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arXiv:2512.10558

Quantum-Amplified M/G/1/K Simulation: A Comparator-Controlled Framework for Arbitrary Service Distributions

Or Peretz, Michal Koren, Nir Perel

quant-phSubmitted Dec 11, 2025• Announced Dec 12, 2025• 0 citations
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Abstract

Finite-capacity single-server queues with general service-time distributions form the backbone of numerous real-world systems, yet classical simulation of performance metrics such as blocking probabilities and delay becomes computationally prohibitive as service variability or required precision increases. This work presents the first coherent quantum circuit for simulating an M/G/1/K queue under arbitrary service-time laws. The circuit encodes the service distribution through a logarithmic-depth ladder of $R_y$ rotations and enforces buffer constraints via a comparator-controlled phase gate, while preserving the quadratic speed-up of amplitude amplification. Grover iterations center on estimating the expected number of customers in the system, yielding provable $O(\sqrt{N})$ variance reduction and closed-form confidence bounds, where $N$ denotes the number of shots. Empirical evaluations on IBM quantum simulators across four service distributions and three traffic intensities demonstrate fidelity above 0.99 with four qubits and above 0.76 with ten qubits, with Jensen-Shannon divergence below 0.11. Waiting-time estimation errors decrease by an order of magnitude as system load approaches capacity and remain within 3% in high-traffic regimes using registers of up to 63 qubits. These results establish the first end-to-end quantum simulation framework for finite-buffer, non-Markovian queueing systems and provide a concrete foundation for quantum-accelerated performance analysis in service-oriented architectures.

