Routing entanglement in the quantum internet
Mihir Pant, Hari Krovi, Don Towsley, Leandros Tassiulas, Liang Jiang, Prithwish Basu, Dirk Englund, Saikat Guha
RRouting entanglement in the quantum internet
Mihir Pant,
1, 2, ∗ Hari Krovi, Don Towsley, Leandros Tassiulas, Liang Jiang,
5, 6
Prithwish Basu, Dirk Englund, and Saikat Guha
2, 8 Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA Quantum Information Processing group, Raytheon BBN Technologies, 10 Moulton Street, Cambridge, MA College of Information and Computer Sciences, University of Massachusetts, Amherst, MA School of Engineering and Applied Science, Yale University, 17 Hill House Avenue, New Haven, CT Departments of Applied Physics and Physics, Yale University, New Haven, CT Yale Quantum Institute, Yale University, New Haven, CT Advanced Networking Systems, Raytheon BBN Technologies, 10 Moulton Street, Cambridge, MA College of Optical Sciences, University of Arizona, 1630 East University Boulevard, Tucson, AZ
Remote quantum entanglement can enable numerous applications including distributed quantumcomputation, secure communication, and precision sensing. In this paper, we consider how a quan-tum network—nodes equipped with limited quantum processing capabilities connected via lossyoptical links—can distribute high-rate entanglement simultaneously between multiple pairs of users(multiple flows). We develop protocols for such quantum “repeater” nodes, which enable a pair ofusers to achieve large gains in entanglement rates over using a linear chain of quantum repeaters,by exploiting the diversity of multiple paths in the network. Additionally, we develop repeater pro-tocols that enable multiple user pairs to generate entanglement simultaneously at rates that can farexceed what is possible with repeaters time sharing among assisting individual entanglement flows.Our results suggest that the early-stage development of quantum memories with short coherencetimes and implementations of probabilistic Bell-state measurements can have a much more profoundimpact on quantum networks than may be apparent from analyzing linear repeater chains. Thisframework should spur the development of a general quantum network theory, bringing togetherquantum memory physics, quantum information theory, and computer network theory. A quantum network can generate, distribute and pro-cess quantum information in addition to classical data [1].The most important function of a quantum network isto generate long distance quantum entanglement, whichserves a number of tasks including the generation ofmultiparty shared secrets whose security relies only onthe laws of physics [2, 3], distributed quantum comput-ing [4], improved sensing [5, 6], blind quantum comput-ing (quantum computing on encrypted data) [7], and se-cure private-bid auctions [8].Recent experiments have demonstrated entanglementlinks , viz., long-range entanglement established betweenquantum memories separated by a few kilometers us-ing a point-to-point optical link [9]. As illustrated inFig. 1, measurements performed at nodes in a quantumnetwork can be used to glue together small entanglementlinks into longer-distance clusters. The nodes containquantum memories that store qubits up to their coher-ence time, sources that generate photons entangled withthe quantum memory to be sent to neighboring nodes,and local quantum processors that can perform multi-qubit joint measurements. Entanglement attempts be-tween neighboring nodes are synchronized on a globalclock. The quantum routing protocol dictates the mea-surements to be performed locally at each node in orderto obtain the desired entanglement topology. Possiblegoals of a routing protocol could be to enable high rateentanglement among multiple user-pairs simultaneously, ∗ [email protected] or to generate multi-partite entanglement (entanglementbetween three or more parties).The development of network algorithms and protocolsfor routing and scheduling information flows was criti-cal for the creation of today’s Internet. We expect asimilar development in algorithm/protocol design to becritical to design a versatile and high performance quan-tum network. Some results and intuitions from the the-ory of classical networks carry over into quantum net-working. However, many new challenges arise due tothe idiosyncrasies of quantum information. Unlike clas-sical communications, where the rate can be increasedby increasing transmit power, photon loss fundamentallylimits the entanglement rate over any single link, whichmust decay exponentially with the length of optical fiber,regardless of the choice of quantum source, the trans-mit power or the detection strategy [10, 11]. Whereascopying of bits at a network node is common in mul-tipath routing in classical networks [12, 13], copying aqubit is impossible because of the quantum no-cloningtheorem [14, 15]. Unlike classical information flow, anentanglement flow does not have directionality. Rather,entanglement is generated across links all over the net-work and pieced together to form long-range entangle-ment. Quantum memories are much shorter lived andexpensive compared to their classical counterparts mak-ing classical routing strategies such as disruption toler-ant routing [16, 17]—where a packet is held by a nodefor until the desired next-hop link is up—much harderto mimic. Finally, distilling and shaping entanglementamong a desired set of nodes from many copies of large a r X i v : . [ qu a n t - ph ] S e p FIG. 1. Examples of fusing small entangled clusters intolarger ones using projective quantum measurements (greenovals) at nodes of a quantum network. Red circles representqubits and black lines represent entanglement. (a) Two-qubitmeasurement (Bell state projection) used to connect two en-tangled links into a longer entangled link; (b) a three-qubitmeasurement (a GHZ state projection) fuses three clusters(two 2-qubit entangled links and one 3-qubit linear cluster)into one 4-qubit entangled cluster; (c) two adjacent nodes ina network performing a three-qubit GHZ measurement and atwo-qubit Bell state measurement simultaneously. The mea-sured qubits are lost, whereas the final entangled state of theunmeasured qubits upon successful completion of both mea-surements is the same regardless of the order or the simultane-ity of the measurements. A quantum measurement at a nodemay succeed only with a probability, which is a function of theclass of optical devices employed to realize the measurement(e.g., linear optics, single photon sources, and single photondetectors) and losses in devices. This figure does not show“failure outcomes”, i.e., the resulting entangled state if oneor both measurements fail. (potentially random) entangled clusters is a purely quan-tum problem that has no classical analogue.In this paper, we present protocols for repeater nodesto support multiple simultaneous entanglement flowswhen every node is limited to the same quantum pro-cessing used in repeater chains: (probabilistic) two-qubitBell state measurement (BSM), also called entanglementswapping. BSMs have been experimentally demonstratedin many physical systems [18–23]. Entanglement at-tempts between repeater nodes are probabilistic becausethey are connected via lossy optical links. In every clockcycle, pairs of neighboring repeater nodes attempt en-tanglement generation. The result of whether an entan-glement generation was successful is transmitted back tothe corresponding pair of nodes. The repeater nodes thenmake their local BSM decisions based on this ‘local’ link-state knowledge, i.e., the successes and failures of entan-glements established across their nearest neighbor links.Measurements at different nodes can all be done in par-allel because BSMs commute with one another.One of our interesting findings is that multi-path rout-ing , i.e. using multiple paths for routing entanglementin a quantum network, can enable long distance entan-glement generation with a superior rate-vs.