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All photonic quantum repeaters

Koji Azuma, Kiyoshi Tamaki, Hoi-Kwong Lo

arXiv:1309.7207v1quant-ph

TL;DR

Long-distance quantum communication is hindered by photon loss, and conventional repeaters are widely believed to require demanding matter quantum memories. This paper introduces time-reversed all-photonic repeaters based on flying qubits and shows that their communication efficiency scales polynomially with channel distance.

  • Problem

    Photon losses cause communication efficiency to decrease exponentially with distance, while conventional quantum repeaters require demanding matter quantum memories.

  • Method

    The paper proposes a time-reversed all-photonic repeater using cluster-state flying qubits, single-photon sources, loss-tolerant measurements, and fast active feedforwards.

  • Results

    The all-photonic quantum repeater has communication efficiency that scales polynomially with channel distance.

  • Takeaways & Limitations

    Quantum repeaters can be pursued using efficient single-photon sources rather than matter quantum memories.

Abstract

from arXiv · show

Quantum communication holds promise for unconditionally secure transmission of secret messages and faithful transfer of unknown quantum states. Photons appear to be the medium of choice for quantum communication. Owing to photon losses, robust quantum communication over long lossy channels requires quantum repeaters. It is widely believed that a necessary and highly demanding requirement for quantum repeaters is the existence of matter quantum memories at the repeater nodes. Here we show that such a requirement is, in fact, unnecessary by introducing the concept of all photonic quantum repeaters based on flying qubits. As an example of the realization of this concept, we present a protocol based on photonic cluster state machine guns and a loss-tolerant measurement equipped with local high-speed active feedforwards. We show that, with such an all photonic quantum repeater, the communication efficiency still scales polynomially with the channel distance. Our result paves a new route toward quantum repeaters with efficient single-photon sources rather than matter quantum memories.

I. INTRODUCTION

Photon losses make long-distance quantum communication inefficient, while conventional quantum repeaters rely on demanding matter quantum memories. The paper proposes time-reversed all-photonic repeaters using flying qubits instead.

  • Motivation: Photon transmission efficiency decreases exponentially with communication distance because optical fibers introduce losses.This limits direct quantum communication to hundreds of kilometers.
  • Conventional requirements: Conventional quantum repeaters use matter quantum memories to generate adjacent entanglement and perform heralded entanglement swapping.The memories must couple to photons and preserve entanglement while heralding signals are exchanged.
  • Conventional limitations: Finite memory coherence and probabilistic Bell measurements can make simple repeater protocols scale exponentially with the square root of communication distance.Long-distance heralding signals cause waiting, during which matter memories undergo dephasing or depolarization.
  • Conventional requirements: Matter-qubit repeater proposals can require universal-computation capabilities plus scalability, making them potentially more difficult than universal quantum computation.Efficient photon–matter coupling also remains challenging for many matter qubits.
  • Proposed approach: The paper introduces all-photonic quantum repeaters using flying qubits, single-photon sources, linear optics, detectors, and fast active feedforward.Heralding signals remain within repeater nodes, so their transmission distance and time can be zero in principle.
  • Proposed approach: The protocol applies time reversal to conventional repeaters, performing entanglement swapping before entanglement generation and using cluster-state flying qubits.The paper identifies this time-reversed construction as an innovative part of the proposal.
  • Result: The all-photonic protocol follows polynomial scaling because it is the time-reversed version of a conventional quantum repeater with polynomial scaling.This is the paper’s stated scaling conclusion.

II. CONVENTIONAL QUANTUM REPEATERS

Conventional quantum repeaters synchronize probabilistic entanglement preparations before swapping, but the required heralding communication creates waiting times that constrain memories and repetition rates. The proposed time-reversed approach can make this waiting time zero in principle.

  • Entanglement swapping: Entanglement swapping connects two shorter entangled pairs into one longer pair by performing a Bell measurement at their shared node.Pairs separated by l can produce a pair separated by 2l.
  • Conventional protocol: Conventional repeater rounds repeatedly prepare specific entangled pairs until entanglement generation and earlier swapping steps succeed.This repetition reflects the probabilistic nature of the preparation process.
  • Parallel preparation: Parallel preparation synchronizes entanglement successes across many qubits, but heralding transmission still takes at least adjacent-node communication time.With probabilistic swapping, the waiting can extend to the communication time across the total distance.
  • Protocol limitation: Because heralding is probabilistic, conventional repeaters require memory time during waiting, limiting their repetition rate.The limitation arises from transmitting heralding signals before subsequent operations.
  • Time-reversed approach: A time-reversed all-photonic repeater addresses heralding delays by arranging the protocol so the waiting time could be zero in principle.For photonic qubits, eliminating waiting also avoids the corresponding loss during storage.

