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Quantum Repeater with Encoding

Liang Jiang, Jacob M. Taylor, Kae Nemoto, William J. Munro, Rodney Van Meter, Mikhail D. Lukin

arXiv:0809.3629v2quant-ph

TL;DR

Long-distance quantum communication suffers exponentially decreasing key-generation rates from fiber attenuation. The paper proposes encoded Bell-pair generation with small CSS codes, simultaneous encoded entanglement connection, and classical error correction; it reports communication over thousands to millions of kilometers with fast key generation, subject to final-station memory assumptions for on-demand Bell pairs.

  • Problem

    Fiber attenuation makes quantum-key-generation rates decrease exponentially with distance, motivating repeaters that reduce this scaling while preserving practical communication rates.

  • Method

    The protocol fault-tolerantly generates encoded Bell pairs, connects them simultaneously at intermediate stations, and uses classical error correction with CSS codes.

  • Results

    F* = 0.999 and L* ≈ 730 are achieved with q = 0.3% using the [[23, 1, 7]] Golay code.

  • Takeaways & Limitations

    The protocol can provide secure quantum communication over thousands or millions of kilometers with key generation above 100 bits/sec using simple CSS encoding.

  • Takeaways & Limitations

    On-demand distant Bell-pair generation requires good quantum memory at the final stations with coherence time longer than the communication time.

Abstract

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We propose a new approach to implement quantum repeaters for long distance quantum communication. Our protocol generates a backbone of encoded Bell pairs and uses the procedure of classical error correction during simultaneous entanglement connection. We illustrate that the repeater protocol with simple Calderbank-Shor-Steane (CSS) encoding can significantly extend the communication distance, while still maintaining a fast key generation rate.

I. INTRODUCTION

The paper addresses distance-dependent losses in quantum communication by proposing an encoded quantum repeater that uses small CSS codes and classical error correction. It aims to extend communication to 10^3–10^6 km while maintaining key generation above 100 bits/sec with logarithmically scaling local resources.

  • Motivation: Fiber attenuation causes quantum-key-generation rates to decrease exponentially with distance, motivating quantum repeaters.Repeaters store intermediate quantum states to reduce this scaling to polynomial scaling.
  • Contribution: The proposed repeater encodes logical qubits with small CSS codes and performs entanglement connection at the encoded level.Classical error correction boosts the fidelity of the connection while avoiding long-distance entanglement purification and quantum error correction.
  • Results: 10^3–10^6 km communication distance can be reached while maintaining a key generation rate above 100 bits/sec.The protocol uses finite local resources rather than requiring unbounded station resources.
  • Results: 30–150 qubits per station scale logarithmically with communication distance.This resource scaling supports the proposed long-distance operating regime.
  • Organization: The paper develops idealized, repetition-code, and general CSS-code repeater protocols before analyzing their fidelity and key-generation performance.The CSS-code treatment is intended to suppress both bit-flip and dephasing errors.

II. FAST QUANTUM COMMUNICATION WITH IDEAL OPERATIONS

The idealized repeater creates neighboring Bell pairs, connects them simultaneously through intermediate Bell measurements, and determines the final Pauli frame from announced classical outcomes. This establishes a fast one-way baseline while motivating treatment of entanglement, operational, and memory imperfections.

  • Idealized protocol: Neighboring Bell pairs are independently generated and verified, then connected by Bell measurements at intermediate stations.Each intermediate station announces two classical bits that determine the Pauli frame of the remaining pair.
  • Parallel connection: Entanglement connection can be applied simultaneously at all intermediate stations because Bell-measurement circuits do not depend on the Pauli frame.The Pauli frame is selected after collecting the announced classical outcomes.
  • Idealized protocol: L = 5 repeater stations illustrate generation, connection, and Pauli-frame determination for creating one remote Bell pair.Intermediate stations each hold two physical qubits and are measured in the Bell basis during connection.
  • Key generation: The idealized repeater supports secret-key extraction when the outer stations measure in matching random X or Z bases.The cycle time is then set by entanglement generation rather than sequential long-distance connection.
  • Imperfections: Three practical imperfections remain: imperfect generated entanglement, local-operation errors, and quantum-memory decoherence.Memory errors are modeled through a storage-time-dependent error probability, while local gates and measurements have their own error probabilities.
  • Encoded extension: The new protocol replaces physical qubits with encoded qubits, generates encoded Bell pairs, connects them simultaneously, and determines an encoded Pauli frame.The construction targets the imperfections identified in the idealized analysis.

