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LP-Based Algorithms for Scheduling in a Quantum Switch

R. Srikant

Abstract

We consider scheduling in a quantum switch with stochastic entanglement generation, finite quantum memories, and decoherence. The objective is to design a scheduling algorithm with polynomial-time computational complexity that stabilizes a nontrivial fraction of the capacity region. Scheduling in such a switch corresponds to finding a matching in a graph subject to additional constraints. We propose an LP-based policy, which finds a point in the matching polytope, which is further implemented using a randomized decomposition into matchings. The main challenge is that service over an edge is feasible only when entanglement is simultaneously available at both endpoint memories, so the effective service rates depend on the steady-state availability induced by the scheduling rule. To address this, we introduce a single-node reference Markov chain and derive lower bounds on achievable service rates in terms of the steady-state nonemptiness probabilities. We then use a Lyapunov drift argument to show that, whenever the request arrival rates lie within the resulting throughput region, the proposed algorithm stabilizes the request queues. We further analyze how the achievable throughput depends on entanglement generation rates, decoherence probabilities, and buffer sizes, and show that the throughput lower bound converges exponentially fast to its infinite-buffer limit as the memory size increases. Numerical results illustrate that the guaranteed throughput fraction is substantial for parameter regimes relevant to near-term quantum networking systems.

LP-Based Algorithms for Scheduling in a Quantum Switch

Abstract

We consider scheduling in a quantum switch with stochastic entanglement generation, finite quantum memories, and decoherence. The objective is to design a scheduling algorithm with polynomial-time computational complexity that stabilizes a nontrivial fraction of the capacity region. Scheduling in such a switch corresponds to finding a matching in a graph subject to additional constraints. We propose an LP-based policy, which finds a point in the matching polytope, which is further implemented using a randomized decomposition into matchings. The main challenge is that service over an edge is feasible only when entanglement is simultaneously available at both endpoint memories, so the effective service rates depend on the steady-state availability induced by the scheduling rule. To address this, we introduce a single-node reference Markov chain and derive lower bounds on achievable service rates in terms of the steady-state nonemptiness probabilities. We then use a Lyapunov drift argument to show that, whenever the request arrival rates lie within the resulting throughput region, the proposed algorithm stabilizes the request queues. We further analyze how the achievable throughput depends on entanglement generation rates, decoherence probabilities, and buffer sizes, and show that the throughput lower bound converges exponentially fast to its infinite-buffer limit as the memory size increases. Numerical results illustrate that the guaranteed throughput fraction is substantial for parameter regimes relevant to near-term quantum networking systems.

Paper Structure

This paper contains 11 sections, 4 theorems, 67 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. Markovian Scheduling and Throughput Optimality
  4. Proposed Algorithm and Its Achievable Throughput
  5. Proposed Scheduling Algorithm
  6. Solving the LP and Decomposing the Solution Into Matchings
  7. Achievable Throughput
  8. A Lower Complexity Algorithm
  9. Throughput Performance as a function of Switch Parameters
  10. Numerical results
  11. Conclusion

Key Result

Lemma 1

Let $\mathbb{P}(L_u > 0, L_v > 0)$ be the stationary probability that the buffers at both endpoints of edge $e=(u,v)$ are non-empty under the randomized decomposition policy with fixed weights $\{w_e\}$. Let $C_u$ and $C_v$ be the steady-state availabilities of the reference chains $\tilde{L}_u$ and

Figures (2)

  • Figure 1: Results for Algorithm I.
  • Figure 2: Results for Algorithm II.

Theorems & Definitions (8)

  • Lemma 1: Availability Lower Bound via Coupling
  • proof
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2: Exponential rate in $B$
  • proof