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IEEE Transactions on Quantum Engineering

Quantum Error Correction for Second-Generation Quantum Repeaters

Dawei Jiao, Mahdi Bayanifar, Alexei Ashikhmin, Olav Tirkkonen

Published Dec 31, 2025• 1 citation
Publisher

Abstract

We consider 2nd generation (2G) Quantum Repeaters (QR) for creating long-distance entanglement in quantum networks. Combining a distance-dependent depolarizing error model with the non-local Bell-state purification procedure required by 2G QRs leads to an error model consisting of correlated and biased errors. To correct correlated errors, non-symmetric CSS codes with joint decoding between stations can be used. The dominating errors are biased, such that different repeater stations suffer from different types of errors. To mitigate this, different quantum codes can be used at the stations, optimized for the specific error model of the station. To comply with the 2G QR procedure, the codes used in neighboring stations must allow for the transversal implementation of non-local logical CNOT gates across the two stations, or alternatively, non-local CZ gates combined with logical Hadamards. We provide a complete characterization of pairs of Calderbank-Steane-Shor (CSS) codes that allow CNOT- or CZ-transversality, and examine an explicit family of mirrored CSS codes allowing CZ-transversality. We verify the Hadamard gate transversality using our framework and show the importance of the mapping matrix. Also, we conclude that using different QECCs can not achieve transversal universal computation with Clifford +$T$-gate set. Finally, we study the entanglement generation rate (EGR) in 2G QRs with limited quantum memory, minimizing the number of intermediate stations for a given fidelity and EGR. By simulation we observe that non-symmetric and mirrored structure QECCs outperform the conventional approach of using symmetric CSS codes at the repeater stations.

