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Ultrafast and Fault-Tolerant Quantum Communication across Long Distances

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

arXiv:1310.5291v2quant-ph

TL;DR

Long-distance quantum repeaters must handle photon loss and operational errors while avoiding the communication overhead of remote entanglement protocols. This paper proposes small-block QPC repeaters using fault-tolerant preparation and TEC, finding poly-logarithmic resource scaling with distance and giving optimized spacing examples.

  • Problem

    Long-distance quantum communication requires repeater schemes that address photon loss, operational errors, and the rate limitations of two-way classical communication.

  • Method

    The scheme uses small QPC encoding blocks, fault-tolerant encoded-state preparation, and TEC at each repeater station to correct photon loss and operational errors.

  • Results

    The cost coefficient has poly-logarithmic scaling with total distance up to Ltot = 10^4 km, including coupling losses up to pc = 10%.

  • Takeaways & Limitations

    The scheme can tolerate coupling loss pc ≲10% and use approximately hundreds of qubits per repeater station while supporting long-distance secret-key generation.

  • Takeaways & Limitations

    The cost function excludes additional qubit overhead for fault-tolerant encoded-state preparation, whose optimal preparation scheme remains future work.

Abstract

from arXiv · show

Quantum repeaters (QRs) provide a way of enabling long distance quantum communication by establishing entangled qubits between remote locations. We investigate a new approach to QRs in which quantum information can be faithfully transmitted via a noisy channel without the use of long distance teleportation, thus eliminating the need to establish remote entangled links. Our approach makes use of small encoding blocks to fault-tolerantly correct both operational and photon loss errors. We describe a way to optimize the resource requirement for these QRs with the aim of the generation of a secure key. Numerical calculations indicate that the number of quantum memory bits required for our scheme has favorable poly-logarithmic scaling with the distance across which the communication is desired.

SUPPLEMENTAL MATERIAL

The supplemental material reviews three quantum-repeater classes and develops the fault-tolerant encoding and correction procedures underlying the proposed scheme.

  • The supplemental material surveys three classes of quantum repeaters and their approaches to long-distance communication.
  • The first two repeater classes require heralded neighboring EPR pairs and two-way classical communication, limiting key-generation rates.
  • The proposed procedure uses one-way classical communication, while encoded-state photon loss can still cause secret-key-generation failure.
  • CSS encoding supports fault-tolerant state preparation and transversal encoded CNOT gates needed for error correction at repeater stations.

FAULT-TOLERANT PREPARATION OF THE ENCODED QUANTUM STATES

The scheme fault-tolerantly prepares QPC-encoded states using GHZ-state verification and repeated stabilizer measurements, then constructs an encoded Bell state from prepared blocks.

  • FAULT-TOLERANT PREPARATION OF THE ENCODED QUANTUM STATES: QPC is a special CSS code whose logical operators and stabilizer structure enable efficient fault-tolerant state preparation.
  • FAULT-TOLERANT PREPARATION OF THE ENCODED QUANTUM STATES: The (n, m)-QPC encodes one logical qubit using nm −1 independent stabilizers, with code distance d = min(n, m).
  • FAULT-TOLERANT PREPARATION OF THE ENCODED QUANTUM STATES: The required encoded states are |0⟩L, |+⟩L, and the encoded Bell state 1/√2(|00⟩L + |11⟩L).
  • FAULT-TOLERANT PREPARATION OF THE ENCODED QUANTUM STATES: |+⟩L is prepared as n copies of m-qubit GHZ states, whose ZZ-syndrome measurements are repeated r′ times to suppress measurement errors.
  • FAULT-TOLERANT PREPARATION OF THE ENCODED QUANTUM STATES: |0⟩L is prepared by repeated measurements of Si,0 using fault-tolerantly prepared 2m-qubit GHZ states, with syndrome history used to determine errors.
  • FAULT-TOLERANT PREPARATION OF THE ENCODED QUANTUM STATES: The encoded Bell state is produced by applying transversal encoded CNOT gates between separately prepared |+⟩L and |0⟩L blocks.

TELEPORTATION-BASED ERROR CORRECTION

Teleportation-based error correction combines encoded Bell-state preparation, transversal CNOT operations, and logical measurements to correct photon loss and operational errors at each repeater station.

