On noise in swap ASAP repeater chains: exact analytics, distributions and tight approximations

Kenneth Goodenough; Tim Coopmans; Don Towsley

doi:10.22331/q-2025-05-15-1744

On noise in swap ASAP repeater chains: exact analytics, distributions and tight approximations

Kenneth Goodenough¹, Tim Coopmans², and Don Towsley¹

¹College of Information and Computer Science, University of Massachusetts Amherst, 140 Governors Dr, Amherst, Massachusetts 01002, USA
²Leiden Institute of Advanced Computer Science, Leiden University, Leiden, The Netherlands

Published:	2025-05-15, volume 9, page 1744
Editor:	Alessio Benavoli
Eprint:	arXiv:2404.07146v5
Doi:	https://doi.org/10.22331/q-2025-05-15-1744
Citation:	Quantum 9, 1744 (2025).

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Abstract

Losses are one of the main bottlenecks for the distribution of entanglement in quantum networks, which can be overcome by the implementation of quantum repeaters. The most basic form of a quantum repeater chain is the swap ASAP repeater chain. In such a repeater chain, elementary links are probabilistically generated and deterministically swapped as soon as two adjacent links have been generated. As each entangled state is waiting to be swapped, decoherence is experienced, turning the fidelity of the entangled state between the end nodes of the chain into a random variable. Fully characterizing the (average) fidelity as the repeater chain grows is still an open problem. Here, we analytically investigate the case of equally-spaced repeaters, where we find exact analytic formulae for all moments of the fidelity up to 25 segments. We obtain these formulae by providing a general solution in terms of a $\textit{generating function}$; a function whose n'th term in its Maclaurin series yields the moments of the fidelity for n segments. We generalize this approach as well to a $\textit{global cut-off}$ policy – a method for increasing fidelity at the cost of longer entanglement delivery times – allowing for fast optimization of the cut-off parameter by eliminating the need for Monte Carlo simulation. We furthermore find simple approximations of the average fidelity that are exponentially tight, and, for up to 10 segments, the full distribution of the delivered fidelity. We use this to analytically calculate the secret-key rate, both with and without binning methods.

Featured image: A depiction of how entanglement can spread in a swap ASAP repeater chain. Adjacent links are swapped as soon as possible, culminating in entanglement between Alice and Bob. The noise will vary per instance, and will depend on the sum of the absolute value of time differences of adjacent links.

Popular summary

Quantum repeater schemes are essential for distributing bipartite entanglement between remote parties. One of the simplest examples is the “swap ASAP” scheme, which immediately swaps (connect two short-range Bell pairs into a single long-range one) whenever two adjacent segments have successfully generated entanglement. Such swap ASAP schemes are amongst the simplest ways to distribute entanglement, and have thus received much attention in the literature. Yet quantifying the average fidelity of the delivered states for swap ASAP schemes has remained elusive: closed‑form results were previously known only for up to eight segments and for limited noise models.

We develop a single generating function whose coefficients yield closed‑form expressions for any number of repeater segments. Moreover, the singularities of this generating function govern the asymptotics of the noise, yielding exponentially tight approximations. Our generating function allows us to explicitly find the fidelity for up to 25 segments for the experimentally relevant class of qudit Pauli channels, which includes previously-used and common noise models such as depolarization and dephasing.
Our work ties together quantum repeater schemes, statistical physics and analytic combinatorics, providing a unifying language for future analyses.

► BibTeX data

@article{Goodenough2025noiseinswapasap,
  doi = {10.22331/q-2025-05-15-1744},
  url = {https://doi.org/10.22331/q-2025-05-15-1744},
  title = {On noise in swap {ASAP} repeater chains: exact analytics, distributions and tight approximations},
  author = {Goodenough, Kenneth and Coopmans, Tim and Towsley, Don},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1744},
  month = may,
  year = {2025}
}

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