III. TIME-REVERSED ALL-PHOTONIC QUANTUM REPEATERS

The protocol replaces stationary repeater memories with flying-qubit cluster states, using photonic Bell measurements, loss-tolerant measurements, and local active feedforward to connect distant nodes.

  • Cluster-state construction: The complete-like cluster state uses 2m arms, each containing a 1st-leaf and 2nd-leaf qubit.The 1st-leaf qubits correspond to candidate memory qubits, while 2nd-leaf photons mediate connections between adjacent nodes.
  • Time-reversed construction: Because every pair of 1st-leaf qubits is already entangled, the protocol avoids a separate Bell measurement to identify candidate qubits.The remaining task is to perform X-basis measurements selected by later heralding signals.
  • Receiver operation: A successful Bell measurement on neighboring 2nd-leaf photons entangles their corresponding 1st-leaf qubits; failure instead triggers Z-basis measurements that remove those arms.This converts probabilistic photonic connections into a cluster-state procedure that can continue using other arms.
  • Loss tolerance: Loss-tolerant encoded measurements can succeed with arbitrarily high probability when physical-qubit loss is below 50%, corresponding to 15 km of optical fiber.The repeater therefore adjusts adjacent-node transmission distances so 1st-leaf-qubit loss remains below this threshold.
  • Measurement requirements: The measurement scheme is especially suitable because the repeater requires only faithful Z-basis and X-basis measurements, unlike universal optical computation, which also needs non-Pauli measurements.For the repeater application, the scheme can remain robust against general errors while avoiding the larger overhead associated with non-Pauli measurements.
  • Network layout: The repeater alternates source and receiver nodes, with source nodes preparing encoded cluster states and sending left and right arms to adjacent receivers.The protocol then applies Bell measurements to the received 2nd-leaf photon pairs and loss-tolerant X- or Z-basis measurements to the 1st-leaf qubits.

IV. APPLICATIONS

The all-photonic repeater supports QKD and other protocols whose entanglement is consumed to produce classical outputs without quantum memories. For strictly quantum outputs, memory is still needed, but the required time scales linearly with distance.

  • QKD operation: Each repeater trial requires only one round of local active feedforward because the subsequent loss-tolerant X- and Z-basis measurements require no active feedforward.Alice and Bob’s raw key can be treated as measurements on entangled pairs prepared before the repeater protocol.
  • Classical-output protocols: The repeater supports QKD, non-local measurements, and position-based cryptography protocols without quantum memories when entanglement is consumed immediately to generate classical output strings.Pauli corrections can be handled offline during classical communication.
  • Quantum-output protocols: Strictly quantum output states still require effectively quantum memories for a duration comparable to the classical communication time between Alice and Bob.This boundary applies to applications such as quantum teleportation when Bob must retain the final state.
  • Quantum-output protocols: In quantum teleportation, the required memory time scales linearly with communication distance L, unlike the polynomial or subexponential scaling stated for conventional repeaters.The paper associates this scaling with greater suppression of quantum-memory errors.

V. SCALING AND PERFORMANCE

The protocol’s average photon consumption scales polynomially with total distance, while its entangled-pair production rate is set by the slowest photonic device.

  • Scaling: The average total photon number consumed to produce an entangled Alice–Bob pair scales polynomially with the total distance.This is the protocol’s principal long-distance efficiency result.
  • Performance: The average entangled-pair production rate is on the order of the repetition rate of the slowest single-photon source, detector, or active-feedforward device.Performance analysis evaluates photon consumption, rate, and fidelity under fiber transmission and small connection errors.

VI. DISCUSSION

The paper argues that quantum repeaters can be built entirely from photonic components, reducing requirements relative to matter-memory approaches while retaining a rigorously analyzed cluster-state implementation.

  • Discussion: The proposed repeater removes demanding matter-memory requirements and uses single-photon sources, linear optics, detectors, and fast active feedforward.The paper presents this as evidence that all-photonic repeaters can be easier than universal quantum computation.
  • Discussion: The paper concludes that this is the first rigorous proof that a quantum repeater is much simpler than a quantum computer.The authors identify further work including improved sources, telecom-wavelength experiments, noise robustness, and general network topologies.
  • Discussion: The detailed protocol analysis prepares encoded complete-like cluster states locally and uses Varnava et al.’s encoding for nearly deterministic, faithful X- and Z-basis measurements under loss and general errors.The analysis includes state preparation and a comparison with the speediest protocol of Munro et al.

1. Basic results [27]

The loss-tolerant measurement replaces a measured qubit with a tree cluster encoding and uses indirect measurements to recover outcomes despite photon loss. Recursive success probabilities can approach unity when the individual loss probability is below 50%.