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•Jan 1, 2021•150 citations
Finite-capacity single-server queues with general service-time distributions form the backbone of numerous real-world systems, yet classical simulation of performance metrics such as blocking probabilities and delay becomes computationally prohibitive as service variability or required precision increases. This work presents the first coherent quantum circuit for simulating an M/G/1/K queue under arbitrary service-time laws. The circuit encodes the service distribution through a logarithmic-depth ladder of RyR_yRy​ rotations and enforces buffer constraints via a comparator-controlled phase gate, while preserving the quadratic speed-up of amplitude amplification. Grover iterations center on estimating the expected number of customers in the system, yielding provable O(N)O(\sqrt{N})O(N​) variance reduction and closed-form confidence bounds, where NNN denotes the number of shots. Empirical evaluations on IBM quantum simulators across four service distributions and three traffic intensities demonstrate fidelity above 0.99 with four qubits and above 0.76 with ten qubits, with Jensen-Shannon divergence below 0.11. Waiting-time estimation errors decrease by an order of magnitude as system load approaches capacity and remain within 3% in high-traffic regimes using registers of up to 63 qubits. These results establish the first end-to-end quantum simulation framework for finite-buffer, non-Markovian queueing systems and provide a concrete foundation for quantum-accelerated performance analysis in service-oriented architectures.
Quantum computing is revolutionizing various fields, including operations research and queueing theory. This study presents a quantum method for simulating single-server Markovian (M/M/1) queues, making quantum computing more accessible to researchers in operations research. We introduce a dynamic amplification approach that adapts to queue traffic, potentially improving simulation efficiency, and design custom-parameterized quantum gates for arrival and service processes. This flexible framework enables modeling of various queueing scenarios while bridging quantum computing and classical queueing theory. Notably, our quantum method shows potential advantages over classical simulations, particularly in high-traffic scenarios. This quantum simulation approach opens new possibilities for analyzing complex queueing systems, potentially outperforming classical methods in challenging scenarios and paving the way for quantum-enhanced operations research. The method was implemented and tested across low-, moderate-, and high-traffic scenarios, comparing quantum simulations with both theoretical formulas and classical simulations. Results demonstrate high agreement between quantum computations and theoretical predictions, with relative errors below 0.002 for effective arrival rates in high-traffic scenarios. As the number of qubits increases, we observe rapid convergence to theoretical values, with relative errors decreasing by up to two orders of magnitude in some cases. Sensitivity analysis reveals optimal parameter regions yielding errors lower than 0.001.
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.
Quantum computing is revolutionizing various fields, including operations research and queueing theory. This work presents a transformative quantum simulation framework that addresses the long-standing challenge of accurately modeling single-server Markovian queues, a fundamental problem in both quantum computing and stochastic modeling. Our approach introduces a dynamic amplitude amplification technique that leverages custom parameterized quantum gates to model Poisson arrivals and exponential service times in real time. The core innovation lies in the adaptive adjustment of these gate parameters, which selectively amplifies critical quantum states based on evolving queue conditions. This dynamic mechanism not only accelerates convergence towards theoretical predictions but also significantly reduces computational overhead compared to static quantum methods. Empirical evaluations demonstrate accuracy across diverse traffic scenarios, with relative errors dropping below 0.002 in high-traffic regimes. By effectively bridging the gap between classical queueing theory and quantum simulation, our method opens new avenues for leveraging quantum resources to tackle complex stochastic processes in operations research.
Quantum Amplitude Amplification (QAA), the generalization of Grover's algorithm, is well-positioned for combinatorial optimization and is particularly promising for Quadratic Unconstrained Binary Optimization (QUBO) problems. QAA is appealing due to its ability to drive the quantum system to a target state, yielding the globally optimal solution with probability over 909090+%. However, realizing QAA for application-scale optimization currently exceeds quantum hardware capacity, which is further compounded by unresolved algorithmic challenges regarding the choice of free parameters. In this study, we address these issues by utilizing a realistic traffic flow QUBO problem to investigate the implementation of QAA as a heuristic solver. Specifically, allowing the diffusion operator parameter to take non-πππ values expands the capabilities of QAA. This unlocks a new multiple-shot, low-iteration strategy that aims for a high cumulative probability rather than maximizing the probability of a single basis state. Using 252525-qubit simulations, our results demonstrate that heuristic QAA addresses two of the main challenges cited in previous studies. Firstly, heuristic QAA works even at a single iteration, reducing circuit depth by orders of magnitude compared to standard QAA. And secondly, we show that the problem-dependent parameters necessary for reaching optimal algorithmic performance can be reliably approximated with minimal upfront classical computing overhead.
The convergence of quantum computing and classical service system modeling presents a transformative opportunity. This paper proposes the formalization of Quantum Queueing Theory (QQT), a novel framework for modeling, analyzing, and optimizing service systems where quantum processors act as servers, quantum algorithms constitute service tasks, and quantum communication channels facilitate arrivals and departures. We outline the fundamental challenges posed by quantum mechanics (superposition, entanglement, measurement, decoherence) to classical queueing paradigms. Key research directions include defining quantum analogues of arrival processes, service disciplines, and performance metrics (quantum fidelity, task success probability, decoherence-limited waiting time). We explore modeling approaches leveraging quantum stochastic processes, quantum walks, and modified Lindblad master equations. The paper details critical areas for future research: hybrid quantum-classical queueing networks, resource allocation under decoherence, stability analysis in the quantum regime, and the development of quantum-aware scheduling policies. QQT is poised to become essential for designing efficient and scalable quantum computing data centers, quantum cloud services, and integrated quantum-classical computing infrastructures.
We introduce a variant of Quantum Amplitude Estimation (QAE), called Iterative QAE (IQAE), which does not rely on Quantum Phase Estimation (QPE) but is only based on Grover’s Algorithm, which reduces the required number of qubits and gates. We provide a rigorous analysis of IQAE and prove that it achieves a quadratic speedup up to a double-logarithmic factor compared to classical Monte Carlo simulation with provably small constant overhead. Furthermore, we show with an empirical study that our algorithm outperforms other known QAE variants without QPE, some even by orders of magnitude, i.e., our algorithm requires significantly fewer samples to achieve the same estimation accuracy and confidence level.