-distance scal-ing than a single linear repeater chain along the short-est path connecting Alice and Bob [24]. The rate alonga single repeater chain falls off exponentially with dis-tance [25, 26]. Multi-path routing reduces the exponent,resulting in an exponential increase in rate with increas- ing distance. The quantum network we propose uses thesame basic elements and operations (probabilistic BSMs)as a linear repeater chain but uses more repeaters. Notethat increasing the number of repeaters in a repeaterchain would not always increase the rate: given the totalend-to-end distance, and given the losses at each node,there is an optimal number of repeaters between the endpoints of a flow that maximizes the rate, i.e., insertingmore nodes along that linear path can actually diminishthe rate [25, 26].Another interesting result is that if the repeater nodeshave ‘global’ link-state knowledge (knowledge of the stateof all links in the network) and the entanglement gen-eration probability is above a (percolation) threshold,multi-path routing enables long distance entanglement-generation at a rate that depends only linearly on thetransmissivity η of a single link in the network, whereasthe rate achieved by a linear repeater chain connect-ing Alice and Bob along the shortest path would de-cay as η n SP where n SP is the length of the shortestpath. Even a linear repeater chain can attain a rate thatis proportional to η , but that requires repeater nodesequipped with error-corrected quantum processors [27].We achieve the same feature (rate proportionality to η )by multipath routing with percolation, with a much sim-pler repeater. We also analyze repeater protocols to sup-port multiple entanglement-generation flows. This anal-ysis reveals that simple space-time-division multiplexingstrategies that use local link-state knowledge at nodescan outperform the best rate-region (the set of rates si-multaneously achievable by different flows) attainable byrepeater nodes that simply time share among assistingindividual flows.Our work also opens up a number of new questionsthat remain unanswered. We abstract off the entangle-ment routing problem to the following parameters: G (network topology), p (probability of successful creationof an entangled pair across one link in a given time step), q (probability of a successful Bell measurement when at-tempted), S (number of parallel links across a networkedge) and T (the number of time slots that a memorycan hold a qubit before it decoheres). Even in this sim-plified model, finding the rate-region optimizing routingprotocol remains open. The aforesaid abstraction applieswhen the only source of imperfection at each component(including the quantum memories) is pure loss. Sinceour protocol only requires a quantum memory to holda qubit for one entanglement attempt between neigh-boring stations ( T = 1), photon loss would indeed bethe major source of imperfection in many implementa-tions of the protocol. Accounting for more general errorswould require purification of entanglement [28–30], whichwill require us to introduce the Fidelity of entanglementduring intermediate steps of the routing protocol as anadditional parameter, as was done by Jiang et al. [31].Furthermore, we restricted our analysis only to 2-qubitmeasurements at repeater nodes. Multi-qubit unitary op-erations and multi-qubit measurements at repeater nodes FIG. 2. Schematic of a general quantum repeater network. The large (green) circles represent ‘trusted’ nodes, which areconnected via a classical network. The blue circles denote repeater stations, and the red circles inside them represent quantummemories. The dashed lines connecting the red circles are independent lossy optical (fiber) channels. In principle, all nodesin the network could be equipped with quantum repeaters (i.e., no trusted nodes), in which case depending upon the need, anode can be a consumer of shared entanglement, or act a router to conduit entanglement flows between other nodes. (e.g., a 3-qubit GHZ projection across three locally heldqubits) would require more complex repeaters than thosein repeater chains, but may improve the achievable rates.Finally, it will be interesting to consider repeater proto-cols for distillation of multi-partite entanglement sharedbetween more than two parties, and a repeater net-work that supports multiple simultaneous flows of multi-partite entanglement generation.
I. BACKGROUND
Let us consider a quantum network with topology de-scribed by a graph G ( V, E ). Each of the N = | V | nodesis equipped with a quantum repeater, and each of the M = | E | edges is a lossy optical channel of range L i (km) and power transmissivity η i ∝ e − αL i , i ∈ E . Con-sider K source-destination (Alice-Bob) pairs ( A j , B j ),1 ≤ j ≤ K , situated at (not necessarily distinct) nodesin V , each of which would like to generate maximally-entangled qubits (i.e., ebits) between themselves (andthus by definition not entangled with any other party,due to the monogamy property of entanglement), atthe maximum rates possible R j (ebits per channel use).The high-level objective is: Given a class of quantumand classical operations at each of the repeater nodes ofthe underlying network, what operations should be per-formed at the repeater stations to maximize the rate re-gion ( R , R , . . . , R K ) simultaneously achievable by theentanglement flows? More importantly, one would liketo address networking questions such as: (a) what is themaximum rate-region attainable, (b) what is the trade-off between sum throughput and latency of the K en-tanglement flows, and (c) where should repeater nodesbe placed, with constraints on devices (e.g., memories,sources, and detectors), to maximize the attainable rateregion; all being subject to various practical considera- tions. Ultimately one would like to develop explicit andefficient practical quantum routing protocols that employquantum operations implemented via lossy and noisy de-vices, while only requiring local link-state knowledge andlimited knowledge of the global network topology, anal-ogous to the classical internet.The entanglement-generation rate across a link oftransmissivity η , in the absence of any repeater media-tion, is limited to − log (1 − η ) ebits per mode, amountingto ≈ . η ebits per mode when η (cid:28) η ∼ e − αL where L is the length of opticalfiber, the ebits-per-mode rate also falls off exponentiallywith range L . Most analyses of repeater networks havebeen limited to linear chains, with the objective of out-performing the repeater-less