III. QUANTUM REPEATER WITH REPETITION CODE

The repetition-code repeater demonstrates encoded Bell measurement using three-qubit blocks and classical majority correction. It suppresses single bit-flip errors while retaining simultaneous encoded connections and a final encoded Bell pair.

  • Repetition-code construction: A 3-qubit repetition code encodes one logical qubit and illustrates the encoded repeater construction.The example is designed to correct bit-flip errors rather than all error types.
  • Scope: The repetition code corrects one bit-flip error but cannot fix all imperfections considered in the idealized protocol.The example supplies the key protocol elements for generalization to CSS codes.
  • Encoded generation: Each station uses three memory qubits and three ancillary qubits in the illustrated encoded-generation procedure.Ancillary Bell pairs implement transversal teleportation-based CNOT gates between encoded states.
  • Encoded connection: Encoded Bell measurement applies three pairwise CNOT gates between the two encoding blocks at an intermediate station.The physical qubits are then measured in complementary bases to infer the logical outcomes.
  • Error correction: Majority voting identifies and corrects one erroneous output when a physical qubit in the measured block suffers a bit-flip.Only classical error correction is required for this correction step.
  • Encoded connection: Encoded connections at all intermediate stations remain simultaneous, producing an encoded Bell pair whose Pauli frame is determined by 2(L −2) announced classical bits.Classical correction yields an effective logical error probability described as Q ∼ q^(t+1) in the generalized construction.

IV. QUANTUM REPEATER WITH CSS CODE

The protocol generalizes from repetition coding to [[n, k, 2t + 1]] CSS codes that correct bit-flip and dephasing errors. Transversal encoded operations and classical decoding produce low-error logical Bell measurements with logarithmically scalable station resources.

  • CSS-code framework: A [[n, k, 2t + 1]] CSS code encodes k logical qubits into n physical qubits and corrects up to t bit-flip and dephasing errors.Examples include the [[5, 1, 3]], [[7, 1, 3]], and [[9, 1, 3]] codes for k = 1.
  • CSS-code framework: CSS-code logical X and Z measurements are obtained from physical measurements in the corresponding bases and classical error correction.The classical codes C_X and C_Z correct up to t errors in the measured output bits.
  • Encoded operations: Encoded CNOT gates use n pairwise CNOT gates that do not propagate errors within an encoding block.The same transversal construction supports preparation of encoded Bell pairs.
  • Resource requirements: Approximately 6n physical qubits per station are required, including 2n memory qubits and about 4n ancillary qubits.The ancillary qubits support fault-tolerant encoded-state preparation and neighboring-station encoded Bell-pair generation.
  • Protocol cycle: Encoded Bell pairs are generated between neighboring stations using fault-tolerant initialization, purified physical Bell pairs, and n teleportation-based CNOT gates.The resulting encoded pairs are then connected through simultaneous encoded Bell measurements at intermediate stations.
  • Logical error suppression: Classical correction yields encoded Bell-measurement outcomes with high accuracy and an effective logical error probability scaling as O(...).Each intermediate station communicates two classical bits, after which the final stations choose the encoded Pauli frame.

V. ERROR ESTIMATE

The protocol evaluates encoded Bell-pair fidelity by accounting for logical errors across repeater connections and uses CSS codes to suppress effective errors. For sufficiently low physical error probability, larger codes can make the logical error arbitrarily small and support scalable long-distance communication.

  • Entanglement fidelity: The generalized entanglement fidelity accounts for small errors outside the logical subspace and calibrates protocol security.It can bound information leaked from the final stations and can be obtained from correlation measurements.
  • Error model: Fault-tolerant initialization, transverse CNOT gates, and encoded measurements make individual physical-qubit errors effectively uncorrelated.The effective physical-qubit error probability is then estimated from the operational error model.
  • Logical error suppression: The effective logical error probability is caused by more than t errors in an encoding block and is suppressed by the code.For encoded Bell measurements, the protocol obtains Q ∼ q^(t+1) under the stated small-error approximation.
  • Connection limit: For q < 0.03, the maximum number of connections scales as 1/q^(t+1) at target fidelity F* = 0.95.This scaling is estimated for various CSS codes from the fidelity and connection formulas.
  • Fidelity scaling: For small Q, the final entanglement infidelity satisfies 1 − F ≈ 2LQ, linking repeater length to accumulated logical errors.The fidelity expression treats logical errors at repeater stations as affecting the final encoded Bell pair.
  • Scalability: Numerically, qc ≈ 5% corresponds to approximately 1% per-gate error rates, and CSS codes with n ≲ 19t exist for arbitrarily large t.These results support a scalable approach to long-distance quantum communication.

