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A quantum network stack and protocols for reliable entanglement-based networks

A. Pirker, W. Dür

arXiv:1810.03556v2quant-phcs.NI

TL;DR

Entanglement-based quantum networks need architectures that support multipartite graph states while organizing devices and handling failures. The paper introduces a hierarchical, technology-independent stack with protocols for reliable intra-network operation and graph-state generation across network boundaries. Its conclusion is that layering breaks large-scale network complexity into independently studyable responsibilities, although full realization remains challenging in the near term.

  • Problem

    Quantum networks need to organize devices and establish entanglement efficiently while supporting arbitrary multipartite graph states rather than only Bell pairs.

  • Method

    The paper constructs a hierarchical quantum network stack, classifies devices by layer, and proposes reliability, graph-state, routing, and network-connection protocols.

  • Results

    The proposed protocols support device-failure compensation within networks and arbitrary graph-state generation across network boundaries.

  • Takeaways & Limitations

    Layering breaks large-scale quantum-network complexity into manageable responsibilities that can be studied and evolved independently.

Abstract

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We present a stack model for breaking down the complexity of entanglement-based quantum networks. More specifically, we focus on the structures and architectures of quantum networks and not on concrete physical implementations of network elements. We construct the quantum network stack in a hierarchical manner comprising several layers, similar to the classical network stack, and identify quantum networking devices operating on each of these layers. The layers responsibilities range from establishing point-to-point connectivity, over intra-network graph state generation, to inter-network routing of entanglement. In addition we propose several protocols operating on these layers. In particular, we extend the existing intra-network protocols for generating arbitrary graph states to ensure reliability inside a quantum network, where here reliability refers to the capability to compensate for devices failures. Furthermore, we propose a routing protocol for quantum routers which enables to generate arbitrary graph states across network boundaries. This protocol, in correspondence with classical routing protocols, can compensate dynamically for failures of routers, or even complete networks, by simply re-routing the given entanglement over alternative paths. We also consider how to connect quantum routers in a hierarchical manner to reduce complexity, as well as reliability issues arising in connecting these quantum networking devices.

I. INTRODUCTION

The paper proposes a layered, technology-independent stack for entanglement-based quantum networks, addressing network organization, reliability, graph-state generation, and inter-network routing. It classifies network devices and introduces protocols for reliable intra-network operation and graph-state generation across network boundaries.

  • Motivation: Quantum networks should support arbitrary multipartite graph states, not only Bell pairs, because many applications require multipartite entanglement.The paper identifies graph states as an important subclass of multipartite entangled states.
  • Motivation: The paper identifies network organization and efficient dynamic entanglement establishment as two central problems in quantum networking.These problems span device coordination, systematic task execution, and entanglement distribution between networking devices.
  • Stack model: The proposed quantum network stack uses hierarchical layers with distinct responsibilities, from connectivity and intra-network graph-state generation to inter-network requests.The model is technology independent and separates layer-specific protocols and device roles.
  • Reliability: The paper introduces adaptive-phase techniques that maintain intra-network reliability when some networking devices fail without re-establishing entanglement.Reliability is defined as compensating for device failures during network operation.
  • Inter-network operation: A proposed adaptive-phase protocol generates arbitrary graph states across network boundaries and addresses complexity and reliability when connecting regions.The paper also discusses reachability within and between entanglement-based quantum networks.
  • Classical reference model: The paper relates the quantum stack to classical layered networking, where lower layers provide transport and higher layers define logical networks, routing, and applications.Classical examples include link-state routing with Dijkstra’s algorithm and separate physical, data-link, network, transport, session, representation, and application responsibilities.

B. Dijkstra’s algorithm and Steiner trees

This section introduces Dijkstra’s algorithm and Steiner-tree construction as graph tools used to determine paths and connect selected vertices with low-cost trees. The paper applies these tools to quantum-network routing and virtual network-state generation, while noting the computational difficulty of exact Steiner trees.

  • Dijkstra’s algorithm: Dijkstra’s algorithm finds shortest paths between graph vertices using a nonnegative edge-cost function.The paper also uses a generalized form to find shortest paths from one vertex to any vertex in a target subset.
  • Steiner trees: A Steiner tree connects a selected vertex subset with minimal cost and may include additional non-selected vertices.When the selected subset equals all graph vertices, the construction becomes a minimum spanning tree.
  • Steiner trees: The paper uses an approximation algorithm that repeatedly connects the closest not-yet-included selected vertex to the current tree.The procedure relies on Dijkstra distances and returns a tree after all selected vertices are included.
  • Computational boundary: The exact Steiner-tree problem is NP-complete in the rectilinear case, motivating the use of an approximation procedure.The computational difficulty is stated as a limitation of determining Steiner trees.

