Rev. Mod. Phys. 95, 045006 (2023) - Quantum repeaters: From quantum networks to the quantum internet

Quantum repeaters: From quantum networks to the quantum internet

Koji Azuma, Sophia E. Economou, David Elkouss, Paul Hilaire, Liang Jiang, Hoi-Kwong Lo, and Ilan Tzitrin
Rev. Mod. Phys. 95, 045006 – Published 20 December 2023
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  • Introduction
  • Preliminaries
  • Quantum Repeaters
  • Milestones: Outperforming Point-to-Point…
  • Experimental Progress Toward Repeaters
  • Quantum Internet
  • Concluding Remarks
  • List of Symbols and Abbreviations
  • ACKNOWLEDGMENTS
    • References

    Abstract

    A quantum internet is the holy grail of quantum information processing, enabling the deployment of a broad range of quantum technologies and protocols on a global scale. However, numerous challenges must be addressed before the quantum internet can become a reality. Perhaps the most crucial of these is the realization of a quantum repeater, an essential component in the long-distance transmission of quantum information. As the analog of a classical repeater, extender, or booster, the quantum repeater works to overcome loss and noise in the quantum channels constituting a quantum network. Here the conceptual frameworks and architectures for quantum repeaters, as well as the experimental progress toward their realization, are reviewed. Various near-term proposals to overcome the limits to the communication rates set by point-to-point quantum communication are also discussed. Finally, the manner in which quantum repeaters fit within the broader challenge of designing and implementing a quantum internet is overviewed.

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    • Received 20 October 2021

    DOI:https://doi.org/10.1103/RevModPhys.95.045006

    © 2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum communicationQuantum memoriesQuantum networksQuantum repeaters
    1. Physical Systems
    Internet
    Quantum Information, Science & Technology

    Authors & Affiliations

    Koji Azuma*

    • NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan and NTT Research Center for Theoretical Quantum Information, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan

    Sophia E. Economou†

    • Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA

    David Elkouss‡

    • QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands and Networked Quantum Devices Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan

    Paul Hilaire§

    • Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA and Quandela SAS, 10 Boulevard Thomas Gobert, 91120 Palaiseau, France

    Liang Jiang∥

    • Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA

    Hoi-Kwong Lo

    • Quantum Bridge Technologies, Inc., 100 College Street, Toronto, Ontario M5G 1L5, Canada, Department of Physics, University of Hong Kong, Pokfulam, Hong Kong, and Center for Quantum Information and Quantum Control, Department of Physics and Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario M5S 3G4, Canada

    Ilan Tzitrin**

    • Department of Physics, University of Toronto, Toronto, Ontario, Canada

    • *koji.azuma@ntt.com
    • †economou@vt.edu
    • ‡david.elkouss@oist.jp
    • §paul.hilaire@quandela.com
    • ∥liang.jiang@uchicago.edu
    • hklo@ece.utoronto.ca
    • **itzitrin@physics.utoronto.ca

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    Vol. 95, Iss. 4 — October - December 2023

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    • Figure 1
      Figure 1

      Bloch sphere representation of a qubit. The (x,y,z) components of a Bloch vector (displayed as an arrow) give the expectation values of the Pauli observables X, Y, and Z. For instance, points (0,0,1), (1,0,0), and (0,1,0) correspond to eigenstates |0⟩,|+⟩=(|0⟩+|1⟩)/2 and |+i⟩=(|0⟩+i|1⟩)/2 of the Pauli operators Z, X, and Y with the eigenvalue of +1, respectively.

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    • Figure 2
      Figure 2

      Graphical rules for operations on graph states. The effects of Pauli operations on the connections in the graph states are shown.

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    • Figure 3
      Figure 3

      Examples of implementations of Bell measurements. (a) Bell measurement for polarization-encoded qubits, spanned by horizontally and vertically polarized single-photon states |H⟩ and |V⟩. This is implemented using the application of a 50∶50 beam splitter (BS) on optical modes, followed by a polarization beam splitter (PBS) on each of the two output modes and then by photon counting at all the output modes. Clicks in detectors DcH and DcV or in DdH and DdV project the received pair of the qubits into the Bell state |Ψ+⟩=(|H⟩|V⟩+|V⟩|H⟩)/2, while clicks in detectors DcH and DdV or in DcV and DdH project the received pair of the qubits into the Bell state |Ψ−⟩=(|H⟩|V⟩−|V⟩|H⟩)/2. Notice that this Bell measurement can succeed only when the input two optical pulses have two or more photons in total. (b) Bell measurement for Fock-encoded qubits, spanned by the vacuum state |0⟩ and the single-photon state |1⟩. This is implemented by the application of a 50∶50 BS on optical modes, followed by photon counting at the output modes. A click in the detector Dc (or Dd) at the constructive-interference (destructive-interference) mode projects the received pair of the qubits into the Bell state |Ψ+⟩=(|0⟩|1⟩+|1⟩|0⟩)/2 [|Ψ−⟩=(|0⟩|1⟩−|1⟩|0⟩)/2]. Both implementations can distinguish |Ψ±⟩ from the other states only, and the success probabilities are thus 1/2 even in the ideal cases.

