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Adaptive Entanglement Distillation

Sijie Cheng, Narayanan Rengaswamy

TL;DR

This paper introduces an adaptive distillation framework for quantum repeaters that alternates among stabilizer codes across distillation rounds to improve end-to-end rate and fidelity. By coupling QEC-based distillation with an efficiency metric that jointly accounts for rate and distillable entanglement, it identifies regimes where switching codes yields performance gains over fixed schemes. The analysis also extends to non-QEC purification (DEJMPS/BBPSSW) and to hybrid 1G+2G strategies, revealing “checkpoints” where the optimal protocol changes as network operating points vary. The results highlight conditions under which QEC-based distillation is advantageous and emphasize the importance of adaptive design for second-generation repeaters, with practical implications for network planning and resource budgeting. The framework provides a path toward scalable, adaptive purification strategies across realistic repeater networks and motivates further fault-tolerant and noise-robust implementations.

Abstract

Quantum network applications impose a variety of requirements on entanglement resources in terms of rate, fidelity, latency, and more. The repeaters in the quantum network must combine good methods for entanglement generation, effective entanglement distillation, and smart routing protocols to satisfy these application requirements. In this work, we focus on entanglement distillation in a linear chain of quantum repeaters. While conventional approaches reuse the same distillation scheme over multiple hop lengths after entanglement swaps, we propose a novel adaptive quantum error correction (QEC) scheme that boosts end-to-end metrics. Specifically, depending on the network operating point, we adapt the code used in distillation over successive rounds to monotonically increase the rate while also improving fidelity. We demonstrate the effectiveness of this strategy using three codes, with parameters [[9,1,3]], [[9,2,3]], [[9,3,3]], and a new performance metric, efficiency, that incorporates both overall rate and fidelity. Since the minimum input fidelity for QEC-based distillation is high, we then extend our study to include non-QEC-based purification protocols, specifically DEJMPS since it outperforms others. We compare the performance of end-to-end DEJMPS against adapting from DEJMPS to QEC once DEJMPS improves the initial fidelity to the threshold for QEC. Through a refined efficiency metric, we illuminate the regime where QEC is beneficial. These results provide a detailed outlook for entanglement purification and distillation in first and second generation quantum repeaters.

Adaptive Entanglement Distillation

TL;DR

This paper introduces an adaptive distillation framework for quantum repeaters that alternates among stabilizer codes across distillation rounds to improve end-to-end rate and fidelity. By coupling QEC-based distillation with an efficiency metric that jointly accounts for rate and distillable entanglement, it identifies regimes where switching codes yields performance gains over fixed schemes. The analysis also extends to non-QEC purification (DEJMPS/BBPSSW) and to hybrid 1G+2G strategies, revealing “checkpoints” where the optimal protocol changes as network operating points vary. The results highlight conditions under which QEC-based distillation is advantageous and emphasize the importance of adaptive design for second-generation repeaters, with practical implications for network planning and resource budgeting. The framework provides a path toward scalable, adaptive purification strategies across realistic repeater networks and motivates further fault-tolerant and noise-robust implementations.

Abstract

Quantum network applications impose a variety of requirements on entanglement resources in terms of rate, fidelity, latency, and more. The repeaters in the quantum network must combine good methods for entanglement generation, effective entanglement distillation, and smart routing protocols to satisfy these application requirements. In this work, we focus on entanglement distillation in a linear chain of quantum repeaters. While conventional approaches reuse the same distillation scheme over multiple hop lengths after entanglement swaps, we propose a novel adaptive quantum error correction (QEC) scheme that boosts end-to-end metrics. Specifically, depending on the network operating point, we adapt the code used in distillation over successive rounds to monotonically increase the rate while also improving fidelity. We demonstrate the effectiveness of this strategy using three codes, with parameters [[9,1,3]], [[9,2,3]], [[9,3,3]], and a new performance metric, efficiency, that incorporates both overall rate and fidelity. Since the minimum input fidelity for QEC-based distillation is high, we then extend our study to include non-QEC-based purification protocols, specifically DEJMPS since it outperforms others. We compare the performance of end-to-end DEJMPS against adapting from DEJMPS to QEC once DEJMPS improves the initial fidelity to the threshold for QEC. Through a refined efficiency metric, we illuminate the regime where QEC is beneficial. These results provide a detailed outlook for entanglement purification and distillation in first and second generation quantum repeaters.

Paper Structure

This paper contains 41 sections, 2 theorems, 69 equations, 24 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Stabilizer Formalism
  4. Error Syndromes
  5. Depolarizing Noise Channel
  6. Werner States
  7. Distillable Entanglement
  8. Distillation based on Error Correction
  9. Protocol Rate
  10. Adaptive Error Correction Protocol
  11. Stabilizer Codes
  12. Decoder Based on Lookup Table
  13. Number of Distillation Rounds
  14. Simulation Setup
  15. Assumptions
  16. ...and 26 more sections

Key Result

Theorem 1

Consider the BBPSSW protocol. Let $P_{I_{\rm in}}=a_0$, $P_{X_{\rm in}}=b_0$, $P_{Y_{\rm in}}=c_0$, $P_{Z_{\rm in}}=d_0$, where the subscript denotes the number of rounds. Assume Then, as $n\to\infty$, we have

Figures (24)

  • Figure 1: Protocol diagram for three repeaters.
  • Figure 2: Three Rounds of Distillation. Legend entry "No Coding" means $F_{\rm out}=F_{\rm in}$, that is, no coding is used.
  • Figure 3: Protocol diagram for a single repeater, where we still perform three rounds of distillation.
  • Figure 4: Protocol diagram for five repeaters, where we perform only three rounds of distillation.
  • Figure 5: The $F_{\rm in}$-$F_{\rm out}$ map for the 0-repeater case. As an example, the plot in the $F_{\rm in}$-$F_{\rm out}$ map is shown for $\llbracket 9,1,3\rrbracket$, $\llbracket 9,2,3\rrbracket$ and $\llbracket 9,3,3\rrbracket$. Additionally, the no-coding error rates for $n=1$, $n=2$ and $n=3$ are provided as references.
  • ...and 19 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2