bound [25, 26, 30, 31, 33–36]. Pirandola analyzed entanglement-generation capac-ities of repeater networks assuming ideal repeater nodes,i.e., those equipped with fully-error-corrected quantumprocessors and argued that for a single flow ( K = 1), themaximum entanglement-generation rate R reduces tothe classical max-flow min-cut problem with edge e beingassociated with capacity C ( e ) = − log (1 − η ( e )) ebits perchannel use [37], where η ( e ) is the transmissivity of edge e . Pirandola subsequently argued that classical cut-setbounds with the above link capacity give outer boundsto the K -flow capacity region, but again, for ideal re-peater nodes. Azuma et al. independently established anupper bound [38] to the entanglement generation boundto the rate which has the same asymptotic scaling butis not tight. Azuma has also looked at an “aggregated”protocol in which repeater protocols run in parallel [39].Schoute and co-authors [40] developed routing protocolson specific network topologies and found scaling laws asfunctions of N , the number of qubits in the memories at FIG. 3. Schematic of a square-grid topology. The blue circlesrepresent repeater stations and the red circles represent quan-tum memories. Every cycle (time slot) of the protocol consistsof two phases. (a) In the first (external) phase, entanglementis attempted between neighboring repeaters along all edges,each of which succeed with probability p (dashed lines). (b)In the second (internal) phase, entanglement swaps are at-tempted within each repeater node based on the successesand failures of the neighboring links in the first phase—withthe objective of creating an unbroken end-to-end connectionbetween Alice and Bob. Each of these internal connectionssucceed with probability q . Memories can hold qubits for T ≥ nodes, and the time and space consumed by the rout-ing algorithms, under the assumption that each link gen-erates a perfect, lossless EPR pair in every time slot,and that the nodes’ actions are limited to (perfect) Bell-state measurements (BSMs). Ac´ın and co-authors [41]have considered the problem of entanglement percola-tion where neighboring nodes share a perfect, losslesspure state. Further, van Meter and co-authors developedexplicit networking protocols also restricted to pair-wiseEPR pair generation and BSMs, but accounting for im-perfect fidelities of the EPR pairs (and thus requiringpurification over multiple imperfect pairs), and finite co-herence times of the qubit memories [42]. There has alsobeen previous work on quantum network coding [43–46]and linear-optic quantum routers [47]. II. ENTANGLEMENT ROUTING PROTOCOLSA. Problem statement and notation
Consider a graph G ( V, E ) that denotes the topologyof the repeater network. See Fig. 2 for an illustration.Each node v ∈ V is a repeater (blue circles), and eachedge e ∈ E is a physical link connecting two repeaternodes. S ( e ) ∈ Z + is an integer edge weight, which corre-sponds to the number of parallel (single spatial, spectralor polarization mode) channels across the edge e (shownusing blue lines). The number of memories at node v is (cid:80) e ∈N ( v ) S ( e ) (see Fig. 2), where the sum is over N ( v ),the set of nearest neighbor edges of v , with d ( v ) = |N ( v ) | the degree of node v .Time is slotted. We assume that each memory canhold a qubit perfectly for T ≥ t , t = 1 , , . . . , is divided into two phases: the“external” phase and the “internal” phase, which occurin that order. During the external phase, each of the S ( e )pairs of memories across an edge e attempts to establish ashared entangled (EPR) pair. An entanglement attemptacross any one of the S ( e ) parallel links across edge e succeeds with probability p ( e ) ∼ η ( e ) [10, 11], where η ( e ) ∼ e − αL ( e ) is the transmissivity of a lossy opticalchannel of length L ( e ). Using two-way classical com-munication over edge e ( u, v ), neighboring repeater nodes u, v learn which of the S ( e ) parallel links (if any) suc-ceeded in the external phase.Let us assume that neighboring repeaters pick upto onesuccessfully-created ebit (i.e., ignore multiple successes ifany) as in Ref. [25, 26], in which case the probability thatone ebit is established successfully across the edge e dur-ing the external phase is given by: p ( e ) = 1 − (1 − p ) S ( e ) .Let us also assume S ( e ) = S, ∀ e ∈ E , which in turn givesus p ( e ) = p, ∀ e ∈ E . While our results in this papercan be adapted to any network topology, we will hence-forth use the 2D regular square grid topology (Fig. 3) toillustrate the performance of our routing algorithms.One instance of the resulting external links created be-tween repeater nodes after the external phase is shownin Fig. 3(b) using solid lines. In the internal phase,entanglement swap (BSM) operations are attempted lo-cally at each repeater node between pairs of qubit memo-ries. We associate these BSM attempts as internal links ,i.e. links between memories internal to a repeater node,shown using dotted lines inside repeaters in Fig. 3(b).If T >
1, a repeater node can attempt a BSM betweenqubits held in two memories that were entangled withtheir respective neighboring node’s qubits in two differ-ent time slots. For minimizing the demands on memorycoherence time [25, 26], we will assume T = 1. So, BSMswill always be attempted between two qubits in distinctmemories that were entangled with their respective coun-terparts at their respective neighboring nodes in the sametime slot. Each of these internal-link attempts succeedwith probability q . Therefore, after the conclusion of onetime slot, along a path comprising k edges (and thus k − p k q k − . Themaximum number of ebits that can be shared betweenAlice (say, node a ) and Bob (say, node b ) after one timeslot is min { d ( a ) , d ( b ) } , assuming S is the same over alledges. For the square-grid topology shown, the maximumnumber of ebits that can be generated between Alice andBob in each time slot is 4.The remainder of the paper is dedicated to finding theoptimal strategy for each repeater node in order to de-cide which locally-held qubits to attempt BSM(s) on dur-ing the internal phase of a time slot, based ideally onlyon knowledge of the outcomes (success or failure) of thenearest neighbor links, i.e., local link-state knowledge,during the respective preceding external phases. We willassume that each repeater node is aware of the overallnetwork topology as well as the locations of the K Alice-Bob pairs. The goal of the optimal repeater strategy willbe to attain the maximum entanglement-generation rate(if there is a single Alice and Bob, i.e., K = 1) or themaximum rate-region for multiple flows (i.e., K >
B. Multipath routing of a single entanglement flow
1. Entanglement routing with global link-state information
