Optimizing quantum codes with an application to the loss channel with partial erasure information

Benjamin Desef; Martin B. Plenio

doi:10.22331/q-2022-03-11-667

Optimizing quantum codes with an application to the loss channel with partial erasure information

Institute for Theoretical Physics & IQST, University of Ulm

Published:	2022-03-11, volume 6, page 667
Eprint:	arXiv:2105.13233v3
Doi:	https://doi.org/10.22331/q-2022-03-11-667
Citation:	Quantum 6, 667 (2022).

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Abstract

Quantum error correcting codes (QECCs) are the means of choice whenever quantum systems suffer errors, e.g., due to imperfect devices, environments, or faulty channels. By now, a plethora of families of codes is known, but there is no universal approach to finding new or optimal codes for a certain task and subject to specific experimental constraints. In particular, once found, a QECC is typically used in very diverse contexts, while its resilience against errors is captured in a single figure of merit, the distance of the code. This does not necessarily give rise to the most efficient protection possible given a certain known error or a particular application for which the code is employed.
In this paper, we investigate the loss channel, which plays a key role in quantum communication, and in particular in quantum key distribution over long distances. We develop a numerical set of tools that allows to optimize an encoding specifically for recovering lost particles both deterministically and probabilistically, where some knowledge about $what$ was lost is available, and demonstrate its capabilities. This allows us to arrive at new codes ideal for the distribution of entangled states in this particular setting, and also to investigate if encoding in qudits or allowing for non-deterministic correction proves advantageous compared to known QECCs. While we here focus on the case of losses, our methodology is applicable whenever the errors in a system can be characterized by a known linear map.

Featured image: After preparation at Alice’s site, $s$ qubits are sent though the channel, but only $r$ arrive. All possible configurations, corresponding to the loss maps $\{M_i\}_i$, are independent and equally likely. A local distillation map $E$ then aims at producing a high-quality Bell pair between Alice and Bob with probability pdist; this map does not know which particles were lost. We visualize the important initial and final correlations by wavy lines, but suppress them at the intermediate stages. While we name the individual parties “Alice” and “Bob,” they may also be repeaters.
In our paper, we aim at finding simultaneously optimal preparation states on Alice's site as well as the corresponding optimal distillation maps, given an arbitrary but fixed success probability $p_{\mathrm{dist}}$.

► BibTeX data

@article{Desef2022optimizingquantum,
  doi = {10.22331/q-2022-03-11-667},
  url = {https://doi.org/10.22331/q-2022-03-11-667},
  title = {Optimizing quantum codes with an application to the loss channel with partial erasure information},
  author = {Desef, Benjamin and Plenio, Martin B.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {667},
  month = mar,
  year = {2022}
}

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