VI. EXAMPLE IMPLEMENTATIONS

The implementation estimates maximum connection counts, communication distances, local resources, and key-generation rates for encoded repeater designs. Under the stated parameters, CSS encoding reaches intercontinental-scale distances while retaining a fast key-generation rate.

  • Implementation estimates: Given q and target fidelity F*, Eq. (12) determines the maximum number of connections, which defines a unitless distance scale for creating Bell pairs.The estimates use Eq. (13) and compare different CSS codes at F* = 0.95.
  • Local resources: Table I estimates 6n qubits per station and uses q = 0.3%, F* = 0.95, and l0 = 10 km to calculate maximum communication distances.Double-bracket CSS codes suppress both bit-flip and dephasing errors, whereas repetition codes suppress only one error type.
  • Maximum distance: At q = 0.3%, L* is approximately 9 without encoding, 1.4×10^2 with the [[7, 1, 3]] Hamming code, and 3.7×10^4 with the [[23, 1, 7]] Golay code.With l0 = 10 km, these correspond to 90 km, 1.4×10^3 km, and 3.7×10^5 km, respectively.
  • Key-generation rate: The cycle time is approximately τc ≈ 0.9κ ms for l0 = 10 km, attenuation length about 20 km, propagation speed about 2×10^5 km/s, and η ≈ 0.3.κ represents the time overhead required to obtain n purified Bell pairs between neighboring stations.
  • Key-generation rate: Approximately 6n qubits per station give τc ≈ 7 ms and a quantum key-generation rate of 100 bits/sec over long distances under the considered parameters.These parameters include κ ≈ 8 for β = δ = 10^-3 and F0 = 0.95, with purified-pair fidelity 0.9984 after three purification levels.

VII. DISCUSSION

The protocol replaces distant entanglement purification with local CSS encoding and classical error correction, enabling fast long-distance operation. Its resource requirements and achievable distances depend on code choice and initialization complexity.

  • Protocol advantages: The protocol is faster because local CSS encoding and classical error correction replace time-consuming distant entanglement purification.It operates in one-way communication mode, so the key generation rate is independent of communication distance and is limited by encoded Bell-pair generation and connection.
  • Rate improvements: Using CSS codes with k > 1, higher efficiency, improved F0, smaller station separation, or more station qubits can further increase the key generation rate.The protocol is described as supporting higher-rate operation through these design changes.
  • Code scaling: CSS codes can yield encoding blocks whose size grows only logarithmically with communication distance.Asymptotically, codes with n ≲19t support arbitrarily large t, with n ∝t ∼ln L when q ≲5%.
  • Code choices: Repetition codes offer small encoding blocks and efficient initialization; at q = 0.3% and F* = 0.95, L* is estimated as 1.0 × 10^3 for 3-qubit and 1.0 × 10^5 for 5-qubit codes.These estimates concern repetition codes used when dephasing errors dominate.
  • Code choices: The [[23, 1, 7]] Golay code can achieve F* = 0.999 at L* ≈730 with q = 0.3%.The resulting high-fidelity entanglement is discussed for quantum state teleportation and distributed quantum computation.
  • Operational conditions: On-demand distant Bell-pair generation requires good quantum memory at the final stations and incurs a classical-communication delay l0L/c.The stated condition is coherence time longer than the communication time.

VIII. CONCLUSION

The paper proposes a fast CSS-encoded quantum repeater for intercontinental quantum key distribution. It reports long-distance secure communication with a key rate above 100 bits/sec under the stated error and fidelity targets.

  • VIII. CONCLUSION: The protocol fault-tolerantly generates a backbone of CSS-encoded Bell pairs and applies classical error correction during entanglement connection.The protocol targets quantum key distribution over intercontinental distances.
  • VIII. CONCLUSION: Secure quantum communication can reach thousands or even millions of kilometers with 0.3% effective error probability per physical qubit and 0.95 target fidelity.These values are reported for the simple CSS-code protocol.
  • VIII. CONCLUSION: The quantum key generation rate can exceed 100 bits/sec and is limited by Bell-pair generation between neighboring stations.The stated rate limitation concerns the neighboring-station generation process.