  • Basic encoding: The encoded qubit is the root of a tree cluster state, with advance X-basis measurements linking each 1st-level qubit to the qubits formerly connected to it.The example uses branching parameters b0 = b1 = 2.
  • Success probability: PL is obtained by recursively solving the success-probability equations, and can be made arbitrarily close to unity when ǫ0 < 0.5.The recursion uses Rl+1 := 0 and bl+1 := 0 as boundary conditions.
  • Indirect measurements: Indirect Z-basis measurement infers a lost qubit’s outcome from a neighboring X-basis measurement and Z-basis measurements on its descendants.The inferred outcome is obtained from the parity of the neighboring X observable and descendant Z observables.
  • Parallel recovery: Parallel independent schemes allow indirect Z-basis measurement to succeed when at least one neighboring scheme succeeds.Their independence also supports majority voting to improve measurement fidelity.

2. Error analysis

The error analysis models independent depolarization of tree-cluster qubits and evaluates loss-tolerant Z-, X-, and general-observable measurements. Majority voting suppresses errors for Z and X measurements, but not generally for arbitrary observables.

  • Error model: The model assumes independent depolarizing channels on all tree-cluster qubits and evaluates their effects on loss-tolerant Z- and X-basis measurements.For a Pauli measurement, the direct measurement error is related to the depolarization error ed through em.
  • Z-basis analysis: Majority voting combines independent indirect Z-basis estimates to improve the guessing probability and suppress depolarization-induced errors.The average error probabilities are derived recursively from the indirect-measurement relations.
  • X-basis analysis: Loss-tolerant X-basis measurement succeeds by identifying one parity among the 1st-level branches, whose stabilizer backaction matches direct X-basis measurement on the encoded qubit.The remaining branches become decoupled from the connected qubits after the parity is identified.
  • General observables: PL ≤ PZ and PL ≤ PX for general observable ˆA(α), while its average error probability is of order em because direct ˆA(α) measurement contributes directly.Z- and X-basis errors are instead greatly suppressed by majority voting.
  • Scope of robustness: The protocol’s special robustness applies to Z- and X-basis measurements, whereas quantum computation additionally requires general single-qubit measurements.This distinction is identified as a notable difference between quantum repeaters and quantum computing.

3. Numerical examples

Numerical examples show that tree encodings can achieve very high Z- and X-measurement success under loss and small measurement error, while general-observable measurement remains substantially less successful.

  • Numerical examples: For ǫ0 ≃ 0.20 and em = 2.8 × 10−5, the {16, 14, 1} encoding uses QL = 464 qubits and gives 1 − PZ = 1.8 × 10−6 and 1 − PX = 5.9 × 10−5.The alternative {11, 11, 1} encoding uses QL = 253 qubits and gives 1 − PZ = 2.5 × 10−5 and 1 − PX = 3.2 × 10−4.
  • Numerical examples: For the same parameters, the general loss-tolerant measurement has 1 − PL = 0.43 for the {16, 14, 1} encoding and 1 − PL = 0.35 for the {11, 11, 1} encoding.

Appendix B: Preparation of an encoded complete-like cluster state | ¯Gm

The encoded complete-like cluster state is prepared locally from photonic tree and star-like cluster states using single-photon sources, linear optics, detectors, and active feedforward. Its preparation resource bound scales polynomially with photon number and is independent of Alice–Bob distance.

  • Resource requirements: The expected photon number and preparation time depend on source efficiency, detector efficiency, fusion success, and feedforward timing.The preparation uses single-photon sources, linear optical elements, photon detectors, and high-speed active feedforward.
  • State construction: The encoded star-like state replaces each 1st-leaf qubit with the root of a QL-qubit tree cluster state, enabling loss-tolerant measurements on those leaves.The root and 0-level qubits receive advance X-basis measurements to complete the encoding.
  • State construction: The encoded star-like state is similar to a tree cluster state with branching parameters {2m, 2, b0, . . . , bl} and can be prepared using the Varnava et al. protocol.The construction begins with three-qubit GHZ states, forms 2-trees by type-II fusion, and builds the target tree state.
  • Scaling and locality: The preparation bound scales polynomially with the total photon number of the encoded state, 2m(QL + 3) + 2.The local preparation makes the total photon number required independent of the distance between Alice and Bob.
  • State conversion: A Y-basis measurement on the redundantly encoded root, followed by local unitaries and X-basis measurements on specified leaves, prepares the encoded complete-like cluster state.The Y-basis measurement corresponds to a Bell measurement on a two-dimensional subspace.
  • Operating assumptions: The repeater analysis assumes effective photon loss ǫ0 < 0.5 and individual depolarization during transmission between adjacent nodes.The loss threshold is imposed on photons in the local preparation or adjacent-node transmission.