  • TELEPORTATION-BASED ERROR CORRECTION: TEC prepares an encoded Bell state and performs an encoded Bell measurement using fault-tolerant transversal CNOT gates followed by logical X and Z measurements.
  • TELEPORTATION-BASED ERROR CORRECTION: Successful recovery requires at least one arriving qubit in every sub-block and at least one sub-block arriving without loss.
  • TELEPORTATION-BASED ERROR CORRECTION: TEC protects against operational errors as well as photon loss errors.
  • TELEPORTATION-BASED ERROR CORRECTION: For a (3, 3)-QPC, missing photons are omitted from the CNOT, after which encoded measurements update the Pauli frame.
  • TELEPORTATION-BASED ERROR CORRECTION: Logical X measurement uses majority voting among complete sub-blocks after computing each complete sub-block's X outcome.
  • TELEPORTATION-BASED ERROR CORRECTION: Logical Z measurement similarly infers the encoded operator from sub-block outcomes and majority voting.
  • TELEPORTATION-BASED ERROR CORRECTION: At the physical level, TEC couples incoming photons, local qubits, and outgoing photons before measuring the incoming photon in X and the local atom in Z.
  • TELEPORTATION-BASED ERROR CORRECTION: Cavity-QED implementations realize TEC by decomposing CNOT into Hadamard and CPHASE gates.

ERROR MODEL & PROBABILITY DISTRIBUTIONS

The error model represents transmission loss, depolarization, gate, and measurement imperfections, then propagates their effects from physical qubit pairs to encoded measurement outcomes.

  • ERROR MODEL & PROBABILITY DISTRIBUTIONS: The model assumes independently prepared R and S blocks, with noisy CNOT operations correlating their errors during encoded Bell measurement.
  • ERROR MODEL & PROBABILITY DISTRIBUTIONS: The Pauli set {I, X, Y, Z} describes the depolarization components of the error model.
  • ERROR MODEL & PROBABILITY DISTRIBUTIONS: For transmitted qubits, η is the transmission probability, 1−η is photon-loss probability, and ϵd is the depolarization probability.
  • ERROR MODEL & PROBABILITY DISTRIBUTIONS: The effective qubit error ϵ incorporates measurement error ϵm and gate error ϵg.
  • ERROR MODEL & PROBABILITY DISTRIBUTIONS: Local S-block qubits undergo depolarization but no photon loss because they do not traverse the channel.
  • ERROR MODEL & PROBABILITY DISTRIBUTIONS: The probability analysis proceeds at the physical qubit-pair, intermediate row-pair, and logical encoded measurement levels.
  • ERROR MODEL & PROBABILITY DISTRIBUTIONS: The resulting distributions account for correlated errors and normalize the encoded error probabilities to unity.

Probability distribution for qubit-pair measurement

The qubit-pair measurement models ideal outcomes as affected by erasure, spin-flip, spin-and-phase-flip, and phase-flip errors, alongside faithful measurements. Their probabilities are constrained to sum to unity.

  • In the ideal case, qubit-pair outcomes are (ri,j, si,j), with ri,j and si,j equal to ±1.
  • With errors, outcomes become (αri,j, βsi,j), where (α, β) includes erasure and sign-flip combinations.
  • Erasure on Ri,j has probability ϵe, while spin-flip and spin-and-phase-flip errors have probabilities ϵX and ϵY, respectively.
  • ϵX is given as 1/2ηϵ, where ϵ is the effective qubit error probability.
  • The phase-flip error has probability ϵZ, and faithful measurement has probability ϵI.
  • The error probabilities satisfy ϵe + ϵI + ϵX + ϵY + ϵZ = 1, while ϵY is correlated with ϵX and ϵZ.

Probability distribution for row-pair measurement

The row-pair measurement extends the error model to outcomes with values 0 and ±1, summing over patterns of photon loss, sign errors, phase errors, and faithful measurements.

  • In the ideal case, row-pair outcomes are (ri, si), where each component is 0 or ±1.
  • With errors, outcomes become (αri, βsi), with each factor drawn from (0, ±1) ⊗ (0, ±1).
  • The probability qα,β does not depend on the row index i because all rows share the same probability distribution.
  • The row-pair distribution sums over patterns containing photon loss, spin-flip, spin-and-phase-flip, phase-flip, and faithful measurements.
  • The first output component is 0 when a ≥1; otherwise it is +1 or −1 according to the parity of c + d.
  • The second output component is determined by comparing 2(b + c) with m − a.

Probability distribution for encoded-pair

The encoded-pair measurement derives logical outcomes from row-pair error patterns, distinguishing ideal conditions, photon-loss cases, and parity-dependent sign assignments.

  • The measurement distribution is formed by summing over the possible encoded-pair error patterns.
  • In the ideal case, the encoded measurement outcomes use ˜XR and ˜ZS with values ±1.
  • With errors, the encoded outcome takes values in (0, ±1) ⊗ (0, ±1).
  • The encoded-pair measurement considers all row-pair error patterns, represented by counts a through i that sum to n.
  • The second encoded output is determined by comparing 2(b + e + h) with n − (a + d + g).
  • The first encoded output is 0 when a + b + c ≥1; otherwise its sign depends on whether g + h + i is even or odd.