We begin with the assumption that global link-stateknowledge is available at each repeater node, i.e., thestate of every external link in the network after the ex-ternal phase is known to every repeater in the networkand can be used to determine the choice of which in-ternal links to attempt within the nodes. Each memorycan only be part of one entanglement swap, i.e., each rednode can only be part of one internal edge. Consider thefollowing greedy algorithm to choose the internal links:consider the subgraph induced by the successful externallinks and the repeater nodes (at the end of the externalphase), and find in it the shortest path connecting Al-ice and Bob. If no connected path between Alice andBob exists, no shared ebits are generated in that timeslot. If a shortest path of length k is found, all internallinks along the nodes of that path are attempted, andthe probability a shared ebit is generated by this path isthe probability that all k − q k − . We then remove all the (externaland internal) links of the above path from the subgraph,and find a shortest path connecting Alice and Bob inthe pruned subgraph. Note that instead of removing thelinks of the first path from the subgraph, we could sim-ply search for a shortest path in the original subgraphbut one that is edge disjoint from the previous path. Ifsuch a path exists, we again attempt all internal links atthe nodes of this path, so the probability the path con-tributes to the generation of an ebit between Alice andBob is q k − where k is the length of the second path;and so on.The entanglement generation rate achieved using thisgreedy algorithm R g is the sum of expected rates (in ebitsper time slot) from these paths. Given the degree-4 nodesin a square grid topology, there can be a maximum offour edge disjoint paths between Alice and Bob. Fig. 3(b)illustrates our greedy algorithm. Given the set of externallinks created, the shortest path has length k = 4, thenext path has length k = 6, and no further paths canbe found. The two edge-disjoint paths are highlightedin green. Hence, the internal links depicted with thedashed lines in Fig. 3(b) are attempted and the expectednumber of shared ebits generated in this time cycle is: q k − + q k − . The net entanglement generation rate isthe expectation of sums like the above (with up to four terms) over many random instantiations of the ( p, − p ) external-link creations during the external phase ofmany time slots. Evaluating this expected rate R g ( p, q )achieved by the above routing strategy analytically as afunction of the Alice-Bob distance ( X , X ) is difficult,even for a square-grid topology.The intuition behind this simple greedy algorithm isthat the entanglement generation rate along a path oflength k decays exponentially as q k − , suggesting thatattempting internal links to facilitate connections alongthe shortest path first would optimize the expected rate.However, it is possible to draw random instances of suc-cesses of external links, where either one of the twopossible options—(1) picking the shortest path (whichdisrupts all other paths) and (2) picking two edge dis-joint (but longer) paths—could yield either a larger ora smaller expected rate than the other, depending uponthe value of q . If q is larger than a threshold, option(2) would have a larger expected rate and vice versa.Finding the global optimal rule remains an open prob-lem. It is easy however to prove that the greedy al-gorithm achieves a rate within a factor of four of theoptimum algorithm employing global link-state knowl-edge, R opt ( p, q ). Let us denote the length of the shortestpath between Alice and Bob with Manhattan distance( X, Y ) in the induced subgraph after the external phase,as n SP ( p ). This quantity is of interest in percolation the-ory, and is not completely understood analytically. Itundergoes a sharp transition (i.e., starts out large andsuddenly jumps to a much smaller value) as p crosses p c from below to above. Clearly, R g ( p, q ) ≥ E [ q n SP ( p ) − ]since using the shortest path is the first step of thegreedy algorithm. Furthermore, since the optimal rulecan create entanglement over a maximum of four edge-disjoint paths in each time step, each of which must havea length no less than the length of the shortest path, R opt ( p, q ) ≤ E [4 q n SP ( p ) − ] (cid:44) R (UB)opt ( p, q ). Therefore, R opt ≥ R g ≥ R opt /
4, i.e., the greedy rule will achievethe same rate-vs.-distance scaling as the optimal algo-rithm that employs global link-state knowledge, and canbe worse only by a small constant factor.In Fig. 4(a) we plot R g ( p, q ) as a function of the Alice-Bob Manhattan distance ( X, Y ) on the square grid (mea-sured in number of hops) with q = 1. When p > p c ,the bond percolation threshold of the underlying network( p c = 0 . giant connected com-ponent is formed by the external links alone at the end ofthe first (external) phase of a time slot. Recall that therate along a length k path is p k q k − , where p ∼ η is thetransmissivity of each link. In the network case, when p > p c and q = 1, the p k portion of the rate expressionbecomes immaterial for scaling with Alice-Bob distance,since percolation guarantees a connected path to existbetween Alice and Bob along successful external links ineach time slot. So, if q = 1, R g ( p, q ) remains essentiallydistance invariant. When p < p c , the rate falls off expo-nentially with distance (even when q = 1). It is instruc-tive to note here that the optimal rate (entanglement- FIG. 4. Entanglement generation rate as a function of the Alice-Bob separation along X and Y (on a square grid) as a functionof ( p, q ); (a) R g ( p, q ) is the rate attained by a global-knowledge-based protocol we propose where each node, in each time step,knows whether any link in the entire network succeeded or failed to establish entanglement. For the case of q = 1, R g is distanceindependent when p is greater than the bond percolation threshold (0.5 for the square lattice) and decays exponentially if itis below the threshold. (b) R (UB) (0.6) is the distance-independent Pirandola rate upper bound for p = 0 .
6, achieving whichrequires perfect quantum processing at repeater nodes. R g (0 . ,
1) is also distance independent, and within a factor 3.6 of R (UB) (0 . q <
1, e.g., R g (0 . , . R (UB)opt is an upper bound on the rateattainable with global-knowledge by any protocol. (c) R loc is attained by a protocol we propose where each node, in eachtime step, only needs to know the link state of neighboring edges. The rate-distance scaling exponent of R loc is clearly worsethan R g , but is significantly superior to that of a linear repeater chain along the shortest path, R lin , demonstrating multi-pathrouting advantage even with local link-state knowledge. (d) Contour plot of the entanglement generation rate with the localrule when p = 0 . q = 0 .
9. Although the Alice to Bob distance along the network links is X + Y , there is a noticeableenhancement in the rate along the X = Y direction because of more Alice-Bob paths of similar length. generation capacity) achievable on a single length k pathdoes not depend on k , and only on the transmissivityof the lossiest link in the path, i.e., C ∼ η [37], butachieving this requires infinite-coherence-time quantummemories and ideal quantum operations at nodes. Themulti-path gain in the p > p c regime lets us achieve adistance-independent rate, but with memories whose co-herence times are no more than one time slot. The ratesare calculated using monte-carlo simulations which re-sults in some numerical noise that is insignificant com-pared to the difference between the plots, but is visiblein R g (0 . , R (UB) ( p ) = − log [(1 − p ) ]. R (UB) (0 .
