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Efficient long distance quantum communication

Sreraman Muralidharan, Linshu Li, Jungsang Kim, Norbert Lütkenhaus, Mikhail D. Lukin, Liang Jiang

arXiv:1509.08435v1quant-ph

TL;DR

Efficient quantum communication beyond 1000 km is limited by fiber attenuation and accumulated operation errors. The paper classifies quantum repeaters into three generations and systematically compares their temporal and physical resource costs across experimental parameters. It identifies parameter regions where different repeater architectures are most favorable, providing guidance for quantum-network design.

  • Problem

    Efficient quantum communication over distances exceeding 1000 km remains challenging because fiber attenuation and accumulated operation errors limit transmission.

  • Method

    The paper classifies quantum repeaters by their loss- and operation-error-correction methods and compares them using a cost function for temporal and physical resources.

  • Results

    For high gate error probability, the first generation dominates; intermediate regimes favor the second generation without encoding, while favorable coupling, speed, and gate-error conditions favor the third generation.

  • Takeaways & Limitations

    The identified parameter regions provide guidance for selecting quantum-repeater architectures and designing efficient long-distance quantum networks.

Abstract

from arXiv · show

Despite the tremendous progress of quantum cryptography, efficient quantum communication over long distances (>1000km) remains an outstanding challenge due to fiber attenuation and operation errors accumulated over the entire communication distance. Quantum repeaters, as a promising approach, can overcome both photon loss and operation errors, and hence significantly speedup the communication rate. Depending on the methods used to correct loss and operation errors, all the proposed QR schemes can be classified into three categories (generations). Here we present the first systematic comparison of three generations of quantum repeaters by evaluating the cost of both temporal and physical resources, and identify the optimized quantum repeater architecture for a given set of experimental parameters. Our work provides a roadmap for the experimental realizations of highly efficient quantum networks over transcontinental distances.

I. INTRODUCTION

Long-distance quantum communication is hindered by photon loss and operation errors, motivating quantum repeaters that divide transmission into shorter segments. The paper classifies repeaters into three generations and compares their temporal and physical resource costs across experimental regimes.

  • Motivation: Fiber attenuation causes loss errors that make quantum communication rates decay exponentially with distance, while quantum states cannot be amplified without disturbance.Operation errors from channels, measurements, and gates add a second challenge over long distances.
  • Motivation: Quantum repeaters divide communication into shorter segments and use active mechanisms at repeater stations to correct loss and operation errors.Loss correction can use heralded entanglement generation or quantum error correction.
  • Three generations: The three repeater generations differ in how they suppress loss and operation errors, with each performing best in a specific regime of local gate speed, gate fidelity, and coupling efficiency.The paper evaluates these generations to identify efficient architectures for different parameter regimes.
  • Three generations: The first generation uses heralded entanglement generation and purification, reducing direct-transfer overhead from exponential to polynomial but retaining polynomially decreasing rates with distance.Temporal, spatial, and frequency multiplexing can boost its communication rate.
  • Three generations: The second generation combines heralded entanglement generation with quantum error correction, replacing purification and avoiding two-way signaling between non-adjacent stations.Its rate remains limited by adjacent-station signaling and local gate operations.
  • Three generations: The third generation uses quantum error correction for both loss and operation errors, needs only one-way signaling, and can achieve rates limited mainly by local operation delay.The second and third generations are faster over long distances but require higher-fidelity gates and operate within stricter loss and error constraints.
  • Comparison framework: The comparison uses a cost function based on qubit memories required to achieve a target transmission rate, capturing both temporal and physical resources.The cost coefficient represents qubits × time for creating one secret bit over 1 km and depends on attenuation, coupling efficiency, gate error, and gate time.

A. Coupling efficiency

Coupling efficiency strongly influences which quantum-repeater generation is optimized: the third generation is favored at high efficiency, while HEG-based generations remain preferable below about 90%. Gate errors, distance, and gate speed further shift the optimized choice.

