Quantum Estimation of Delay Tail Probabilities in Scheduling and Load Balancing

Estimating delay tail probabilities in scheduling and load balancing systems is a critical but computationally prohibitive task due to the rarity of violation events. Quantum Amplitude Estimation (QAE) offers a generic quadratic reduction in sample complexity 1/sqrt(p) vs 1/p, but applying it to steady-state queueing networks in challenging: classical simulations involve unbounded state spaces and random regeneration cycles, whereas quantum circuits have fixed depth and finite registers. In this paper, we develop a framework for quantum simulation of delay tail probabilities based on truncated regenerative simulation. We show that regenerative rare-event estimators can be reformulated as deterministic, reversible functions of finite random seeds by truncating regeneration cycles. To control the resulting bias, we use Lyapunov drift and concentration arguments to derive exponential tail bounds on regeneration times. This allows the truncation horizon–and hence the quantum circuit depth–to be chosen such that the bias is provably negligible compared to the statistical error. The proposed framework enables quantum estimation in models with countably infinite state spaces, avoiding the challenge of determining the sufficient mixing time required for direct finite-horizon simulation. We provide bounds on qubit and circuit complexity for a GI-GI-1 queue, a wireless network under MaxWeight scheduling, and a multi-server system with Join-the-Shortest-Queue (JSQ) routing.

💡 Research Summary

The paper tackles the notoriously difficult problem of estimating very small delay‑tail probabilities (e.g., P{delay ≥ d} ≈ 10⁻⁶–10⁻⁹) in queueing, scheduling, and load‑balancing systems. Classical Monte‑Carlo (CMC) requires O(1/p) samples to achieve a fixed relative error, which is infeasible for such rare events. While variance‑reduction techniques (importance sampling, splitting, cross‑entropy, etc.) can dramatically cut the sample size, they usually demand problem‑specific large‑deviation analyses and careful tuning, limiting their applicability to complex, high‑dimensional networks.

Quantum Amplitude Estimation (QAE) offers a generic quadratic speed‑up: estimating a probability p with error ε needs only O(1/√p · log(1/ε)) oracle calls, independent of the underlying distribution. However, applying QAE directly to steady‑state queueing models faces two fundamental obstacles. First, queueing dynamics are defined on continuous time, have countably infinite state spaces, and involve random regeneration cycles of unbounded length. Second, QAE assumes a fixed‑depth, reversible unitary oracle that maps a finite seed to a value in