OVERHEAD FROM FAULT-TOLERANT STATE PREPARATION

Fault-tolerant state preparation trades qubit overhead against time overhead, with parallel stabilizer measurement and alternative GHZ-state reuse strategies.

  • For a (4, 4) QPC, stabilizers can be measured in two parallel time steps.Rows {1, 2} and {3, 4} are measured simultaneously, followed by rows {2, 3}.
  • The alternative GHZ-reuse approach requires an overhead of 4m qubits.
  • The 4m-qubit approach scales the time overhead by a factor of (n −1) relative to the previous encoded-EPR-pair preparation scheme.
  • The cost function counts qubits required to create the encoded Bell pair but excludes additional fault-tolerant preparation qubits.
  • The best fault-tolerant preparation scheme for the QPC remains for future work to determine, considering both qubit and time overhead.

FAULT TOLERANT PROPERTIES OF QPC

The QPC-based repeater architecture suppresses encoded errors while accounting for both operational faults and photon losses. Numerical results show suppression to approximately 10^-14, including loss levels up to 10%.

  • Fault-tolerant error model: Photon loss is a defining fault-tolerance concern for quantum repeaters, alongside operational errors.The encoded error rate includes both heralded failure and quantum bit error rates.
  • Encoded-error suppression: Encoded error rates can be suppressed to approximately 10^-14 with suitable QPC encodings.The effective encoded error rate incorporates both failure and bit-error probabilities.
  • Loss tolerance: 10% loss errors remain compatible with arbitrarily suppressing the encoded error rate to approximately 10^-14.Below 10^-14, numerical errors begin to affect the calculations.
  • Code requirements: Table V lists QPC codes required to achieve an encoded error rate of approximately 2 × 10^-14 across physical-error rates and varying losses.The table concerns code choices in the presence of losses specified in percent.

DETAILS OF THE OPTIMIZATION ALGORITHM

The optimization procedure searches repeater parameters to minimize the cost coefficient for long-distance quantum communication. It begins from specified total-distance, repeater-spacing, and encoding-size values.

  • Initialization: The search starts with Ltot = 500 km, L0 = 1 km, and encoding dimensions m = n = 2.These values initialize the optimization search.
  • Optimization procedure: The algorithm searches for optimized quantum-repeater parameters using a flow chart designed to minimize the cost coefficient.The units of Ltot and L0 are kilometers and are omitted from the figure for convenience.

SCALING OF THE COST COEFFICIENT

Quantum repeaters change the cost coefficient’s distance dependence from exponential without repeaters to favorable poly-logarithmic scaling. This behavior persists numerically up to 10^4 km, including coupling losses up to 10%.

  • Distance scaling: Without a quantum repeater, the cost coefficient scales exponentially with the communication distance.This is the baseline distance dependence for the desired communication span.
  • Distance scaling: 10^4 km is the numerical distance range over which poly-logarithmic cost scaling is indicated with and without coupling losses up to pc = 10%.The scaling is examined through C′(Ltot) and its poly-logarithmic indication.
  • Photon-loss-dominated regime: O(logD)2 scaling applies when ε < 10^-3 and photon loss dominates the QBER and success probability without coupling losses.As operational-error contributions increase, the quadratic scaling breaks while poly-logarithmic behavior remains indicated.

GENERALIZED COST COEFFICIENT

The generalized cost coefficient incorporates a tunable qubit-cost exponent, allowing optimization to reflect scenarios ranging from free to fully counted qubit resources. Cheaper qubits can support larger repeater spacings in the reported examples.

  • Definition: The generalized cost coefficient introduces a constant k satisfying 0 ≤ k ≤ 1 to represent qubit-cost assumptions.The definition is intended to remain unitless and polynomial in the number of qubits.
  • Interpretation of k: k = 0 treats qubits as costless, whereas k = 1 corresponds to the original coefficient that accounts for qubit cost.Comparisons for different k values are shown in Fig. 13.
  • Repeater spacing: Cheaper qubits permit higher repeater spacings in the reported optimization results.For k = 0, the possible spacing can increase by extending the search range.
  • Numerical examples: 4.3 km spacing across 1000 km requires 800 qubits per station, while 6.3 km requires 8500 qubits per station.These estimates use ε = 10^-3 and pc = 0.
  • Numerical examples: 4.1 km spacing across 10,000 km requires 1000 qubits per station, while 5.5 km requires 9100 qubits per station.These estimates also use ε = 10^-3 and pc = 0.
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