  • Coupling efficiency: ηc above 90% gives the third generation an obvious advantage by eliminating two-way classical signaling.Below approximately 90%, the optimization selects the first or second generation, and C′ is proportional to ηc^-2 for HEG protocols.
  • Coupling efficiency: 10^-2 gate error compromises the third generation’s ability to correct loss, whereas HEG-based first- and second-generation repeaters work well at low coupling efficiencies.This contrast reflects the third generation’s dependence on QEC across the lossy channel.
  • Gate speed: For fast gates, the third generation has the lowest optimized cost coefficient, with C′ proportional to t0; for slower gates, the second generation becomes optimal with nearly constant C′.The transition occurs as local gate time becomes comparable to two-way signaling delay between adjacent stations.
  • Gate fidelity: The first generation has an operation-error threshold up to about 3%, while the second and third generations have thresholds of approximately 1%.The second generation’s threshold is slightly lower because teleportation-based non-local CNOT and entanglement-swapping operations require extra gates.
  • Gate fidelity: At ηc = 100% and t0 = 1µs, the third generation is optimized for moderate gate errors, while lower coupling efficiencies favor the first and second generations.For 1000km and 10000km, the second generation is more favorable below gate-error transitions of about 0.8% and 0.6%, respectively.

IV. OPTIMUM GENERATION OF QRS

The optimized quantum-repeater generation depends strongly on coupling efficiency, gate time, and gate-error probability. The resulting parameter-space classification links these regimes to architectural choices and experimental implementations.

  • Parameter-space optimization: The bubble plot maps optimized repeater protocols by parameter-space region, with bubble color indicating protocol and diameter proportional to cost coefficient.The comparison varies coupling efficiency, gate time, and gate infidelity across the three-dimensional parameter space.
  • Parameter-space regions: For high gate-error probability (ϵG ≳1%), the first generation dominates.
  • Parameter-space regions: For intermediate or low gate-error probability with poor coupling efficiency or slow local operation, the second generation is more favorable under the stated regimes.The paper distinguishes encoded and non-encoded second-generation regimes according to the operational parameters.
  • Parameter-space regions: For high coupling efficiency, fast local operation, and low gate-error probability, the third generation becomes the most favorable scheme by cost coefficient.
  • Experimental implications: The classification provides a guideline for matching repeater architectures to physical platforms, including atomic ensembles, trapped ions, NV centers, quantum dots, and nanophotonic devices.The analysis uses simplified standard HEG, HEP, CSS-code, and teleportation-based QEC procedures; more realistic cost functions and protocol variants remain possible.

Supplementary Material

The supplementary material models two-qubit gate and measurement imperfections, photon-mediated elementary entanglement generation, and memory-coherence requirements across repeater generations.

  • Operation and measurement errors: Gate infidelity ϵG models a desired two-qubit operation occurring with probability 1 −ϵG and a maximally mixed output otherwise.
  • Operation and measurement errors: Measurement infidelity ξ is the probability of a wrong projective measurement outcome.
  • Operation and measurement errors: An ancillary-qubit comparison converts mismatched measurement outcomes into loss errors and yields a reduced effective measurement error when outcomes match.
  • Memory requirements: The analysis assumes perfect memory qubits, while first-generation repeaters require coherence times approaching the two-way Alice–Bob communication time.
  • Memory requirements: Second-generation repeaters have substantially less demanding coherence times, whereas third-generation repeaters are not limited by two-way communication time.
  • Elementary entanglement generation: Neighboring-station entanglement generation uses two-photon detection, with coupling efficiency ηc accounting for photon emission, fiber transfer, and final detection.

Fidelity

This section represents imperfect elementary entangled pairs in the Bell basis, describes purification and its resource trade-offs, and identifies fidelity limits imposed by imperfect operations and measurements.

  • Bell-state representation: Imperfect neighboring-station entangled pairs are represented by a density matrix in the Bell-state basis, with fidelity defined from its Bell-state components.
  • Purification: First- and second-generation repeaters purify elementary pairs before connecting them, trading reduced pair numbers for higher fidelity.
  • Fidelity limits: Imperfect quantum operations and measurements impose an upper bound on purified-pair fidelity that depends on the raw-pair density matrix, ϵG, ξ, and the purification protocol.
  • Fidelity limits: The calculations assume elementary pairs already reach the asymptotic fidelity and exclude the temporal and physical resources consumed during elementary-level purification.
  • Purification protocols: Deutsch purification reaches higher fidelities with fewer rounds, while D¨ur purification saves qubits by retaining an auxiliary pair but costs time and restarts failed rounds.
  • Purification protocols: The purification analysis computes success probability and the Bell-basis diagonal elements of the output state from two input density matrices under ϵG and ξ.

c) Entanglement Swapping

Entanglement swapping extends elementary links across repeater levels, but imperfect operations and input states reduce fidelity; the first-generation optimization varies nesting, purification, and protocol choices.

