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arXiv:2412.01434 · Physical Review Research

Feasibility of logical Bell state generation in memory assisted quantum networks

Vladlen Galetsky, Nilesh Vyas, Alberto Comin, Janis Nötzel

Published May 16, 2025• 2 citations
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Abstract

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.

DOI
10.1103/rsrk-c7yg

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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.
This study explores the feasibility of utilizing quantum error correction (QEC) to generate and store logical Bell states in heralded quantum entanglement protocols, crucial for quantum repeater networks. Two lattice surgery-based protocols (local and non-local) 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 non-destructive 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. Pseudo-thresholds are identified, with physical error rates above perr∼ 10−3 offering no advantage over unencoded Bell states under depolarizing noise. Pseudo-thresholds are also reevaluated in terms of gate error rates perr⁡H,perr⁡CX and perr⁡M{p_{{{\operatorname{err} }_H}}},{p_{{{\operatorname{err} }_{CX}}}}{\text{ and }}{p_{{{\operatorname{err} }_M}}}perrH​​,perrCX​​ and perrM​​. 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 (0.1perr⁡H,0.1perr⁡CX, and 0.1perr⁡M)\left({0.1{p_{{{\operatorname{err} }_H}}},0.1{p_{{{\operatorname{err} }_{CX}}}},{\text{ and }}0.1{p_{{{\operatorname{err} }_M}}}}\right)(0.1perrH​​,0.1perrCX​​, and 0.1perrM​​). These results highlight the need for significant hardware improvements to implement logical Bell state protocols with quantum memories. Additionally, the non-local 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.
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.
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.
Modular architectures are a promising approach to scaling quantum computers to fault tolerance. Small, low-noise quantum processors connected through relatively noisy quantum links are capable of fault-tolerant operation as long as the noise can be confined to the interface. Finding protocols that implement the quantum links between modules as efficiently as possible is essential because inter-module entanglement is challenging to produce at a similar rate and fidelity as local entanglement. We introduce a protocol for lattice surgery on surface codes in which all non-local operations are Bell measurements. The protocol simultaneously confines the link noise and requires only half as many module-crossing gates as previously proposed protocols. To mitigate distance-reducing hook errors, we introduce a strategy of alternating the gate sequence between rounds of syndrome measurement, which prevents multiple hooks from simultaneously aligning with a logical operator in the code. We evaluate our protocol's performance when two logical qubits on separate modules are prepared in a logical Bell state. Circuit-level simulations under depolarizing noise show that the logical error suppression for a given entanglement rate between modules is consistently stronger compared to the best-performing alternative protocols for a wide range of link noise, with a typical 40% entanglement resource saving for a constant logical error rate. Our approach to protocol design is applicable to any quantum circuit that must be divided across processor modules and can therefore guide development of resource-efficient modular quantum computation beyond the surface code.
Advances in single-photon creation, transmission, and detection suggest that sending quantum information over optical fibers may have losses low enough to be correctable using a quantum error correcting code (QECC). Such error-corrected communication is equivalent to a novel quantum repeater scheme, but crucial questions regarding implementation and system requirements remain open. Here we show that long-range entangled bit generation with rates approaching 10 8 entangled bits per second may be possible using a completely serialized protocol, in which photons are generated, entangled, and error corrected via sequential, one-way interactions with as few matter qubits as possible. Provided loss and error rates of the required elements are below the threshold for quantum error correction, this scheme demonstrates improved performance over transmission of single photons. We find improvement in entangled bit rates at large distances using this serial protocol and various QECCs. In particular, at a total distance of 500 km with fiber loss rates of 0.3 dB km −1 , logical gate failure probabilities of 10 −5 , photon creation and measurement error rates of 10 −5 , and a gate speed of 80 ps, we find the maximum single repeater chain entangled bit rates of 51 Hz at a 20 m node spacing and 190 000 Hz at a 43 m node spacing for the and QECCs respectively as compared to a bare success rate of 1 × 10 −140 Hz for single photon transmission.
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.