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    • Figure 4
      Figure 4

      Idealized quantum repeater protocol. Three quantum repeater nodes (corresponding to the case of NQR=3) are located at regular intervals between Alice and Bob, who are separated by a distance L, with L0=L/4. The protocol starts by entanglement generation (EG) based on the application of the linear-optical Bell measurement of Fig. 3 to polarized single photons from adjacent repeater nodes with success probability pg(L0)=e−L0/Latt/2, followed by entanglement swapping (ES) with success probability ps. The EG protocol establishes a Bell pair between adjacent repeater nodes after a number of trials of the order of ⟨Tg(L0)⟩=pg−1(L0). Given halves of a pair of Bell states, the ES protocol succeeds in swapping the entanglement after a number of trials of the order of ⟨Ts⟩=ps−1. If a trial of ES fails, we need to start again from EG to go back to the trial. Therefore, in this figure the average of the total number of trials Ttot(3) needed to establish a Bell pair between Alice and Bob is ⟨Ttot(3)⟩∼⟨Tg(L0)⟩⟨Ts⟩2=ps−2pg−1(L0)=2ps−2eL/(4Latt). This is of the order of the square root of ⟨Ttot(1)⟩, which is further of the order of the square root of ⟨Ttot(0)⟩.

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    • Figure 5
      Figure 5

      First-generation repeater protocol [BDCZ scheme ([86])]. (a) In a realization based on the pumping protocol with N+1=9 nodes, the number of qubits per node is bounded by 2log22N=8. Each orange oval surrounding two vertices (or two qubits) describes an application of a Bell measurement to the two qubits for entanglement swapping. (b)–(d) Two entangled pairs with distance 1 are connected through entanglement swapping (orange oval) at node 1 to produce an entangled state with distance 2 that is stored in the qubits (as described by the purple arrows) at a higher level. (e)–(g) Another entangled state with distance 2 is produced to purify the entangled state (as described by purple arrows) stored in qubits at a higher level. Similarly, entangled states with distance 2n can be connected to produce entangled states with distance 2n+1, which can be further purified, as indicated in (a). From [255].

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    • Figure 6
      Figure 6

      Bubble plot comparing various QR protocols in the three-dimensional parameter space spanned by coupling efficiency ηc, gate error probability εG, and gate time t0 for (a) Ltot=1000  km and (b) Ltot=10 000  km. The bubble color indicates the associated optimized QR protocol, and the bubble diameter is proportional to the cost coefficient. Region plots show the distribution of different optimized QR protocols in the three-dimensional parameter space for (c) Ltot=1000  km and (d) Ltot=10 000  km. A yellow region of the second generation with encoding is contained in (c), which can be verified in a bubble plot with a finer discretization of εG. From [365].

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    • Figure 7
      Figure 7

      Left graphic: clique. Right graphic: biclique. In the clique, each vertex is connected with every other. In the biclique, each vertex from the left set is connected with a vertex on the right, but the sets are internally disconnected. These graphs can underlie repeater graph states. See Sec. 2d1 for more on graph states.

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    • Figure 8
      Figure 8

      Encoded RGS proposed by [32]. Left graphic: the two layers of the RGS. The inner layer is composed of core qubits (large red vertices, closer to the center), while the outer layer is composed of outer qubits (or leaves) (small blue vertices, farther from the center). Each vertex in the clique (left graphic) is a logical qubit, which can be encoded in, for instance, the Varnava tree code (right graphic) ([508]) to protect itself from loss (as well as general errors under the restriction of Pauli measurements). Displayed are the levels and branching parameters {b0,b1,…,bd−1} of the tree (d=2 here). Note the root and zeroth-level qubits (the two upper red qubits) in the tree will be measured out in the X basis, connecting the qubits in the first level with all of the neighbors of the root qubit. The inner logical qubits, which are conduits for the entanglement swapping, are connected to outer unencoded physical leaf qubits, which help effect the entanglement generation.

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    • Figure 9
      Figure 9

      Summary of the original all-photonic repeater scheme ([32]). Alice (A) and Bob (B) want to establish one entangled pair; each prepares m Bell pairs (m=3 here) and sends them to a nearby receiver. Repeater graph states are created at C1s and C2s and their qubits are sent to adjacent receivers C1r and C2r and the ones C2r and C3r, respectively. The receivers perform m simultaneous Bell-state measurements on the outer qubits. In every receiver node, X-basis measurements are performed on a pair of inner qubits adjacent to outer qubits, to which the Bell measurement is successfully applied, while Z-basis measurements are conducted on the other inner qubits. From [32].