  • c) Entanglement Swapping: Entanglement swapping deterministically connects adjacent Bell pairs into pairs spanning twice the distance at each nesting level.
  • c) Entanglement Swapping: Deterministic swapping is crucial because otherwise the probability of entangling qubits separated by Ltot decreases exponentially with distance.
  • First-generation repeater optimization: First-generation repeaters address photon loss with HEG and operation errors with HEP by dividing Ltot into 2^n segments.
  • c) Entanglement Swapping: Imperfect CNOT operations, measurements, and input Bell pairs reduce swapped-pair fidelity, potentially requiring purification at every level.
  • First-generation repeater optimization: The optimization varies nesting level N, purification rounds M⃗, and the Deutsch-versus-D¨ur protocol choice for a fixed gate error rate ϵG.
  • First-generation repeater optimization: The resource calculation allows arbitrary purification rounds, uses the four-state secure-fraction protocol, and includes deterministic swapping and gate-operation time t0.

Deutsch et al. entanglement purification protocol

The Deutsch-protocol resource analysis accounts for purification time, signaling, success probabilities, secure key rate, and memory-qubit cost in the repeater architecture.

  • Temporal resources: The temporal resource TDeu for one raw-key bit is assembled from Bell-pair preparation, purification signaling, gate-operation time, and purification success probabilities.
  • Secure key rate: The secure key generation rate Rsecure is obtained from the temporal resource and the asymptotic secure fraction.
  • Secure key rate: The four-state protocol estimates the secure fraction from QX/Z through the average QBER and binary entropy function h(Q).
  • Physical resources: Physical resource is measured by the number of memory qubits consumed at half a repeater station.
  • Overall cost: The cost function combines the temporal and physical resource measures used to compare repeater schemes.

D¨ur et al. entanglement purification protocol

The D¨ur purification protocol’s temporal resource accounts for Bell-pair preparation, gate operations, and two-way signaling, with success probabilities varying by purification round and nesting level.

  • Bell-pair preparation time includes an additional entanglement-pumping term in AD¨ur[i].
  • BD¨ur[i] includes gate-operation time and two-way classical signaling for confirming purification success.
  • PD¨ur(i, j) denotes the success probability of purification round i at nesting level j.
  • Secure-key-rate and cost-function derivations follow the approach used in the preceding section.

b) Second generation without encoding

Second-generation repeaters without encoding generate elementary entangled pairs between neighboring stations and use entanglement swapping, with resources optimized over memory, spacing, and generation-trial parameters.

  • For gate error rates ϵ > 10^-3 and distances around 10^3 km, encoding may be unnecessary and elementary-pair generation with swapping can save resources.
  • The optimization varies memory qubits per half station N, neighboring-station spacing L0, and elementary entanglement-generation rounds nE.G.
  • The secure key-generation rate is expressed in terms of the elementary-generation success probability, trial count, and timing quantities.
  • At least one neighboring-station entangled pair occurs with probability 1 − Prob(0, nE.G.).
  • Frequency multiplexing during flying-qubit transmission and spatial multiplexing during entanglement swapping may be incorporated.

c) Second generation with encoding

Second-generation repeaters with encoding create encoded Bell pairs between neighboring stations and perform encoded entanglement swapping, using error-correcting codes to suppress physical imperfections.

  • Encoded Bell pairs are generated between neighboring stations before encoded entanglement swapping across repeater stations.
  • Any [[N, k, 2t + 1]] CSS code corrects up to t X-errors and t Z-errors.
  • The optimization considers the Steane [[7,1,3]], Golay [[23,1,7]], and QR [[103,1,19]] codes.

d) Third generation QRs

Third-generation repeaters relay encoded qubits between stations and perform local teleportation-based error correction, with performance analyzed under independent-error assumptions and bounded code sizes.

  • Third-generation repeaters perform one memory–fiber upload and download, giving neighboring-photon transmission probability ηc e^-L0/Latt.
  • Teleportation-based error correction is performed locally within every repeater station rather than through encoded entanglement swapping.
  • The analysis assumes errors do not propagate between repeater stations and each qubit has an independent error.
  • Majority voting can herald failure, correctly decode, or incorrectly decode the logical qubit.
  • The success probability accounts for avoiding heralded failure across all R repeater stations.
  • The cost function for (n, m) quantum parity codes depends on local-operation time t0 and the number of repeater segments.
  • The comparison restricts third-generation repeaters to at most 200 qubits and searches quantum parity codes with 2 ≤ (m, n) ≤ 20.
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