Distributed Quantum Computing

• Distributed quantum computing is a paradigm where multiple quantum processors collaborate via quantum networks to execute complex algorithms beyond single-device capacity.
• It employs stochastic programming to optimize resource allocation and scheduling by balancing pre-reserved and on-demand QPU resources under uncertainty.
• Simulation studies demonstrate that adaptive scheduling and topology-aware resource allocation outperform deterministic strategies in cost management and task scaling.

Distributed quantum computing (DQC) is a paradigm in which multiple quantum processing units (QPUs)—potentially separated by significant physical distances and using heterogeneous hardware—collaborate through quantum networks to execute quantum algorithms that exceed the capacity of any individual device. DQC is motivated by both the intrinsic scaling limits in single-device architectures and the emerging practicality of quantum communication protocols, especially entanglement generation and gate teleportation, to overcome the no-cloning constraint on quantum information. Research in DQC addresses algorithmic, architectural, and resource management challenges, seeking both theoretical scalability and high-fidelity operation under the noisy, resource-constrained conditions characteristic of near-term quantum devices.

1. Architectural Principles and Physical Constraints

DQC architectures comprise clusters of quantum processors (QPUs) interconnected by quantum channels capable of supporting entanglement distribution and classical side channels for feed-forward operations. The central architectural constraint is the impossibility of simply moving or copying quantum information between arbitrary QPUs due to the no-cloning theorem. Instead, quantum teleportation and gate teleportation protocols—using entangled states such as Bell pairs—form the foundation for all inter-QPU quantum data and gate transfers.

The architecture often features two classes of qubits:

• Data (computing) qubits: Core registers in each QPU where quantum computations are performed.
• Communication qubits: Interface qubits responsible for generating, storing, and managing entangled links to remote QPUs.

In advanced pooling frameworks (e.g., shared quantum gate processing units such as S-QGPU), communication resources are centralized, reducing per-node hardware requirements and enabling dynamic resource allocation that scales more efficiently under bursty remote gate requests (Du et al., 2023).

Resource management, connectivity, and physical separation collectively lead to problems in decoherence, link infidelity, and probabilistic entanglement generation. These fundamental constraints necessitate sophisticated strategies for resource allocation, circuit scheduling, and topological adaptation.

2. Resource Allocation and Scheduling under Uncertainty

Resource allocation in DQC must balance computational requirements, network constraints, and the uncertain quality of quantum resources. A salient model is the two-stage stochastic programming formulation (Ngoenriang et al., 2022), which abstracts resource assignment into:

• First-stage decisions: Pre-task reservations of QPUs for future jobs, expressed as binary variables $x_j \in \{0,1\}$ with associated deployment costs $c_j^{(dep)}$.
• Second-stage decisions: Operational choices made after actual network and computational scenario $\omega$ is revealed—selecting which reserved QPUs $\tilde{x}_j(\omega)$ and on-demand resources $y_r(\omega)$ to activate, subject to computing power and fidelity constraints.

The overall objective is to minimize the expected total deployment cost: $$\min_{x_j, \tilde{x}_j(\omega), y_r(\omega)} \sum_j c^{(dep)}_j x_j + \mathbb{E}\left[ \mathcal{Q}(\tilde{x}_j, y_r, \omega) \right]$$ where $\mathcal{Q}$ encapsulates operational costs (computing, Bell pair establishment, on-demand activation).

Critically, this model incorporates three classes of uncertainty:

1. Demand uncertainty – Task qubit requirements, modeled as a random variable $\tilde{n}(\omega)$.
2. Available computing power uncertainty – Each machine’s $k_j(\omega)$ may fluctuate due to concurrent tasks or degraded performance.
3. Quantum channel fidelity uncertainty – Transmission or teleportation quality, quantified by the scenario-dependent fidelity $\tilde{q}_{i,j}(\omega)$ of shared Bell pairs.