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    • Figure 10
      Figure 10

      The concept of memory-assisted MDI QKD. In this protocol, once node C confirms the arrival of an optical polarization qubit either from Alice’s side or from Bob’s side with QND measurement (which is indicated by a red flag on a box labeled “QND”), it keeps it in a quantum memory (QM) until an optical polarization qubit arrives at node C from the other side, followed by its release to be subjected to Bell measurement (BM).

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    • Figure 11
      Figure 11

      Secret key rate (per pulse) of an adaptive MDI QKD protocol based on matter quantum memories with heralding storage and on Alice’s and Bob’s use of ideal single-photon sources. The secret key rate of the ideal BB84, which scales linearly with η=e−L/Latt (Latt=22  km), is also shown as a reference. T2 is the dephasing time for the matter quantum memories, 1/T is the pulse generation rate of Alice and Bob, and e11;X is the phase error rate for Alice’s and Bob’s raw keys. T2/T corresponds to how many attempts, each of which needs time T, are possible for the matter quantum memory to successfully store a single photon within its coherence time T2, that is, the allowed number m of time multiplexing in the protocol. The secret key rate scales linearly with η as long as T2/T≥(η)−1, but it then converges to η as η decreases. This is because the increase of phase error e11;X for the case of T2/T≤(η)−1 nullifies the benefit of time multiplexing from the use of matter quantum memories, as shown. From [389].

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    • Figure 12
      Figure 12

      The concept of all-photonic adaptive MDI QKD. In this protocol, node C first performs QND measurements to confirm the successful arrival of single photons (which is indicated by a red flag on a box labeled QND in the figure), followed by optical switches (SWs) to send the surviving photons to BM modules. Adapted from [33].

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    • Figure 13
      Figure 13

      Secret key rate (per pulse) G of an all-photonic adaptive MDI QKD protocol. η is rephrased by the distance L between Alice and Bob, with η=e−L/Latt (Latt=22  km) and c=2.0×108  m/s. Lines (I)–(IV) represent the performance of the protocol with active optical switches, that of the protocol with a passive Hadamard linear-optical circuit, that of the original MDI QKD protocol ([325]), and that of the TGW bound, respectively ([477]). This graph is described under the following assumptions ([33]): a single active feed forward can be completed within time τa, during which photons run in optical fibers and are subject to the corresponding photon loss; heralded single-photon sources emit pulses with duration τs and efficiency ηs, and they are multiplexed ([352]; [334]; [117]; [128]; [76]) to produce high-fidelity telecom single photons with the repetition rate of the slowest optical device at the expense of the use of at least one active feed forward; single-photon detectors have a quantum efficiency ηd and a dark count rate νd; Bell pairs for the all-photonic QND measurements can be generated in a constant time τa with single-photon sources rather than a Bell-pair photon source by paralleling a probabilistic procedure ([91]) with the active-feed-forward technique. In particular, they are assumed to be ηs=0.90 ([352]; [120]; [202]), τs=100  ps ([461]), ηd=0.93 ([341]), νd=1 s−1 ([460]; [341]), and τa=67  ns ([334]). From [33].

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    • Figure 14
      Figure 14

      Schematic of a TF-type QKD protocol ([136]). Both Alice and Bob choose the Z or X basis randomly. If the Z basis is selected, Alice and Bob prepare coherent state |α⟩ or |−α⟩ at random and send it to the central node C. If the X basis is selected, Alice and Bob prepare a phase-randomized coherent state (PRCS) whose intensity is chosen randomly from a predefined set [so as to be able to use the decoy-state method ([242]; [327]; [523])] and send it to the central node C. Upon receiving pulses from Alice and Bob, the central node C performs the Bell measurement based on single-photon interference [Fig. 3]. The secret key is distilled only from instances where both Alice and Bob choose the Z basis and the Bell measurement at node C succeeds. Adapted from [135].

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    • Figure 15
      Figure 15

      Secret key rates (per pulse) of a TF-type QKD protocol for different dark count rates pd in logarithmic scale as a function of the overall loss between Alice and Bob. The PLOB bound is the private capacity of a lossy bosonic channel ([403]). Assume a misalignment of 2% in each of the channels between Alice and the central node C and between Bob and C, and also the inefficiency function for the error-correction process f=1.16. Adapted from [136].

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    • Figure 16
      Figure 16

      Postpairing measurement-device-independent QKD. In this protocol, Alice and Bob send N pulses to the middle node C (Charlie) to perform the linear-optical Bell measurement of Fig. 3 based on single-photon interference (SPI), and a two-photon Bell state is obtained by postmatching two successful SPI events. Here n represents the number of successes of the Bell measurement based on SPI and η represents the transmittance of pure-loss channels between Alice and Charlie and between Charlie and Bob. Adapted from [534].