1. Total photon number and entanglement generation rate of our repeater protocol

The protocol produces Alice–Bob entanglement using photonic cluster states and loss-tolerant measurements, while its photon consumption scales polynomially with distance. Its rate is set by the slowest device, and teleportation use introduces endpoint memory requirements.

  • Entanglement generation: A successful protocol run leaves Alice’s and Bob’s qubits entangled through X-basis measurements at all receiver nodes.Success requires every receiver node to have at least one successful Bell measurement.
  • Entanglement generation rate: The entanglement-generation rate ¯R is governed by f, the repetition rate of the slowest single-photon source, detector, or active-feedforward device.The protocol can increase f by operating single-photon sources faster rather than adding parallel devices.
  • Application flexibility: For quantum key distribution, the protocol requires no quantum memory, whereas general teleportation requires memories at Alice and Bob.Teleportation itself requires sending Alice’s Bell-measurement outcome to Bob.
  • Application flexibility: Strategy (b) starts teleportation after heralding and minimizes photon consumption, while strategy (a) is effective for short endpoint-memory coherence times.For strategy (b), the required memory time includes the classical communication time from Alice to the receiver node adjacent to Bob.
  • Application flexibility: At L = 1000 km, the protocol requires about 10 ms of memory time in strategy (b) when used for teleportation.The paper contrasts this with additional classical-communication factors in conventional memory-based repeaters.
  • Photon consumption: The average total photon number ¯Q required to produce an entangled pair scales polynomially with distance L.This follows from n = L/L0 − 1.

2. Error analysis for our repeater protocol

The error analysis tracks how loss-tolerant Z- and X-basis measurement errors and transmission depolarization propagate to the final Alice–Bob entangled pair. Under stated robustness and fidelity assumptions, the resulting error increase is polynomial in distance.

  • Error propagation: The analysis derives error probabilities for the entangled pair AB obtained when the repeater protocol succeeds.The success configuration is represented as a transformed cluster state with depolarized 2nd-leaf qubits.
  • Z-basis measurements: Errors in loss-tolerant Z-basis measurements on 2(m−1) 1st-leaf qubits produce a phase-flip channel on the remaining 1st-leaf qubits.The average channel factor is (1−2¯eZ)^(2m−2).
  • X-basis measurements: Errors in adjacent loss-tolerant X-basis measurements propagate through correlated measurement outcomes to phase flips on the neighboring 2nd-leaf qubits.The outcome on qubit a correlates with Z_B, while the outcome on b correlates with Z_A.
  • Final pair: The final entangled pair’s errors combine measurement-induced phase flips with depolarization affecting the 2nd-leaf qubits.The protocol’s success state can be transformed into the configuration used to specify the final error channels.
  • Distance scaling: When loss-tolerant measurement errors remain sufficiently small and adjacent-node entanglement fidelity is sufficiently high, transmission depolarization increases final-pair errors only polynomially with L.The approximation uses (1−2¯eZ)^((2m−2)n) ≃ 1 and (1−2¯eX)^(2n) ≃ 1.

3. Numerical examples

Numerical examples evaluate photon consumption, success probability, rate, and fidelity under specified optical and device assumptions, and compare the protocol with Munro et al.’s scheme. The reported examples reach long distances with polynomial-scaling resource estimates.

  • Evaluation setup: The numerical study estimates ¯Q, P, ¯R, ¯EZ, ¯EX, ¯EY, and ¯F for four parameter cases and compares the protocol with Munro et al.’s speediest protocol.The comparison uses assumptions from the updated version of Munro et al.’s work.
  • Numerical results: For L = 5000 km, one case achieves ¯Q = 7.6 × 10^7, P = 0.65, and ¯R = 65 kHz.The parameters include L0 = 8 km, ηDηS = 0.95, τa = 150 ns, and m = 27.
  • Numerical results: For L = 1000 km, the corresponding case achieves ¯Q = 7.3 × 10^6, P = 0.60, and ¯R = 60 kHz.The same passage reports ¯EY = 7.6 × 10^−3 and ¯F = 0.97 for this case.
  • Numerical results: For the 5000-km and 1000-km cases, the reported average fidelities are ¯F = 0.89 and ¯F = 0.97, respectively.The 5000-km case reports ¯EY = 3.3 × 10^−2, while the 1000-km case reports ¯EY = 7.6 × 10^−3.
  • Comparison: Munro et al.’s comparison assumes 97% photon–matter-qubit coupling and, for L0 = 6.15 km, 2.5 × 10^6 consumed photons per trial.The setup corresponds to 129 repeater nodes and 19,500 matter quantum memories per node.
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