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arXiv:1508.04599 · Physical Review A

Interoperability in encoded quantum repeater networks

Shota Nagayama, Byung-Soo Choi, Simon J. Devitt, Shigeya Suzuki, Rodney Van Meter

Published Apr 26, 2016• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.93.042338

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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{10}^{\ensuremath{-}3}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.
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.
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.
This study explores the feasibility of utilizing quantum error correction to generate and store logical Bell states in heralded quantum entanglement protocols, crucial for quantum repeater networks. Two lattice surgery-based protocols (local and nonlocal) are introduced to establish logical Bell states between distant nodes using an intermediary node. We simulate the protocols using realistic experimental parameters, including ion trap memories, noisy optical channels, frequency conversion, and nondestructive detection of photonic qubits. The study evaluates rotated and planar surface codes alongside Bacon-Shor codes for small code distances ( d = 3 , 5 ) under depolarizing and physical noise models. Pseudothresholds are identified, with physical error rates above p err ∼ 10 − 3 offering no advantage over unencoded Bell states under depolarizing noise. Pseudo-thresholds are also reevaluated in terms of gate error rates p err H , p err C X , and p err M . For a distance of 1 km between the end node and the intermediary, an advantage over unencoded Bell-state heralded protocols requires reducing gate error rates by an order of magnitude ( 0.1 p err H , 0.1 p err C X , and 0.1 p err M ). These results highlight the need for significant hardware improvements to implement logical Bell state protocols with quantum memories. Additionally, the nonlocal protocol rate was analyzed achieving rates up to ( 32.53 ± 1.53 ) Hz over distances of 1 to 80 km between the end node and the intermediary node.
Long-distance entanglement distribution requires error correction protocols to compensate for qubit decoherence in quantum memories and noise introduced during entanglement swapping. We argue that repeater chains with error-correction capabilities should exploit more than one physical platform, combining the complementary strengths of different quantum memory technologies into a single hybrid repeater architecture. An important constituent of such an architecture is a hybrid repeater which combines type-1 memories, characterized by fast entanglement generation rates and suitability for multiplexing, and type-2 memories that offer long coherence times and low two-qubit gate error rates. Taking the resource-intensive nature of hybrid nodes into account, we propose and analyze repeater chains in which only a subset of nodes need to be hybrid, while the remaining nodes are simpler first-generation repeaters with no error-correction capability. Through detailed Monte Carlo simulations of fault-tolerant encoded repeater chain protocols based on the three-qubit phase-flip repetition code, the [[7,1,3]][[7,1,3]][[7,1,3]] Steane code, and the [[9,1,3]][[9,1,3]][[9,1,3]] Shor code, we demonstrate that these hybrid architectures outperform pure architectures based on a single memory platform in terms of end-to-end entanglement distribution rate. In our study we develop a full circuit-level noise model of our architectures and examine the impact of an imperfect interface between the two platforms on our hybrid architecture. We also develop a modified version of the swap-as-soon-as-possible policy with multiplexing, more suited to our architecture where some nodes perform error-correction while others do not. This modified policy significantly reduces the information storage time in memory qubits relative to the previously considered swap policies in encoded repeater chains.
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.
We present an encoded hybrid quantum repeater scheme using qubit-repetition and Calderbank-Shor-Steane codes. For the case of repetition codes, we propose an explicit implementation of the quantum error-correction protocol. Moreover, we analyze the entangled-pair distribution rate for the hybrid quantum repeater with encoding and we clearly identify trade-offs between the efficiency of the codes, the memory-decoherence time, and the local gate errors. Finally, we show that in the presence of reasonable imperfections our system can achieve rates of roughly 24 Hz per memory for 20 km repeater spacing, a final distance of 1280 km, and final fidelity of about 0.95.