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Routing entanglement in the quantum internet
Mihir Pant, Hari Krovi, Don Towsley, Leandros Tassiulas, Liang Jiang, Prithwish Basu, Dirk Englund, Saikat Guha
TL;DR
The paper studies how quantum repeaters in lossy networks can maximize entanglement rates for one or multiple user pairs under limited processing and memory constraints. It develops routing protocols exploiting multiple paths and finds substantial advantages over linear repeater chains, while leaving rate-optimal routing and general-error treatment open.
Problem
The paper asks how networks of limited-capability quantum repeaters can maximize entanglement rates or rate regions for one or multiple user pairs.
Method
The paper analyzes repeater protocols that select internal entanglement swaps using global or local link-state information and exploit multiple network paths.
Results
Multi-path routing can far outperform a linear repeater chain for a single flow and enable distance-independent rates above the percolation threshold with one-slot memory coherence.
Takeaways & Limitations
Non-trivial network topologies can substantially benefit quantum entanglement distribution and more readily exceed repeater-less rate-versus-distance upper limits than linear chains.
Takeaways & Limitations
The rate-optimal protocol remains open, and treating general errors would require entanglement purification and intermediate fidelity accounting.
Abstract
from arXiv · showhide
Remote quantum entanglement can enable numerous applications including distributed quantum computation, secure communication, and precision sensing. In this paper, we consider how a quantum network-nodes equipped with limited quantum processing capabilities connected via lossy optical links-can distribute high-rate entanglement simultaneously between multiple pairs of users (multiple flows). We develop protocols for such quantum "repeater" nodes, which enable a pair of users to achieve large gains in entanglement rates over using a linear chain of quantum repeaters, by exploiting the diversity of multiple paths in the network. Additionally, we develop repeater protocols that enable multiple user pairs to generate entanglement simultaneously at rates that can far exceed what is possible with repeaters time sharing among assisting individual entanglement flows. Our results suggest that the early-stage development of quantum memories with short coherence times and implementations of probabilistic Bell-state measurements can have a much more profound impact on quantum networks than may be apparent from analyzing linear repeater chains. This framework should spur the development of a general quantum network theory, bringing together quantum memory physics, quantum information theory, and computer network theory.
I. BACKGROUND
The paper frames quantum-network routing as a problem of distributing entanglement across lossy links with constrained repeater processing. It motivates protocols that exploit network topology and local information rather than relying only on linear repeater chains.
- The paper seeks practical routing protocols that support multiple entanglement flows with limited quantum processing at repeater nodes.
- The square-grid protocol alternates probabilistic external link generation with internal entanglement swaps, while memories retain qubits for T ≥1 time slots.
A. Problem statement and notation
The model represents a lossy repeater network as a graph with parallel channels, finite-memory storage, and probabilistic link generation and swapping. Its objective is to choose local operations that maximize single-flow rates or the simultaneous rate region for multiple flows.
- Each repeater-network edge e has S(e) parallel channels, and node memory counts are tied to the incident channel multiplicities.
- Each memory stores a qubit for T ≥1 time slots, and each slot has ordered external link-generation and internal swapping phases.
- With per-channel success probability p0, the edge success probability is p(e) = 1−(1−p0)^S(e), simplifying to p when S(e) is uniform.
- The analysis uses T = 1 so swaps combine qubits created during the same preceding external phase.
- Repeater decisions aim to maximize the entanglement rate for K = 1 or the simultaneously achievable rate region for K > 1.
1. Entanglement routing with global link-state information
The global-information protocol selects successive shortest paths in the successful-link subgraph and attempts swaps along them. Multi-path routing can make rates distance independent above the percolation threshold when swaps are perfect, while practical imperfections restore distance decay.
- 1. Entanglement routing with global link-state information: The greedy global-information algorithm repeatedly selects a shortest Alice–Bob path among successful external links and removes its used resources before selecting another.
- 1. Entanglement routing with global link-state information: Ropt ≥ Rg ≥ Ropt/4, so the greedy protocol matches the optimal global-information rate-distance scaling within a constant factor.
- 1. Entanglement routing with global link-state information: A path of length k contributes expected rate p^k q^(k−1), reflecting successful external links and internal swaps.
- 1. Entanglement routing with global link-state information: When q = 1 and p > 0.5, Rg is essentially distance independent; below the square-lattice threshold, it decays exponentially with distance.
- 1. Entanglement routing with global link-state information: The greedy rate is estimated through Monte Carlo simulations, whose numerical noise is reported as insignificant relative to plot differences.
- 1. Entanglement routing with global link-state information: For p = 0.6 and q = 1, Rg is within a factor of approximately 3.6 of the Pirandola upper bound while using one-time-slot memories.
2. Entanglement routing with local link-state information
With only local link-state knowledge, the proposed routing rule uses distance-based decisions to exploit multiple possible paths, achieving better rate-distance scaling than a linear repeater chain.
- Local-information protocol: Rloc uses only adjacent external-link states, while network topology and Alice-Bob positions are known beforehand at every repeater.Each repeater selects internal links for BSM attempts from locally observed neighboring-link successes.
