Bosonic quantum error correction with microwave cavities for quantum repeaters

TL;DR

The article addresses the challenge of long-distance quantum communication by analyzing secret-key rates for a two-segment quantum repeater that uses bosonic Binomial codes to protect lossy memories, focusing on microwave-cavity implementations with a transmon and an all-optical alternative. It develops a detailed framework for encoding, syndrome detection, and entanglement swapping, and compares single- and multiple-time error correction (SEC vs MEC) for both lower- and higher-order Binomial codes. The results show that Binomial-code memories outperform unencoded memories across realistic memory coherences, with MEC offering additional gains and the Higher Binomial Code often providing the strongest protection; a practical microwave-cavity protocol achieving high-fidelity state engineering and 50% all-optical BSM guidance is provided. These findings offer near-term guidance for constructing practical quantum repeaters and help delineate the trade-offs between microwave-cavity and all-optical approaches in bosonic QEC frameworks.

Abstract

Long-distance quantum communication necessitates the use of quantum repeaters, which typically include highly coherent quantum memories. We provide a theoretical analysis of the secret key rates for a quantum repeater system incorporating bosonic error correction and memory components. Specifically, we focus on the application of Binomial codes for two repeater segments. Using these codes, our investigation aims to suppress memory loss errors that commonly affect systems such as atoms and microwave cavities, in contrast to dephasing errors in single-spin memories. We further discuss a physical implementation of such a quantum repeater comprising a microwave cavity and a superconducting transmon, capable of state engineering with high fidelities ($>97\%$) and logical Bell state measurements for successful entanglement swapping. As an alternative approach, we also discuss a realization in the all-optical domain.

Figures (6)

• Figure 1: Two-segment, second-generation quantum repeater, with memory implemented as a microwave cavity (shown as an example). (a) Entanglement between a travelling photon and a microwave cavity occurs at all four memory units. (b) Entanglement within a single repeater segment of distance $L_0$ is established between two memory units via projection measurement (using beam splitters) on travelling photons, which are conducted independently in both repeater segments. (c) Entanglement swapping occurs locally between two memory units at the middle station of the two segments via a logical Bell state measurement (BSM). (d) The final entangled state is achieved between distant memory units 1 and 4. Throughout these steps, the superposition states of the photon are represented by yellow and pink photons, though only one entangled pair of photons is distributed per round of the protocol. Cuboids represent microwave cavities, and the encoded states of the photon are depicted by the thick lines in the harmonic oscillator.
• Figure 2: Comparison of secret key rates with different memory coherence times for unencoded, Lower Binomial Code (LBC) and Higher Binomial code (HBC) when SEC is performed. The memory interface parameter $\eta_m=0.95$.
• Figure 3: Comparison of secret key rates with different memory coherence times for Lower Binomial Code (LBC) and Higher Binomial code (HBC) when SEC is performed. The memory interface parameter $\eta_m=0.9$.
• Figure 4: Comparison of the secret key fraction of LBC with different coherence times for the cases of SEC and MEC. The memory interface parameter $\eta_m=1.0$.
• Figure 5: Comparison of secret key rates with different memory coherence times for unencoded, Lower Binomial Code (LBC) and Higher Binomial code (HBC) when SEC is performed and entanglement swapping is performed using linear optics with $50\%$ efficiency. The memory interface parameter $\eta_m=0.95$