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From Entanglement Purification Scheduling to Fidelity-constrained Multi-Flow Routing

Ziyue Jia, Lin Chen

TL;DR

An optimal entanglement purification scheduling algorithm for the single-hop case and a polynomial-time algorithm constructing an $\epsilon$-optimal fidelity-constrained path for the multi-hop case.

Abstract

Recently emerged as a disruptive networking paradigm, quantum networks rely on the mysterious quantum entanglement to teleport qubits without physically transferring quantum particles. However, the state of quantum systems is extremely fragile due to environment noise. A promising technique to combat against quantum decoherence is entanglement purification. To fully exploit its benefit, two fundamental research questions need to be answered: (1) given an entanglement path, what is the optimal entanglement purification schedule? (2) how to compute min-cost end-to-end entanglement paths subject to fidelity constraint? In this paper, we give algorithmic solutions to both questions. For the first question, we develop an optimal entanglement purification scheduling algorithm for the single-hop case and analyze the \textsc{purify-and-swap} strategy in the multi-hop case by establishing the closed-form condition for its optimality. For the second question, we design a polynomial-time algorithm constructing an $ε$-optimal fidelity-constrained path. The effectiveness of our algorithms are also numerically demonstrated by extensive simulations.

From Entanglement Purification Scheduling to Fidelity-constrained Multi-Flow Routing

TL;DR

An optimal entanglement purification scheduling algorithm for the single-hop case and a polynomial-time algorithm constructing an -optimal fidelity-constrained path for the multi-hop case.

Abstract

Recently emerged as a disruptive networking paradigm, quantum networks rely on the mysterious quantum entanglement to teleport qubits without physically transferring quantum particles. However, the state of quantum systems is extremely fragile due to environment noise. A promising technique to combat against quantum decoherence is entanglement purification. To fully exploit its benefit, two fundamental research questions need to be answered: (1) given an entanglement path, what is the optimal entanglement purification schedule? (2) how to compute min-cost end-to-end entanglement paths subject to fidelity constraint? In this paper, we give algorithmic solutions to both questions. For the first question, we develop an optimal entanglement purification scheduling algorithm for the single-hop case and analyze the \textsc{purify-and-swap} strategy in the multi-hop case by establishing the closed-form condition for its optimality. For the second question, we design a polynomial-time algorithm constructing an -optimal fidelity-constrained path. The effectiveness of our algorithms are also numerically demonstrated by extensive simulations.
Paper Structure (20 sections, 10 theorems, 47 equations, 9 figures, 2 algorithms)

This paper contains 20 sections, 10 theorems, 47 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Entanglement Purification Scheduling
  3. Technical Background on Entanglement Purification
  4. Single-hop Entanglement Purification Scheduling
  5. Multi-hop Entanglement Purification Scheduling
  6. Fidelity-constrained Entanglement Routing
  7. The Case of Single Alice-Bob Pairs
  8. The Case of Multiple Alice-Bob Pairs
  9. Performance Evaluation
  10. Entanglement Purification Scheduling
  11. Fidelity-constrained Entanglement Routing
  12. Related Works
  13. Conclusion and Perspective
  14. Proofs of theorems and lemmas
  15. Proof of Theorem \ref{['thm:opt-single-hop']}
  16. ...and 5 more sections

Key Result

Theorem 1

Let $T'$ denote the tree output by our algorithm and $l'=(b',\hat{f}',\hat{\xi}',T')$ denote the corresponding entry in $\@fontswitch\mathcal{L}$. Under the condition $\Delta\xi\le b'\xi'\epsilon$ and $\Delta f\le b'f'\epsilon$, Algorithm alg:purif2 outputs an $\epsilon-$optimal solution of Problem

Figures (9)

  • Figure 1: Quantum teleportation and entanglement swapping
  • Figure 2: We dispose $4$ EPR pairs of fidelity $0.75$. The left and right sub-figures depict $2$ purification scheduling policies producing $2$ final EPR pairs of different fidelities. Their purification success probabilities are highlighted in red font.
  • Figure 3: swap-and-purify vs. purify-and-swap
  • Figure 4: Illustration of entanglement purification qnetbook. Alice and Bob each holds one half of two EPR pairs, the first pair to be purified and the second the pair to be sacrificed. The second pair is measured. The arrows indicate classical message exchange of the measurement results. If the measurement results are the same, i.e., $00$ or $11$, the purification is successful.
  • Figure 5: Example of a purification tree
  • ...and 4 more figures

Theorems & Definitions (15)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Lemma 2
  • Lemma 3
  • Definition 1
  • Theorem 3
  • ...and 5 more