C. Graph states and GHZ states

Graph states encode multipartite entanglement through a classical graph, while GHZ states are a useful graph-state form that remains closed under Bell-measurement fusion.

  • Graph states: Graph states associate qubits with graph vertices and correlations with graph edges, forming the common +1 eigenstate of correlation operators.The operators use Pauli X and Z matrices acting on specified qubits.
  • Graph-state operations: Local complementation transforms the graph locally by inverting the subgraph induced by a chosen vertex’s neighbourhood.Z-basis measurement removes a vertex, while Y-basis measurement first applies local complementation and then removes it.
  • GHZ states: GHZ states are treated as LU-equivalent star graph states with a chosen root node.They are also LU-equivalent to fully connected graph states.
  • GHZ-state fusion: Bell-measuring one qubit from |GHZ_n⟩ and one from |GHZ_m⟩ produces |GHZ_{m+n−1}⟩ up to local Pauli corrections.Thus, GHZ resources can be fused into a larger GHZ state.

D. Quantum networks

Quantum-network research seeks to move beyond Bell-pair distribution toward arbitrary graph states for remote clients. The paper frames network architectures and pre-established states as resources for that goal.

  • Motivation: Quantum repeater approaches commonly assume networks whose nodes share Bell pairs, but many applications require arbitrary multipartite states between remote clients.Graph states are identified as an important class for such applications.
  • Related network approaches: Prior routing work addresses specific repeater topologies, cluster-type networks, end-to-end paths, entanglement gradients, base graphs, capacities, and configurable two-dimensional routing.These approaches primarily concern routing Bell pairs or restricted network settings.
  • Network architecture: The example network contains three switches and one router connected through decreasing-size GHZ states, with each device connecting three clients through internal GHZ states and Bell pairs.The entanglement structure differs from the physical channel configuration.
  • Universal resources: Universal network and device states must allow any graph state to be generated using LOCC without creating additional entanglement after a request arrives.The target state is not known during the static phase, so multiple copies of each GHZ state are mandatory.
  • Resource construction: The proposed network-state construction is sufficient for arbitrary graph-state generation but is not claimed to be optimal, and target states may require only a subset of its resources.The construction includes both network and device states.

E. Assumptions

The proposal assumes discrete-variable qubit networks with quantum memories, local state preparation, specified Clifford-capable operations, noiseless operations, and deterministic Bell-state measurements, while discussing relaxations for noise and nondeterminism.

  • System model: The framework is restricted to discrete-variable systems represented by qubits.
  • Allowed operations: Devices can apply controlled-phase gates, Bell-state measurements, and single-qubit Pauli matrices and measurements.
  • Noise model: All quantum operations and measurements are assumed noiseless, although the paper discusses ways to relax this assumption.
  • Measurement model: Bell-state measurements are assumed deterministic, while approaches for nondeterministic measurements are discussed elsewhere in the paper.The relevant discussions address both connectivity and physical-layer issues.
  • Device capabilities: Quantum networking devices are assumed to have quantum memories and the ability to create local quantum states.Noise in quantum memory is discussed separately.
  • Fault tolerance: The operations used by network devices are Clifford operations, so the proposal does not require the cited assumption concerning point 4 of the referenced checklist.

F. Relation to other stack and network models

The paper extends prior repeater and single-network models by organizing entanglement-based networking into a hierarchical stack for arbitrary graph-state generation. Its layers separate physical channels, connectivity, intra-network states, and inter-network routing while permitting cross-layer qubit access.

  • Relation to prior models: Unlike the repeater stack of, which establishes single entangled links, this stack targets arbitrary graph states for complex client protocols.The cited examples include conference key agreement and distributed quantum computation.
  • Relation to prior models: Unlike, which addresses single quantum-network architectures, this work adds reliability and mechanisms for connecting different networks.The GHZ architecture from reduces network-device storage requirements by a factor of 2 versus direct Bell-pair architectures.
  • Stack operation: The stack differs from the classical OSI model because neighboring quantum layers can access and combine qubits rather than only passing packets downward.Layer operations may be invoked synchronously or asynchronously.
  • Stack structure: The proposed stack has four layers: physical channel configuration, connectivity through high-fidelity entanglement, intra-network multipartite states, and inter-network routing via routers.
  • Physical layer: The physical layer forwards qubits over channels without error correction or distillation and establishes short-link entanglement.It also interfaces between quantum-memory and quantum-channel technologies.
  • Connectivity layer: Layer-2 devices use quantum channels to create long-distance entangled states whose structure can be independent of the physical channel configuration.