6) is plotted in Fig. 4(b). The known methodsfor achieving R (UB) require infinite coherence time mem-ories and error-corrected quantum processors at eachnode. For our implementation (assuming global linkstate knowledge), R g (0 . ,
1) is also plotted in Fig. 4(b).Although our protocol only requires memories to holdentanglement for one time step, the multi path advan-tage gives us the same constant rate-distance scaling andwithin a factor of ∼ . R (UB) (0 . q <
1, in whichcase R g ( p, q ) falls off exponentially with distance; evenwhen p > p c , as seen in the plot for R g (0 . , . R g , FIG. 5. The entanglement swap rule used at the repeater Cin the dotted box in the case of local link-state knowledge. Aand B are the repeaters closest to Alice and Bob, respectively,with a direct edge to C. (a) If two or three links are up, thememories linked to A and B undergo an entanglement swap.(b) If four links are up, the remaining two memories alsoundergo an entanglement swap. R (UB)opt (0 . , . R g , but larger by a factor less than 4.
2. Entanglement routing with local link-state information R g ( p, q ), the rate attained by the protocol describedin the previous subsection that employs global link-stateknowledge, is re-plotted in Fig. 4(c). We also plot R lin ( p, q ) = p n SP (1) q n SP (1) − , the rate attained by a sin-gle linear repeater chain, where n SP (1) is the shortest-path length between Alice and Bob along the edges ofthe underlying square grid. The assumption of globallink-state knowledge in large networks is unrealistic, asit requires memories whose coherence time increases withthe network size due to the time required for the traver-sal of link-state information across the entire network.In this section, we describe a more realistic protocol inwhich knowledge of success and failure of an externallink at each time slot is communicated only to the two repeater nodes connected by the link, as is the case inthe analysis of many ‘second-generation’ linear repeaterchains [25, 31, 33]. Repeater nodes need to decide onwhich pair(s) of memories BSMs should be attempted(i.e., which internal links to attempt), based only on in-formation about the states of external links adjacent tothem. We assume that network topology and positionsof Alice and Bob are known to each repeater station, andcommunicated classically beforehand.Let us consider a local repeater rule illustrated inFig. 5. The repeater u inside the dotted square boxhas to make a decision regarding which internal edgesto attempt based on the information of which of the fourneighboring external edges have been successfully createdin the external phase. We associate d A and d B as the dis-tance to Alice and Bob, respectively, at every repeaternode. We use the L norm distance to Alice (resp. Bob)as d A (resp. d B ). Of all the nearest neighbor nodes of u whose links to u were successful in that time slot, we la-bel the one that has the minimum d A as v . Similarly, theneighbor with a successful external link with u and theminimum d B is labelled w . An internal link is attemptedbetween the memories connected to v and w respectively,as shown in Fig. 5(a). If v and w are the same node, v (or w ) is replaced by node u ’s nearest-neighbor node with thenext smallest value of d A (or d B ). The choice of whetherto replace v or w is made in a manner that minimizesthe sum of d A and d B from the eventually chosen twoneighbors to connect. If all four external links are suc-cessful, an additional internal link is attempted betweenthe remaining two memories as shown in Fig. 5(b). Ifonly one of the neighboring external links is successful,no internal links are attempted, since this repeater nodecannot be part of a path from Alice to Bob in that timeslot. If two neighbors have the same values of d A and d B , an unbiased coin is tossed to determine the choice of v and w , to preserve symmetry in the protocol.The entanglement generation rate R loc ( p, q ) achievedby the above described local rule is plotted in Fig. 4(c)and compared to R g ( p, q ) and R lin ( p, q ). We use p =0 . q = 0 .
9, the same values used for the global-information rate plots in Fig. 4(b). As one expects, therate-distance scaling of R loc is worse than that of R g .However, the rate-distance scaling exponent achieved bythe local rule is superior to that of a linear chain, eventhough the physical elements employed to build the re-peaters are identical. This is proven analytically in ap-pendix B 2. Note that each of the three rates R g , R loc , R lin fall exponentially with distance, but the exponentsare different. The scaling advantage of R loc over R lin arises because the local rule allows the entanglement-generation flow between Alice and Bob to find different(and potentially simultaneously multiple) paths in differ-ent time slots, and does not have to rely on all links alonga linear chain to be successful. This is analogous to multi-path routing in a classical computer network. The con-tour plot in Fig. 4(d) further illustrates this point: thereis a noticeable enhancement of R loc along the X = Y FIG. 6. (a) Multi-flow routing for two Alice-Bob pairs that liealong the sides of a 6 × ×
100 grid;(b) rate region ( R , R ) with different rules at repeater nodes,each employing local link-state knowledge, for p = q = 0 . line because the diagonal direction contains the largestspatial density of possible paths between Alice and Bob.The scaling advantage over R lin persists in any direction,i.e., along Y = 0 as well.Sweeping over different values of p and q , we findthat the multi-path advantage relative to a linear re-peater chain increases as p decreases from unity, butthere is little relative improvement as q is varied (seeAppendix B 1).Clearly, other distance metrics (e.g., L p norm for p ≥
1) can be used in lieu of the L norm in the algorithmdescribed above. In Appendix A, we present a recursivenumerical evaluation technique to find the rate-optimaldistance metric, which can be applied to any networktopology. For planar network topologies, the L normappears near-optimal for our local routing algorithm.An analytical enumeration of the expected number ofedge-disjoint paths as a function of p between Alice andBob separated by a given distance ( X, Y ) in a bond-percolation instance (i.e., with p > p c ) of a network is anopen question, the solution of which will enable a firmerquantitative understanding of the multi-path advantagein entanglement generation in a repeater network. C. Simultaneous entanglement flows
In this section, we consider simultaneousentanglement-generation flows between two Alice-Bob pairs, using local link state knowledge at all
FIG. 7. A heat map plotting p usage , the probability that agiven repeater node is involved in a successful creation of ashared ebit generated between Alice and Bob, separated by6 hops in an underlying square grid topology, when our localrule is employed. We assume p = 0 . q = 0 . repeater nodes. Consider two pairs Alice 1 - Bob 1 (rednodes) and Alice 2 - Bob 2 (green nodes) as shown in thetwo scenarios in Fig. 6. In Fig. 6(a), the shortest pathsconnecting the two Alice-Bob pairs do not cross, butthey do in Fig. 6(b). In both cases, they are placed atthe four corners of a 6 × R and R the entanglement generation rates achieved by the twoAlice-Bob pairs respectively. We first consider the caseof non-intersecting flows shown in Fig. 6(a). A simplestrategy is for every single repeater node (including thenodes labeled as the two Alices and Bobs) to use thelocal rule described in the previous section tailored tosupport the Alice 1-Bob 1 flow for a fraction, λ , of thetime slots and to support the Alice 2-Bob 2 flow forthe remaining 1 − λ fraction. For p = q = 0 .
