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arXiv:1508.02141 · Physical Review A

Analysis of quantum network coding for realistic repeater networks

Takahiko Satoh, Kaori Ishizaki, Shota Nagayama, Rodney Van Meter

Published Mar 1, 2016• 46 citations
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Abstract

Quantum repeater networks have attracted attention for the implementation of long-distance and large-scale sharing of quantum states. Recently, researchers extended classical network coding, which is a technique for throughput enhancement, into quantum information. The utility of quantum network coding (QNC) has been shown under ideal conditions, but it has not been studied previously under conditions of noise and shortage of quantum resources. We analyzed QNC on a butterfly network, which can create end-to-end Bell pairs at twice the rate of the standard quantum network repeater approach. The joint fidelity of creating two Bell pairs has a small penalty for QNC relative to entanglement swapping. It will thus be useful when we care more about throughput than fidelity. We found that the output fidelity drops below 0.5 when the initial Bell pairs have fidelity $F<0.90$, even with perfect local gates. Local gate errors have a larger impact on quantum network coding than on entanglement swapping.

DOI
10.1103/physreva.93.032302

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Quantum repeater networks have attracted attention for the implementation of long-distance and large-scale sharing of quantum states. Recently, researchers extended classical network coding, which is a technique for throughput enhancement, into quantum information. The utility of quantum network coding (QNC) has been shown under ideal conditions, but it has not been studied previously under conditions of noise and shortage of quantum resources. We analyzed QNC on a butterfly network, which can create end-to-end Bell pairs at twice the rate of the standard quantum network repeater approach. The joint fidelity of creating two Bell pairs has a small penalty for QNC relative to entanglement swapping. It will thus be useful when we care more about throughput than fidelity. We found that the output fidelity drops below 0.5 when the initial Bell pairs have fidelity F<0.90F<0.90F<0.90, even with perfect local gates. Local gate errors have a larger impact on quantum network coding than on entanglement swapping.
Quantum network coding is an effective solution for alleviating bottlenecks in quantum networks. We introduce a measurement-based quantum network coding scheme for quantum repeater networks (MQNC), and analyze its behavior based on results acquired from Monte Carlo simulation that includes various error sources over a butterfly network. By exploiting measurement-based quantum computing, operation on qubits for completing network coding proceeds in parallel. We show that such an approach offers advantages over other schemes in terms of the quantum circuit depth and therefore improves the communication fidelity without disturbing the aggregate throughput. The circuit depth of our protocol has been reduced by 56.5% compared to the quantum network coding scheme (QNC) introduced in 2012 by Satoh et al. For MQNC, we have found that the resulting entangled pairs' joint fidelity drops below 50% when the accuracy of local operations is lower than 98.9%, assuming that all initial Bell pairs across quantum repeaters have a fixed fidelity of 98%. Overall, MQNC showed substantially higher error tolerance compared to QNC and slightly better than buffer space multiplexing using step-by-step entanglement swapping, but not quite as strong as simultaneous entanglement swapping operations.
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.
We evaluate the performance of two-way quantum repeater chains with sequential entanglement swapping. Within the analysis we consider memory decoherence, gate imperfections, and imperfect link-level entanglement generation. Our main results include closed-form expressions for the average entanglement fidelity of the generated end-to-end entangled states. We generalize previous findings for the one-shot fidelity analysis and study the case where repeater chains serve end-to-end requests continuously. We provide solutions to the continuous request scenario by combining results from quantum information theory and queuing theory. Finally, we apply the formulas obtained to analyze the impacts of hardware parameters, i.e., coherence times and gate fidelity, and distance on the entanglement fidelity and secret key rate of homogeneous quantum repeater chains.
A fundamental challenge in generating and distributing high-fidelity entanglement in quantum repeater networks is the management and suppression of quantum errors. In this work, we develop an approach for extending the effective coherence time of shared entanglement between nodes by utilizing compact quantum error correcting (QEC) codes, which are scheduled and run locally on resource-constrained repeaters, to maximize the fidelity of the shared state, while also accounting for the incurred gate noise from QEC circuit operations. This local QEC approach stabilizes the distributed state via local operations, so independent execution on the nodes requires no additional synchronization or protocol overhead. Simulation results show significant improvements in the effective coherence time relative to idle decay, suggesting this approach could be useful in several quantum networking protocols such as entanglement generation, swapping, purification, and teleportation.
This paper investigates the scalability and transmission efficiency of entanglement swapping protocols within multi-node quantum networks. As the realization of a global quantum internet relies heavily on long-distance entanglement distribution, quantum repeaters utilizing entanglement swapping serve as a critical backbone. However, scalability remains a bottleneck due to photon loss, operational infidelities, and quantum memory coherence times. In this study, we analytically model a multi-node quantum chain and evaluate how the end-to-end entanglement fidelity degrades as the number of intermediate nodes scales up. Our analysis highlights the critical thresholds for Bell state measurement (BSM) success rates required to maintain quantum state integrity across large distances. The findings provide optimization constraints for designing scalable and fault-tolerant routing protocols in future quantum information networks.