B. Layer 2 – Connectivity Layer

The connectivity layer establishes long-distance entanglement despite channel imperfections, while higher layers use shared multipartite states to generate graph states and connect quantum networks. Auxiliary protocols support state generation, reachability monitoring, entanglement operations, error correction, and network-status monitoring.

  • Connectivity layer: Layer 2 establishes long-distance quantum communication and addresses errors caused by imperfections in quantum channels.Technologies include quantum repeaters and bipartite or multipartite entanglement schemes.
  • Link layer: The link layer defines a quantum network through a shared distributed multipartite network state established dynamically using connectivity-layer entanglement.After the dynamic phase, link-layer switches share the network state in a static phase.
  • Link layer: Linking protocols transform the network state and device-internal states using LOCC into client-requested graph states, consuming the network-state entanglement.The specific protocol depends on the network state, including GHZ or decorated architectures.
  • Network layer: The network layer uses quantum routers and multipartite entanglement to generate and manipulate graph-state requests spanning multiple quantum networks.Routers may belong to several regions and can also provide entry points into link-layer quantum networks.
  • Auxiliary protocols: Auxiliary protocols generate high-fidelity states, check reachability, perform swapping or merging, support error correction, and monitor network status.Devices select different subsets of these protocol types according to their layer.

IV. LAYER 3 – RELIABLE STATE LINKING

Reliable state linking seeks to preserve usable entanglement for remaining devices when a network device disconnects. Direct GHZ-based network states are fragile because tracing out one qubit can destroy the shared entanglement, motivating failure-tolerant protocols.

  • Reliability objective: Reliability means that parts of a network’s entanglement structure remain usable after a device disconnects without further operation.The definition applies to the static phase of a quantum network using multipartite entanglement.
  • Failure model: A disconnected device is represented by tracing out its qubits from the corresponding GHZ states.The paper also describes local measurements that reduce the sizes of affected GHZ-state connections.
  • Failure model: Tracing out one qubit of a GHZ state produces a separable state, destroying entanglement among the other devices sharing that state.Depending on which device disconnects, all network states may be lost.
  • Design requirement: Because any network device may fail, reliable schemes must preserve functionality for the remaining system after arbitrary device disconnections.The paper presents multipartite-entanglement protocols as proof-of-principle solutions and reports storage benefits over a full bipartite approach.

A. Reliable state linking – Symmetrization

The paper proposes symmetrization and shielding to make multipartite network states resilient to device failures. Symmetrization preserves at least one intact configuration after a single failure, while shielding preserves the remaining network state through additional qubits and measurements.

  • Reliable state linking – Symmetrization: Symmetrization cyclically shifts root-leaf assignments across m configurations, distributing network-state roles among devices.Several copies of each GHZ state are required to support arbitrary graph-state requests.
  • Reliable state linking – Symmetrization: After one device disconnects, at least one symmetrized configuration remains fully connected among the remaining devices, preserving at least n/m copies.Other configurations may remain only partially intact.
  • Reliable state linking – Symmetrization: Full symmetrization over all device permutations is proposed to address arbitrary losses, including failures of multiple network devices.This is described as a further variant beyond cyclic shifting.
  • Reliable state linking – Shielding: Shielding decorates each GHZ-state edge with an additional qubit, except for Bell pairs, and places the shielding qubit at the device holding the GHZ root.The shielded state is represented as a decorated star graph.
  • Reliable state linking – Shielding: After a device failure, remaining devices can measure the shield qubits associated with the disconnected device in the Z basis and retain a full network state.Failure detection uses classical ping messages during network operation.
  • Comparison: Shielding uses more quantum memory than symmetrization but remains more qubit-efficient than a direct bipartite approach.Its storage advantage arises because qubits are added before network-state expansion, while its failure behavior preserves all network states after a device failure.

C. Further considerations

The presented protocols are proof-of-principle approaches to reliability, and alternative techniques may provide other solutions.

  • The protocols in Sections IV A and IV B demonstrate, as proof-of-principle, how multipartite quantum states can support network reliability.
  • Possible alternatives include decorated graph states, network coding, quantum error-correction codes, and other approaches left for future work.

