:88) l =0 P ( l | k )[ H ( x )] l + qx ∞ (cid:88) k = n +1 p k n (cid:88) l =0 (cid:34) n − l (cid:88) l =0 p l + l (1 − p ) k − l − l (cid:18) nl (cid:19) × (cid:18) k − nl (cid:19) [ H ( x )] l [ H ( x )] l + k − n (cid:88) l = n − l p l + l × (1 − p ) k − l − l (cid:18) nl (cid:19)(cid:18) k − nl (cid:19) [ H ( x )] l [ H ( x )] n − l (cid:35) (B3)The average cluster size (cid:104) s (cid:105) diverges when the giant com-ponent appears. (cid:104) s (cid:105) = H (cid:48) (1) = q n (cid:88) k =0 p k k (cid:88) l =0 P ( l | k ) (cid:0) lH (cid:48) (1) (cid:1) + q ∞ (cid:88) k = n +1 p k n (cid:88) l =0 (cid:34) n − l (cid:88) l =0 p l + l (1 − p ) k − l − l (cid:18) nl (cid:19) × (cid:18) k − nl (cid:19)(cid:0) l H (cid:48) (1) + l H (cid:48) (1) (cid:1) + k − n (cid:88) l = n − l p l + l (1 − p ) k − l − l (cid:18) nl (cid:19)(cid:18) k − nl (cid:19)(cid:0) l H (cid:48) (1) + ( n − l ) H (cid:48) (1) (cid:1)(cid:35) (B4) H (cid:48) (1) = q n − (cid:88) k =0 e k k (cid:88) l =0 P ( l | k ) (cid:0) l H (cid:48) (1) (cid:1) + q ∞ (cid:88) k = n e kn − (cid:88) l =0 (cid:34) n − − l (cid:88) l =0 p l + l (1 − p ) k − l − l (cid:18) n − l (cid:19) × (cid:18) k − n + 1 l (cid:19)(cid:0) l H (cid:48) (1) + l H (cid:48) (1) (cid:1) + k − n +1 (cid:88) l = n − l p l + l (1 − p ) k − l − l (cid:18) n − l (cid:19)(cid:18) k − n + 1 l (cid:19) × (cid:0) l H (cid:48) (1) + ( n − − l ) H (cid:48) (1) (cid:1)(cid:35) (B5) FIG. 11. The site-bond region for Poisson-degree distributedrandom graph with mean node degree ( λ = 50) for 10-GHZbrickwork protocol H (cid:48) (1) = q ∞ (cid:88) k = n e k n − (cid:88) l =0 (cid:34) n − − l (cid:88) l =0 p l + l (1 − p ) k − l − l × (cid:18) nl (cid:19)(cid:18) k − nl (cid:19)(cid:0) l H (cid:48) (1) + l H (cid:48) (1) (cid:1) + k − n (cid:88) l = n − l p l + l (1 − p ) k − l − l (cid:18) nl (cid:19)(cid:18) k − nl (cid:19) × n − l l + 1 (cid:0) l H (cid:48) (1) + ( n − − l ) H (cid:48) (1) (cid:1)(cid:35) (B6)Equations (B5) and (B6) form linear system equations in H (cid:48) (1) and H (cid:48) (1) and can be re-written as - H (cid:48) (1) = qS H (cid:48) (1) + qS H (cid:48) (1) + C (B7) H (cid:48) (1) = qS H (cid:48) (1) + qS H (cid:48) (1) + C (B8)The mean cluster size diverges when(1 − qS )(1 − qS ) = q S S The site-bond curve is given by - q ( p ) = − S − S + (cid:112) ( S + S ) + 4( S S − S S )2( S S − S S ) (B9) Appendix C: Rate calculation for 2-GHZ protocol
Consider 2-GHZ protocol on the square-grid network,i.e., the repeaters perform only Bell state measurements(BSMs) on the successful links. Let d AB be the Man-hattan distance between Alice and Bob. For the link2generation probability p , let F ( p ) denote fraction of gridlying in giant connected component for a square lattice.Let the BSM success probability be q . Then the sharedentanglement generation rate R is proportional to theprobability that there exists a path between Alice andBob in the graph generated after performing BSMs. Forthis protocol, the maximum possible achievable rate is 4 ebits/cycle. Hence, we can write, R ≤ F ( p ) q d AB − (C1)This is a very loose upper bound on the rate. But itdecays exponentially with the separation between Aliceand Bob. Hence, it is impossible to achieve distance-independent rate by using only Bell state measurements. [1] M. Takeoka, S. Guha, and M. M. Wilde, Nat. Commun. , 5235 (2014).[2] S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi,Nat. Commun. , 15043 (2017).[3] S. Guha, H. Krovi, C. A. Fuchs, Z. Dutton, J. A. Slater,C. Simon, and W. Tittel, Phys. Rev. A , 022357(2015).[4] S. Muralidharan, L. Li, J. Kim, N. L¨utkenhaus, M. D.Lukin, and L. Jiang, Sci. Rep. , 20463 (2016).[5] M. Pant, H. Krovi, D. Englund, and S. Guha, Phys. Rev.A , 012304 (2017).[6] M. Pant, H. Krovi, D. Towsley, L. Tassiulas, L. Jiang,P. Basu, D. Englund, and S. Guha, npj Quantum Infor-mation , 1 (2019).[7] M. K. Bhaskar, R. Riedinger, B. Machielse, D. S. Levo-nian, C. T. Nguyen, E. N. Knall, H. Park, D. Englund,M. Lonˇcar, D. D. Sukachev, and M. D. Lukin, Nature , 60 (2020).[8] K. R. Brown, J. Kim, and C. Monroe, npj QuantumInformation , 16034 (2016).[9] C. H. Bennett, G. Brassard, and N. D. Mermin, Phys.Rev. Lett. , 557 (1992).[10] A. K. Ekert, Phys. Rev. Lett. , 661 (1991).[11] D. Poulin, Phys. Rev. Lett. , 230504 (2005).[12] M. A. Nielsen, Rep. Math. Phys. , 147 (2006).[13] M. E. Newman, S. H. Strogatz, and D. J. Watts,Phys. Rev. E Stat. Nonlin. Soft Matter Phys. , 026118(2001).[14] G. Grimmett, in Percolation (Springer, 1999) pp. 1–31.[15] F. Ewert and P. van Loock, Phys. Rev. Lett. , 140403(2014).[16] J. M. Hammersley, J. Math. Phys. , 728 (1961).[17] M. E. J. Newman and R. M. Ziff, Phys. Rev. E64