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

TL;DR

This paper develops numerical tools to optimize quantum error correcting codes specifically for loss channels with partial erasure information, leading to new codes tailored for quantum communication and entanglement distribution.

Contribution

The authors introduce a novel numerical optimization approach for designing quantum codes optimized for loss channels with partial erasure knowledge, surpassing traditional QECCs in specific scenarios.

Findings

Optimized codes improve entanglement distribution over loss channels.

Encoding in qudits can be advantageous over qubits.

Probabilistic correction enhances error resilience.

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.