V. LAYER 4 – REGION ROUTING, HIERARCHICAL REGIONS AND RELIABLE REGIONS

Layer 4 addresses inter-network graph-state requests by routing entanglement among quantum-network routers organized into regions. Its routing procedure abstracts networks into a classical graph and uses Steiner-tree construction to generate virtual network states.

  • Layer 4 role: Layer 4 enables inter-network graph-state requests through GHZ-connected router regions and LOCC-based protocols.
  • Region routing: The routing protocol follows classical link-state routing, representing each quantum network through routers and using graph-based path selection.
  • Region routing: A virtual network state connects routers of the networks involved in a request before routers combine it with inner network states.
  • Region routing: Regions may be highly irregular because administrators select which routers connect, allowing network boundaries and inter-network connectivity to be configured.
  • RegionRouting: The protocol transforms networks into a classical graph, repeatedly selects a requesting vertex, constructs a Steiner tree, and generates a GHZ state over the selected set.
  • RegionRouting: Routers track internally selected states and minimize multipartite entanglement consumption for each request.

B. Hierarchical regions

Hierarchical regions limit GHZ-state sizes while connecting routers across multiple levels. Designated routers and symmetrization support hierarchical connectivity and provide a basis for handling regional failures, though the reliability construction is not claimed optimal.

  • Motivation: Connecting n routers in one region requires a largest GHZ state of size n, whose fragility increases with size because of noise and decoherence.
  • Hierarchical regions: The proposed hierarchy uses only limited-size GHZ states and arranges regions hierarchically, avoiding fragile large GHZ states.
  • Hierarchical regions: Hierarchical regions can be arranged on demand and optimized for expected traffic, reducing dependence on a fixed, globally known topology.
  • Designated routers: Designated routers connect regions at different hierarchical levels and enable reachability across those levels.
  • Designated routers: The hierarchy can be constructed automatically when new routers discover existing routers and designated routers determine suitable region membership.
  • Reliable regions: Symmetrization preserves GHZ-type entanglement between regions, while alternative permutations can be reserved for route failures.
  • Reliable regions: The symmetrization approach is presented as proof-of-principle, and its optimality is explicitly not claimed.

VI. AUXILIARY – REACHABILITY PROTOCOLS

The reachability protocols classify quantum-network connectivity checks from classical reachability to trusted and device-independent entanglement verification. Verification can consume entanglement, and the paper presents the categorization as an outline for future detailed analysis.

  • Purpose: Quantum reachability protocols check whether entanglement connectivity between networking devices remains available across multiple stack layers.
  • Verification levels: The proposed categorization includes classical ping, trusted state verification, and device-independent state verification.
  • Verification levels: Trusted state verification can use entanglement witnesses, graph-state correlation operators, or quantum state tomography.
  • Verification levels: Device-independent verification checks entanglement without trusting each network device’s local apparatus and is the strongest reachability level.
  • Verification constraints: Reliable non-destructive verification generally requires consuming network entanglement or using independent copies for prediction after local measurements.
  • Scope: The reachability-level analysis is left for future work, so the section provides an outline rather than a detailed protocol evaluation.
  • Conclusion: The stack model abstracts implementation details into goal-oriented layers and includes protocols for reliability inside and between quantum networks.

Appendix A: An example of region routing

The example shows region routing establishing a virtual network state for clients across four networks, then transforming Steiner-tree constructions into GHZ states. The protocol iteratively processes target networks and combines selected qubits through local GHZ preparation and Bell measurements.

  • Example scenario: Four clients in four networks use region routing to establish a virtual network state for sharing a graph state.The example is illustrated through a Steiner-tree construction for the first router.
  • Iterative routing: Figure 12 depicts the router configuration across regions and the protocol's first step for generating the requested graph state.
  • Steiner-tree transformation: After constructing a Steiner tree, the protocol converts it into a GHZ state by combining the root's selected qubits using local GHZ preparation and Bell measurements.If the root degree exceeds one, the resulting GHZ state has size deg(r) + 1.
  • Steiner-tree transformation: Non-target nodes perform Bell measurements on selected Steiner-tree qubits, while nonterminal target nodes locally create GHZ states, retain one qubit, and measure the others.Some routers may additionally use Z-basis measurements to shape regional GHZ states.
  • Iterative routing: The protocol removes a processed target network and repeats the construction for the remaining target networks in the set.
  • Example scenario: The full-stack example begins with a client requesting a four-qubit cluster state across blue, yellow, and red networks connected through regional GHZ states.
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