DOI
10.1109/tqe.2025.3649561

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We consider 2nd generation (2G) Quantum Repeaters (QR) for creating long-distance entanglement in quantum networks. Combining a distance-dependent depolarizing error model with the non-local Bell-state purification procedure required by 2G QRs leads to an error model consisting of correlated and biased errors. To correct correlated errors, non-symmetric CSS codes with joint decoding between stations can be used. The dominating errors are biased, such that different repeater stations suffer from different types of errors. To mitigate this, different quantum codes can be used at the stations, optimized for the specific error model of the station. To comply with the 2G QR procedure, the codes used in neighboring stations must allow for the transversal implementation of non-local logical CNOT gates across the two stations, or alternatively, non-local CZ gates combined with logical Hadamards. We provide a complete characterization of pairs of Calderbank-Steane-Shor (CSS) codes that allow CNOT- or CZ-transversality, and examine an explicit family of mirrored CSS codes allowing CZ-transversality. We verify the Hadamard gate transversality using our framework and show the importance of the mapping matrix. Also, we conclude that using different QECCs can not achieve transversal universal computation with Clifford +TTT-gate set. Finally, we study the entanglement generation rate (EGR) in 2G QRs with limited quantum memory, minimizing the number of intermediate stations for a given fidelity and EGR. By simulation we observe that non-symmetric and mirrored structure QECCs outperform the conventional approach of using symmetric CSS codes at the repeater stations.
For achieving long-distance quantum communication, Quantum Repeaters (QRs) have to be used, with commu-nication range and reliability increased by using intermediate stations. The physical requirements of second generation QRs may be achievable in the near future. They use Quantum Error Correction Codes (QECC) to protect logical qubits against environmental interaction using physical redundancy. In this work, we study the types of errors that can corrupt quantum codewords in intermediate stations. Our studies show that the errors are distance-dependent, and also consist of correlated errors and biased errors. To mitigate this error model, we use non-symmetric CSS codes as well as mirrored structure coding. We show that using non-symmetric CSS codes results in better performance. Also, we prove the logical CZ gate transversality of the mirrored structure coding. The effectiveness of the proposed methods is verified by numerical simulations.
In this work, we generalize the concept of transversality in quantum error-correcting codes. Unlike conventional methods, we study transversal logical two-qubit gates between two different codes. We consider an application of this concept for quantum networks which consist of multiple intermediate stations equipped with quantum repeaters (QRs). The stations may experience different error models. For instance, in one station, a particular Pauli error may dominate, whereas in its neighboring stations, other types of Pauli errors are more likely. The standard approach of using the same CSS code Q{\mathcal{Q}}Q in different stations does not allow for simultaneously adaptating Q{\mathcal{Q}}Q to be optimized to the errors prevailing in various stations. Considering this fact, we suggest using different CSS codes in each station. In this work, we analyze CNOT and CZ transversality for pairs of CSS codes and provide a complete characterization. We formulate necessary and sufficient conditions for a pair of CSS codes to be CNOT and CZ transversal. In comparison to the conventional approach of having the same CSS code, we show that these conditions are less restrictive.
In this paper, we investigate the transversality of pairs of CSS codes and their use in the second generation of quantum repeaters (QR)s. We show that different stations of quantum link can experience different errors. Considering this fact, we suggest to use different CSS codes in different stations. We also suggest to use [[n,k]][[n, k]][[n,k]] codes with k>1k > 1k>1 as they are more efficient then codes with k=1k = 1k=1. We establish sufficient and necessary conditions for a pair of CSS codes to be non-local CNOT-transversal. We show that in contrast to the well known CNOT transversality which states that two CSS codes should be the same, less restrictive constraints are needed. Next, we establish sufficient and necessary conditions for a code pair to be CZ-transversal.
Quantum repeaters are essential for scalable long-distance quantum networking. As quantum information processing moves toward fault-tolerant and error-corrected operations, it becomes increasingly important to study quantum repeaters that also move beyond raw physical entanglement and towards logical entanglement. In this paper, we implement and simulate the quantum repeater with encoding and classical error correction (QRE-CEC) protocol in SeQUeNCe, a discrete-event simulator of quantum networks. The protocol distributes logical Bell pairs, performs encoded entanglement swapping, and uses classical error correction for the decoding of entanglement swapping measurement outcomes to determine Pauli-frame corrections. For this study, we extend SeQUeNCe with a stabilizer-based backend, add support for CSS code-based encoded operations, and integrate gate, measurement, idle decoherence, and state-initialization noise models. Our simulation results show that QRE-CEC suppresses all modeled errors to the second order. Also, QRE-CEC can distribute logical Bell pairs with 0.91 fidelity over a distance of 2000 km under the parameter regimes we study. Beyond protocol-level performance evaluation, our implementation exposes practical simulator and control-plane challenges that are typically abstracted away in theoretical studies.
The future of quantum repeater networking will require interoperability between various error-correcting codes. A few specific code conversions and even a generalized method are known, however, no detailed analysis of these techniques in the context of quantum networking has been performed. In this paper we analyze a generalized procedure to create Bell pairs encoded heterogeneously between two separate codes used often in error-corrected quantum repeater network designs. We begin with a physical Bell pair and then encode each qubit in a different error-correcting code, using entanglement purification to increase the fidelity. We investigate three separate protocols for preparing the purified encoded Bell pair. We calculate the error probability of those schemes between the Steane [[7,1,3]] code, a distance-3 surface code, and single physical qubits by Monte Carlo simulation under a standard Pauli error model and estimate the resource efficiency of the procedures. A local gate error rate of ${10}^{\ensuremath{-}3}$ allows us to create high-fidelity logical Bell pairs between any of our chosen codes. We find that a postselected model, where any detected parity flips in code stabilizers result in a restart of the protocol, performs the best.
Developing and deploying advanced Quantum Repeater (QR) technologies will be necessary to scale quantum networks at longer distances. Depending on the error mitigation mechanisms adopted to suppress loss and errors, QRs are typically classified into “memory-based” or “all-photonic” QRs, and each type of QR may be best suited for a specific type of underlying quantum technology, a particular scale of quantum network, or a specific regime of operational parameters. Modeling and simulation of quantum repeaters and networks can help in understanding the relative performance and resource requirements of different types of QRs, identify tradeoffs of alternative quantum network architectures, optimize quantum hardware, and assist in the design of a robust control plane. This paper introduces a quantum repeater simulation toolkit based on NetSquid, which includes component models and protocols to study advanced quantum repeater networks. We present our initial research findings on modeling and simulation of “allphotonic” entanglement-based QRs and “memory-based” first generation trapped ion QRs. We study the relative performance and resource requirements of these two different types of QRs and networks. The performance metrics include entanglement generation rate and fidelity between the endpoints of the network. For fair comparison of resource requirements of different types of QRs, entanglement generation rate is normalized by the total number of matter qubits involved.