APPENDIX A: EFFECTIVE ERROR PROBABILITY

The appendix defines an effective physical-qubit error probability by combining imperfections across preparation, gates, and entanglement connection. It also describes CSS-code initialization and resource requirements for suppressing correlated and uncorrelated errors.

  • Error model: The effective error probability q estimates the chance of a wrong physical-qubit output during entanglement connection.It combines imperfections from entanglement generation and entanglement connection, including β, δ, and µ.
  • Error model: Bit-flip and phase errors are characterized separately because the relevant operations do not mix the two error types.Z-basis measurements are sensitive to bit-flip errors, while the protocol tracks probabilities (b, p).
  • CSS-code background: CSS-code stabilizers define the subspace storing logical information through +1 eigenvalues of all stabilizer generators.The appendix describes separate products of X and Z operators and corresponding logical operators.
  • Initialization: The first preparation approach uses state distillation to suppress correlated X and Z errors in initially non-fault-tolerant logical-state copies.The correlated multi-qubit error probability can occur at order O(ε) before distillation.
  • Initialization: State distillation suppresses correlated errors and drives uncorrelated errors toward a steady per-qubit value of order β + δ.The distillation operation does not introduce new correlated errors, according to the passage.
  • Initialization: Fault-tolerant preparation can use stabilizer measurements with GHZ states, followed by state distillation to suppress errors.The second approach avoids correlated errors initially by preparing physical qubits in a product state and measuring stabilizers.
  • Encoded operations: A generalized encoded CNOT propagates stabilizer eigenvalues alongside logical qubits during encoded operations.The control outputs are (x1, z1z2), while the target outputs are (x1x2, z2).
  • Resource requirements: For two encoding blocks and GHZ-state ancillas, each repeater station requires 4n+nGHZ qubits.The resource count includes 2n memory qubits and additional blocks for fault-tolerant preparation and two-level distillation.

APPENDIX C: ENTANGLEMENT FIDELITY AND CORRELATION

Final encoded Bell-pair fidelity and secret-key correlation are reduced by intermediate Bell-measurement errors and unsuccessful local error correction. Their scaling is governed by the effective connection error probability Q.

  • Entanglement fidelity: The final entanglement fidelity is estimated as F ≈(1 −Q)^(2L−2) ≳(1 −Q)^(2L).Intermediate Bell-measurement errors and unsuccessful local error correction are identified as the two fidelity-reducing sources.
  • Secret-key correlation: Secret-key correlation is approximately C ≈(1 −Q)^L because only half of the intermediate classical bits affect X- or Z-basis keys.Successful classical error correction contributes a factor of order (1 −Q)^2.

APPENDIX D: TIME OVERHEAD AND FAILURE PROBABILITY FOR ENTANGLEMENT PURIFICATION

The appendix relates purification failure probability to the number of starting Bell pairs and shows that increasing this number reduces failure probability.

  • More unpurified Bell pairs N0 generally reduce the failure probability Pfail when producing n purified Bell pairs.The calculation also depends on the initial fidelity F0 and local-operation error probabilities β and δ.
  • Pfail decreases exponentially with N0 after N0 surpasses a threshold for n = 7 and n = 23.

1. Failure Probability

The failure-probability analysis builds purified Bell-pair distributions level by level, then evaluates performance and the resources needed to generate them.

  • Failure Probability: The protocol calculates the number distribution of purified Bell pairs obtained from N0 unpurified Bell pairs before evaluating Pfail.
  • Failure Probability: Purified Bell pairs are distinguished by purification level, with each level-(i + 1) pair formed from two level-i pairs.Level-0 pairs are unpurified, while level-l pairs are used for non-local CNOT gates.
  • Failure Probability: 0.9984 is the reported level-3 purified-pair fidelity for F0 = 0.95 and β = δ = 10−3 under depolarizing error.
  • Failure Probability: N0/n ≈15 is sufficient to ensure Pfail < 10−5 across a wide range of n.
  • Time Overhead and Key Generate Rate: Each entanglement-generation attempt takes l0/v and succeeds with probability η^2e−l0/latt, determining the unpurified-pair generation rate R.
  • Time Overhead and Key Generate Rate: Approximately 6n qubits per station can achieve τc ≈7 ms, sufficient for a quantum key generation rate of 100 bits/sec over long distances.
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