- Rate-distance scaling: The local rule achieves a rate-distance exponent superior to a linear chain, although both Rloc and Rlin decrease exponentially with distance.Its advantage comes from finding different, potentially simultaneous paths across time slots instead of requiring every link on one chain to succeed.
- Geometric effects: Rloc is enhanced along the diagonal because that direction contains the greatest spatial density of possible Alice-Bob paths, while the advantage persists along Y = 0.
- Parameter dependence: The multi-path advantage relative to a linear chain increases as p decreases from unity, with little relative improvement as q varies.
- Open question: An analytical enumeration of expected edge-disjoint paths as a function of p remains open, limiting firmer quantitative understanding of the multi-path advantage.
C. Simultaneous entanglement flows
For multiple Alice-Bob pairs, spatially assigning repeaters to flows can outperform time sharing, especially when the flows’ useful repeater regions are nearly disjoint; crossing paths reduce this benefit.
- Spatial usage: The heat map pusage shows that repeaters significantly used by one flow cluster near the straight Alice-Bob line, motivating spatial division between flows.
- Non-crossing flows: For non-crossing flows, spatial division assigns red-region repeaters to flow 1 and green-region repeaters to flow 2, with the boundary determining R1 and R2.
- Non-crossing flows: Spatial division significantly outperforms single-flow time sharing because the two flows’ most useful repeater sets are almost disjoint.The flows can operate with only a very small reduction from their individual best rates.
- Crossing flows: For crossing paths, multi-flow time sharing improves on single-flow time sharing, but its maximum R1 is slightly lower because Alice 2 and Bob 2 do not assist flow 1.
- Crossing flows: For crossing paths, varying the spatial-division angle produces a further rate-region improvement, but less pronounced because useful repeater regions overlap.
III. CONCLUSIONS
The paper develops multi-path and multi-flow quantum repeater protocols using linear-chain operations, showing benefits over linear routing while identifying broader-operation and error-model limitations.
- Conclusions: The protocols use probabilistic Bell-state measurements and account for lossy channels, device inefficiencies, and probabilistic entanglement swaps.
- Conclusions: A single flow can far outperform a linear repeater chain even with local link-state knowledge, due to the multi-path routing advantage.
- Conclusions: Multi-flow routing strategies outperform rate regions obtained when repeaters simply time share among individual flows.
- Conclusions: Two-dimensional network topologies can provide substantial benefits over linear chains under constraints on quantum memories, link losses, and imperfect repeater processing.
- Limitations and open questions: The rate-optimal protocol remains open in the pure-loss abstraction, while general errors would require entanglement purification and broader fidelity accounting.
- Limitations and open questions: The analysis is restricted to two-qubit measurements; multi-qubit operations and measurements may improve achievable rate regions.
Appendix A: Distance metric for the local routing
The local routing protocol uses distance metrics to choose entanglement swaps, with results based on the L2 norm. A recursive numerical method adapts the metric to arbitrary topologies, although optimality remains unresolved.
- The protocol uses neighboring repeaters’ distances from Alice and Bob to select memories for entanglement-swap attempts.
- The paper’s results use the L2 norm as the distance metric, despite its limited generalizability beyond square-grid topologies.
- The local routing rule is not proved rate-optimal, and the optimal distance metric for a given topology remains unclear.
- A recursive numerical method begins with rates computed under the L1 norm and iteratively defines new distance metrics from those rates.
- RL2 has better rate-distance scaling than RL1, while Ri2 and RL1 are nearly indistinguishable and coincide in Fig. 8.
- The improvement ratio f(p, q)/pq increases as p decreases in [pc, 1], whereas changing q has negligible effect.
1. Numerical Evaluation
The numerical evaluation compares the local routing rule with shortest-path linear repeater chains. It shows improved rate-versus-distance scaling and examines how that improvement varies with network distance.
- The evaluation quantifies the improvement in the rate-versus-distance exponent of the local rule over a shortest-path linear chain.
- Fig. 10 depicts the network used to prove that rate scaling with Alice-Bob Manhattan distance is better for local routing than for a shortest-path linear chain.
2. Analytical lower bound on the rate achieved by the local routing rule
The paper derives an analytical lower bound for the L2-based local routing rule by restricting attention to selected multi-path configurations. The bound establishes better rate-versus-distance scaling than a shortest-path linear chain, though it is not tight.
- The analysis derives a lower bound on the L2-based local routing rate using only paths whose external links belong to the black dashed-link set.
- The lower-bound construction accounts for external links succeeding with probability p and internal links succeeding with probability q.
- The construction allows at most one black-link path between Alice and a repeater in a time step because every such path must include link 8.
- The local rule’s selected internal links can produce a path probability p′ = p + p3(1 − p)5q2 greater than p.
- The conditional probability P(v ↔ x|A ↔ v) exceeds P(v ↔ x), because the two events share link 9 and are positively related through it.
- Because β < 1, Rloc scales with distance using a smaller exponent than Rlin, establishing better rate-versus-distance scaling for multi-path routing.
- The derived lower bound is not tight, so it demonstrates the multi-path advantage without fully characterizing the local rule’s achievable rate.