9, therate region attained by varying λ ∈ [0 ,
1] is depictedwith the blue line in Fig. 6(b), which we refer to assingle-flow time-share. However, if every repeater withthe exception of the Alices and Bobs carry out the abovetime-sharing strategy, even when all repeater nodessupport flow 1, there is still some ‘left-over’ non-zero R that is attained. This multi-flow time-share rate regionis shown using the red line in Fig. 6(a).In Fig. 7, for the case that Alice and Bob are separatedby 6 hops on the square grid, we plot a color map of p usage , the probability a given repeater node is involvedin a successful creation of a shared ebit generated be-tween Alice and Bob when our local rule is employed.We observe that only the repeaters lying in a small spa-tial region surrounding the straight line joining Alice andBob are used significantly. This observation motivates amulti-flow spatial-division rule, in which we divide thenetwork between the two flows, as shown in Fig. 6(a).Any repeater in the red shaded region follows the localrule tied to the Alice 1 - Bob 1 flow while repeaters in thegreen region operate with the local rule tied to the Alice 2- Bob 2 flow. The placement of the boundary determinesthe rates R and R . The rate region attained is plottedwith the yellow line in Fig. 6(b). This significantly out-performs time sharing. The two flows can co-exist andoperate with a very small reduction from their individ-ual best rates, because the repeaters they respectivelybenefit from the most form almost disjoint sets.In the other extreme, we consider two Alice-Bob pairs,still separated by six hops, but with their shortest pathscrossing as shown in Fig. 6(c). The rate region attainedby multi-flow time sharing, shown by the line segmentBC, still provides an improvement over single-flow time-sharing, shown by the line segment AD, as shown inFig. 6(d). It is interesting to note that the maximum R under multi-flow time sharing (point B) is slightlylower than maximum R with the single-flow time-sharerule (point A). This happens because unlike in singleflow time-share, the nodes at Alice 2 and Bob 2 do notcontribute to R under multi-flow time-share. A pointalong AB represents time sharing between the strategiesat points A and B. To further increase the rate, we adopta multi-flow spatial division strategy in which nodes inthe red region are configured to assist flow 1 and nodes inthe green region are configured to assist flow 2. Varyingthe angle θ demarcating those regions results in the rateregion shown by the yellow line in Fig. 6(d). This time,the improvement due to the spatial-division rule is notas pronounced, since the spatial regions corresponding to‘useful’ repeater nodes for the two flows are not disjoint. III. CONCLUSIONS
We proposed and analyzed quantum repeater protocolsfor entanglement generation in a quantum network in anarchitecture that uses the same elements as in linear re-peater chains. We accounted for channel losses betweenrepeater nodes and the probabilistic nature of entangle-ment swaps at each repeater stemming from device inef-ficiencies as well as the probabilistic nature of Bell-statemeasurements (e.g., due to inherent limitations of us-ing linear optics and lossy detectors). The rate attainedfor a single entanglement-generation flow can far outper-form that attainable over a linear repeater chain, evenwhen the nodes only have local link-state knowledge, dueto the multi-path routing advantage. We also proposeda modified version of our routing protocol for support-ing simultaneous entanglement generation flows betweenmultiple Alice-Bob pairs. We found multi-flow entangle-ment routing strategies that outperform the rate regionattained when each repeater simply time shares amongeach flow. Our results suggest that building and connect-ing quantum repeaters in non-trivial network topologies could provide a substantial benefit over linear repeaterchains alone. Seen another way, given constraints on thenumber and quality of quantum memories, link lossesbetween nodes, and limited and imperfect processing ca-pabilities at repeater nodes, a 2D network topology canoutperform the repeater-less rate-vs.-distance upper lim-its [10, 11] more easily than a linear repeater chain con-necting the communicating parties.Our work has also opened a number of new questions.Even in our simplified model—an abstraction that ap-plies when the only source of imperfection at each com-ponent (including the quantum memories) is pure loss—the rate-optimal protocol remains open. Since our pro-tocol only requires a quantum memory to hold a qubitfor one entanglement attempt between neighboring sta-tions, photon loss would indeed be the major source ofimperfection in many implementations of the protocol.Accounting for more general errors would require purifi-cation of entanglement [28–30], i.e., converting severalpoorer-quality EPR pairs into a few good ones using lo-cal quantum operations and classical communication, ac-counting which will require us to introduce the Fidelity ofshared entanglement at intermediate steps of the proto-col. Furthermore, we restricted our analysis to the sameoperation used in the nodes of a linear repeater chain: 2-qubit measurements. Being able to perform multi-qubitunitary operations and multi-qubit measurements at re-peater nodes (e.g., a 3-qubit GHZ projection across threelocally held qubits) may improve the achievable rate re-gions. The idea of using a distance metric to choosethe measurements at the repeater station could be usedin protocols that use measurements of more than twoqubits as well. Finally, it will be interesting to considerrepeater protocols for the distillation of multi-partite en-tanglement shared between more than two parties, and arepeater network that can support multiple simultaneousflows of generation of multi-partite entanglement.