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    • Figure 17
      Figure 17

      Level structure and heralded entanglement generation. (a) Lambda-level structure with states |e⟩ (excited), |0⟩ (connected to |e⟩ by horizontally polarized light |H⟩), and |1⟩ (connected to |e⟩ by vertically polarized light |V⟩). (b) Level structure for time-bin entanglement, where |e⟩ is connected only to |0⟩; control of the qubit states is required. (c) Setup for spin-spin heralded entanglement generation. From [472].

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    • Figure 18
      Figure 18

      State-of-the-art cavity-QED devices. (a) A quantum dot coupled deterministically to an open Fabry-Perot cavity. From [491]. (b) A silicon-vacancy center in diamond in a photonic crystal cavity evanescently coupled to a fiber. From [69].

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    • Figure 19
      Figure 19

      A proof-of-principle experiment for an all-photonic quantum repeater. Passive choice measurement (PCM) automatically performs an entangling Bell measurement (in the case of a coincidence detection) or a disentangling local X measurement (in the case of a single-photon detection or the failure of the Bell measurement). From [310].

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    • Figure 20
      Figure 20

      Quantum networks. (a) Shanghai-Beijing QKD network. From [112]. (b) Experimental quantum network composed of N-V centers acting as quantum memories. From [408].

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    • Figure 21
      Figure 21

      Quantum and Bell-pair networks. (a) A quantum network as a graph. A quantum network can be abstracted using a directed graph G=(V,E), with V and E the sets of vertices and edges. We associate with each vertex v∈V a node in the quantum network, and with each edge e∈E a quantum channel Ne. In this example, Alice’s node A and Bob’s node B are part of a network with seven nodes that also include the intermediary nodes C1, C2, C3, C4, and C5. (b) A network of maximally entangled states. One approach to entanglement distribution between distant parties in a quantum network is the aggregated repeater protocol ([27]). In this protocol, adjacent nodes prepare maximally entangled states that can then be transformed into end-to-end entanglement between two distant parties by swapping the entanglement. The quantum network in (a) has been used to generate entanglement between adjacent nodes. Each edge is annotated with a fraction x/y, where the denominator y denotes the number of entangled pairs, while the numerator x denotes the number of entangled states used to establish entanglement between the end parties A and B. In this example, the minimum cut Δ(V) over y is given for the choice of V={A,C1,C3}⊂VA;B, and a total of eight Bell pairs could be distributed between A and B. Adapted from [27].

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    • Figure 22
      Figure 22

      Linear network and general protocol. (a) A repeater chain or linear quantum network is associated with a linear graph, i.e., a graph that can be described by a sequence of edges connecting distinct nodes. The linear network in the panel may be a subnetwork of the network shown in Fig. 21. (b) General adaptive protocol illustrated over the linear network in (a). The goal of the protocol is to distribute Bell pairs between A and B. The protocol begins with the network joint state represented by a separable state and proceeds iteratively until meeting a termination condition. On each round a node transmits a local subsystem through a quantum channel. All nodes then perform an LOCC operation. The LOCC operation, the choice of channel, and the transmitted subsystem can depend on the history of the measurement outcomes [such as k1 and k2 in (b)] of the protocol. The nodes of the linear network can be divided into two disjoint virtual nodes: VA [nodes in the left (pink) box] including A and VB [nodes in the right (green) box] including B. The intuition behind the capacity upper bounds in Eqs. (47) and (49) is that distributing entanglement between these two virtual nodes is an easier task than distributing entanglement between A and B over the network. Adapted from [28].

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    • Figure 23
      Figure 23

      Upper bound on the secret key rate achievable with a noisy linear network. In particular, the upper bound applies to a wide range of protocols that includes DLCZ ([159]) and others ([280]; [444]; [31]) when implemented with matter quantum memories in the presence of dephasing noise. The linear network consists of a chain of repeaters equally separated and connected by an optical fiber with attenuation length 22 km and spanning a total distance of L  (km). The coupling efficiency to an optical fiber is assumed to be 90%. In the calculation, the number of repeater nodes is optimized. The curves labeled (i)–(vi), respectively, correspond with the following coherence times: 1.0×10−2, 5.0×10−3, 2.5×10−3, 1.0×10−3, 5.0×10−4, and 1.0×10−4  s. The upper bound in (vi) scales better than direct transmission and is roughly proportional to the square root of the PLOB bound but equivalent to the intercity QKD protocols in Sec. 4. In consequence, with a coherence time of 1.0×10−4  s there can be no advantage for a DLCZ-type repeater scheme over the simpler intercity QKD protocols. From [28].

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