ACKNOWLEDGMENTS
SG would like to thank Stefano Pirandola, ZacharyDutton, and Dongning Guo for valuable discussions. MP,DE and LJ acknowledge support from the Air Force Of-fice of Scientific Research MURI (FA9550-14-1-0052) andthe Army Research Laboratory (ARL) Center for Dis-tributed Quantum Information (CDQI). SG, HK, PB,MP and DE would like to acknowledge the Office of NavalResearch program
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Lecture Notes in Computer Science (in-cluding subseries Lecture Notes in Artificial Intelligenceand Lecture Notes in Bioinformatics) , Vol. 5555 LNCS(2009) pp. 622–633, arXiv:0908.1457.[45] T. Satoh, F. Le Gall, and H. Imai, Physical Review A , 032331 (2012).[46] T. Satoh, K. Ishizaki, S. Nagayama, and R. Van Meter,Physical Review A , 032302 (2016).[47] K. Lemr, K. Bartkiewicz, A. ˇCernoch, and J. Soubusta,Physical Review A , 062333 (2013).[48] S. Pirandola, R. Garc´ıa-Patr´on, S. L. Braunstein, andS. Lloyd, Physical review letters , 050503 (2009).[49] M. M. Wilde, M. Tomamichel, and M. Berta, IEEETransactions on Information Theory , 1792 (2017),arXiv:1602.08898. Appendix A: Distance metric for the local routingrule using L norm and recursion Our entanglement routing protocol with local link-state information uses the ‘distance’ of neighboring re-peater stations from Alice and Bob to decide which mem-ories at a repeater should undergo entanglement swapattempts. The results presented in the paper use the L norm as the distance metric. While the L norm can beeasily calculated for the square grid, it may not be eas-ily generalizable for other (e.g., non-planar) topologies.Further, even though we do not prove the rate optimalityof our local link-state routing protocol, given a networktopology, it is not clear whether or not the L norm isthe optimal distance metric to be used in our protocol.In order to adapt our algorithm for arbitrary networktopologies, and also to find a near-optimal distance met-ric for our algorithm, we employ the following numericalrecursive method. Our evaluation begins with calculatingR L ( n , n ), the entanglement generation rate achievedwhen our local rule is used to route entanglement betweennodes n and n , using the L norm as the distance met-ric. In Fig. 8, we plot R L ( n , n ) as a function of ( X, Y ), FIG. 8. Entanglement generation rates with different distancemetrics. R L and R L are evaluated using the L and L norms respectively. The distance metric for R i1 (iteration 1)is calculated using R L , and R i2 (iteration 2) is calculatedusing R i1 . R i2 and R L are nearly indistinguishable as theycoincide. where X and Y are the distance (in hops) between n and n along the horizontal and vertical dimensions of thesquare grid, respectively. The rate-distance scaling expo-nent for R L is worse than that of R L , the rate attainedby our protocol, using the L norm as the distance metric.Next, for every repeater node n , we define distances d A and d B to Alice A and Bob B respectively, with respectto the following new distance metric (let us name thismetric i d A := 1 / R L ( n , A ) and d B := 1 / R L ( n , B ).We then calculate R i ( n , n ), the entanglement gener-ation rate achieved when our local rule is used with the i n and n . In Fig. 8, we plot R i ( n , n )as a function of ( X, Y ). We see that the rate-distancescaling achieved by R i is even lower than that of R L .However, when we go through the second iteration of thealgorithm—i.e., define distance metric i
2, under which d A = 1 / R i ( n , A ) and d B = 1 / R i ( n , B ), and use our lo-cal rule to evaluate R i ( n , n ) as a function of ( X, Y )—we find that the resulting rate R i is almost the same(visually indistinguishable in the plot) as R L , the ratewe obtained directly when using the L norm as the dis-tance metric. This suggests that: (a) for the square grid(and presumably for any planar network topology) the L norm metric might be near-optimal for use within ourlocal rule, and that (b) for any given network topology,one could potentially pre-compute the optimal distancemetric by a recursive strategy on the given topology us-ing the L norm as the starting point. However, thereare instances where our local rule does not give the rate-optimal local routing rule. As an example, when p = 1and q = 1, it is possible to find four disjoint paths withoutany link-state knowledge (the links are all deterministic)and the optimal rate is four ebits/cycle for any location2 FIG. 9. f ( p, q ) /pq quantifies the improvement in the scal-ing of R loc ( p, q ) with respect to R lin ( p, q ) with respect to theAlice-Bob Manhattan distance, n . f ( p, q ) /pq increases as p isreduced in [1 , p c ] but changing q has a negligible effect. of Alice and Bob. However, the fact that we are trying toroute every flow through the best possible path withoutany coordination between different flows leads to colli-sions, which results in a rate that is below the optimalrate of four ebits/cycle. Finding the rate-optimal localrouting rule across different parameter values is left forfuture research. Appendix B: Multipath rate advantage1. Numerical Evaluation
The goal this subsection is to quantify the improve-ment in the rate-vs.-distance exponent achieved by ourlocal rule over that of a linear chain along the short-est path, for all possible pairs of values of p and q .Fig. 4(c) shows this improvement, i.e., that of R loc ( p, q )compared to R lin ( p, q ), for p = 0 . q = 0 .
9. Clearly, R lin ( p, q ) = ( pq ) n ( p ) /q ∼ (1 /q )[ pq ] n , where n is the Man-hattan distance between Alice and Bob. We have nu-merically verified that R loc ( p, q ) ∼ g ( p, q )[ f ( p, q )] n for n large. We hence quantify the rate improvement by nu-merically evaluating the ratio f ( p, q ) / ( pq ) exhaustivelyfor all ( p, q ) ∈ [0 , × [0 , ◦ with respect to thegrid axes. We see that f ( p, q ) /pq increases as p decreasesin [ p c , q has a negligible effect on thisratio. FIG. 10. Network used to prove the lower bound on entangle-ment generation rate with our local routing rule which showsthat scaling of the rate with Alice-Bob manhattan distancefor our rule is better than the scaling of the rate along a linearrepeater chain along the shortest path between Alice-Bob.
2. Analytical lower bound on the rate achieved bythe local routing rule
In this subsection, we derive an analytical lower boundon the entanglement generation rate attained by our localrouting rule (using the L norm as the distance metric),with the objective of demonstrating multi-path routingadvantage, i.e., the rate-vs.-distance scaling attained byour local rule is strictly better than that attained by a lin-ear repeater chain along the shortest path between Aliceand Bob.Consider routing entanglement between Alice and Boblocated at ( X, Y ) and ( X + n, Y ) respectively, i.e., n hopsapart along the X dimension of the square lattice. Wewill evaluate a lower bound on R loc by only evaluatingthe rate contributions from paths in which all the (exter-nal) links belong to the set of black dashed links shownin Fig. 10. The choice of internal links made at repeaternodes proceed as usual per our local rule. As a result,there are instances in which our local rule routes en-tanglement through paths comprising not just the blacklinks, resulting in flows that do not contribute to our ratelower bound.We will refer to Fig. 10 for the ensuing discussion.Recall that external links succeed (are ‘up’) with prob-ability p and fail (are ‘down’) with probability 1 − p ,whereas internal links succeed with probability q . Con-sider P ( A ↔ v ), the probability that there is a path be-tween Alice A and repeater v that uses only black links. P ( A ↔ v ) includes the probability of making the requiredinternal links to create a path between A and v , but notthe probability of any internal links at the end points A or v . It is easy to see that in any given time step, there canbe no more than one edge-disjoint path between A and v along the black dashed links, since link 8 must be part ofthe path. Let l (cid:16)(cid:101) l (cid:17) be the event that the external link l is up (down). Further, note that at any given time step,of all the possible (0, 1 or 2) internal links attemptedby our local rule at a repeater node, only one internallink, if successful, contributes to A ↔ v . Let l − m be theevent that the internal link attempted at a repeater nodeto connect external links l and m is successful. If links31 and 8 are both up, node u attempts to connect thosetwo links based on our local rule, regardless of the otherlinks. If links 2, 3, 4 and 8 are up, but 1, 5, 6, 7 and9 are down, u attempts to connect 4 and 8, z attemptsto connect 3 and 4 and y attempts to connect 2 and 3.Considering these two possibilities, we have P ( A ↔ v ) > Pr(1 , , , , , , (cid:101) , (cid:101) , (cid:101) , (cid:101) , (cid:101) , , , (cid:2) p + p (1 − p ) q (cid:3) pq = p (cid:48) pq, (B1)where p (cid:48) = p + p (1 − p ) q > p . P ( v ↔ x ) is the probability that there is a path be-tween v and x that uses only black links (the probabilityof internal link successes at the end points v and x are notincluded). P ( v ↔ x ) and P ( A ↔ v ) are not independentevents because they both involve link 9. P ( v ↔ x | A ↔ v )is the probability that there exists a path along blackdashed lines between v and X given that a path alongblack dashed lines exists between A and v . We now showthat P ( v ↔ x | A ↔ v ) > P ( v ↔ x ). P ( v ↔ x | A ↔ v ) = P ( v ↔ x | A ↔ v, | A ↔ v ) + P (cid:16) v ↔ x | A ↔ v, (cid:101) (cid:17) Pr (cid:16)(cid:101) | A ↔ v (cid:17) = P ( v ↔ x | | A ↔ v ) + P (cid:16) v ↔ x | (cid:101) (cid:17) Pr (cid:16)(cid:101) | A ↔ v (cid:17) = P ( v ↔ x | (cid:16) − Pr (cid:16)(cid:101) | A ↔ v (cid:17)(cid:17) + P (cid:16) v ↔ x | (cid:101) (cid:17) Pr (cid:16)(cid:101) | A ↔ v (cid:17) = Pr (cid:16)(cid:101) | A ↔ v (cid:17) × (cid:16) P (cid:16) v ↔ x | (cid:101) (cid:17) − P ( v ↔ x | (cid:17) + P ( v ↔ x |
9) (B2)where P ( v ↔ x | A ↔ v,
9) = P ( v ↔ x |
9) and P (cid:16) v ↔ x | A ↔ v, (cid:101) (cid:17) = P (cid:16) v ↔ x | , (cid:101) (cid:17) because link 9 be-ing up or down is the only probabilistic event that influ-ences both P ( A ↔ v ) and P ( v ↔ x ). Further, P ( v ↔ x ) = P ( v ↔ x | P (cid:16) v ↔ x | (cid:101) (cid:17) Pr (cid:16)(cid:101) (cid:17) = P ( v ↔ x | (cid:16) − Pr (cid:16)(cid:101) (cid:17)(cid:17) + P (cid:16) v ↔ x | (cid:101) (cid:17) Pr (cid:16)(cid:101) (cid:17) = Pr (cid:16)(cid:101) (cid:17) (cid:16) P (cid:16) v ↔ x | (cid:101) (cid:17) − P ( v ↔ x | (cid:17) + P ( v ↔ x | . (B3)Comparing B2 and B3, Pr (cid:16)(cid:101) | A ↔ v (cid:17) =Pr (cid:16)(cid:101) (cid:17) Pr (cid:16) A ↔ v | (cid:101) (cid:17) / Pr ( A ↔ v ) > Pr (cid:16)(cid:101) (cid:17) becausePr (cid:16) A ↔ v | (cid:101) (cid:17) > Pr ( A ↔ v ) following equation B1.Similarly, (cid:16) P (cid:16) v ↔ x | (cid:101) (cid:17) − P ( v ↔ x | (cid:17) >
0. Hence, P ( v ↔ x | A ↔ v ) > P ( v ↔ x ).From Fig. 10, we can see that in order to get a pathalong black dashed lines from A to x , there must be apath along black dashed lines from A to v and from v to x , and the internal link at v must succeed. Therefore, P ( A ↔ x ) = P ( A ↔ v ) qP ( v ↔ x | A ↔ v ) > P ( A ↔ v ) P ( v ↔ x ) q = ( P ( A ↔ v )) q = ( p (cid:48) pq ) q, (B4)where we use symmetry between A ↔ v and v ↔ x inthe third line. Repeating this for all repeaters betweenAlice and Bob, it is easy to see that R loc > P ( A ↔ B ) > p (cid:48)(cid:100) n/ (cid:101) p (cid:98) n/ (cid:99) q n − (B5) ≥ (cid:16)(cid:112) p (cid:48) p (cid:17) n q n − = (cid:104)(cid:16)(cid:112) p (cid:48) p (cid:17) q (cid:105) n q − = ( pq ) βn q − , where (cid:100) n/ (cid:101) is the smallest integer greater than or equalto n/ (cid:98) n/ (cid:99) is the largest integer smaller than orequal to n/
2. The second inequality uses the fact that p (cid:48) > p and n > β = log (cid:2)(cid:0) √ p (cid:48) p (cid:1) q (cid:3) / log [ pq ] < p < p (cid:48) < q < R loc > ( pq ) βn q − with β < R lin = ( pq ) n q − , the exponent in the scaling with n is smaller in R loc compared to R lin , i.e. the rate-vs.-distance scaling is better with multi-path routing. Usinga similar reasoning, it is easy to see that the same is trueeven when Alice and Bob are at located at different Y coordinates. It should be noted that the lower boundwe derive here is not meant to be tight (see Section B 1for a full numerical evaluation of the exponents for R loc and R linlin