Constraints are enforced to guarantee that deployed and on-demand resources collectively provide sufficient qubit capacity to meet computational demand, and that quantum links have both sufficient capacity and fidelity to support teleportation between chosen QPUs.

This formulation supports adaptive risk-aware resource management, dynamically balancing the low predictable cost of reserved QPUs against the higher, scenario-driven cost of activating expensive on-demand resources only when necessitated by adverse realizations (e.g., low channel fidelity or unexpectedly high task demand).

3. Performance Analysis: Cost, Provisioning, and Adaptivity

The stochastic allocation framework undergoes detailed numerical evaluation in varied task and network scenarios (Ngoenriang et al., 2022), demonstrating salient performance characteristics:

• Task Demand Scaling: Increasing the quantum task size (measured in qubits) directly increases the number of QPUs that need deployment; if fixed QPU resources fall short, the allocation seamlessly transitions to using on-demand resources, maintaining task completion at increased but bounded cost.
• Computing Power Sensitivity: Higher per-node computational power reduces overall required deployments and cost, illustrating the economic value in maximizing per-QPU efficiency.
• Fidelity Dependence: When channel fidelity drops below certain thresholds (e.g., 0.5), remote QPU collaboration becomes infeasible, thereby necessitating fallback to local or on-demand non-collaborative computation at higher cost.
• Scenario and Cost Parameter Effects: The model exhibits cost sensitivity to both the likelihood of “favorable” scenarios (i.e., those with high fidelity and power) and the relative pricing of on-demand services, allowing an operator to tune deployment risk levels.

The stochastic scheduler consistently outperforms deterministic or random assignment strategies, achieving lower expected provisioning costs by dynamically navigating uncertainty and exploiting scenario structure.

4. Model Formulation, Constraints, and Optimization Strategies

The computational heart of the framework is a mixed-integer nonlinear programming (MINLP) problem. Decision variables include first-stage deployment choices $x_j$ (binary), second-stage operational decisions $\tilde{x}_j(\omega)$ (binary, dependent on first-stage assignments), and $y_r(\omega)$ (on-demand activations).

Key constraints include:

• Utilization Constraint: $\tilde{x}_{j}(\omega) \leq x_j$, ensuring only reserved QPUs can be operationalized in the second stage.
• Task Completion Constraint: Ensuring the sum of actual available computing power meets the exponentially-scaling requirement for $n$-qubit computations.
• Link Capacity/Fidelity Constraint: $$C_{i,j} \tilde{q}_{i,j}(\omega) \ge \min\{\tilde{k}_i(\omega)\tilde{x}_i(\omega), \tilde{k}_j(\omega)\tilde{x}_j(\omega)\}$$, guaranteeing successful quantum information transfer.

The MINLP formulation enables practical solution with commercial or open-source optimization solvers, rendering deployment recommendations scalable and reproducible.

Table 1 summarizes the variable and constraint structure:

Variable	Domain	Interpretation
$x_j$	$\{0,1\}$	First-stage: Reserve QPU $j$
$\tilde{x}_j(\omega)$	$\{0,1\}$	Second-stage: Use QPU $j$
$y_r(\omega)$	$\{0,1\}$	Use on-demand QPU $r$
$c_j^{(dep)}$, $c_j^{(com)}$, $c_{i,j}^{(Bell)}$, $c_r^{(ond)}$	$\mathbb{R}_+$	Cost coefficients
$\tilde{k}_j(\omega)$, $g_r$	$\mathbb{N}$	Computing power (scenario/on-demand)
$\tilde{q}_{i,j}(\omega)$, $C_{i,j}$	$[0,1], \mathbb{R}_+$	Link fidelity/capacity

5. Comparative Effectiveness and Trade-offs

Adopting the stochastic programming approach enables the DQC operator to achieve robust, cost-minimizing resource allocation in environments characterized by limited, fluctuating, or uncertain quantum resources. Systematic simulation shows the approach:

• Outperforms deterministic or random allocation strategies in both provisioning cost and adaptability.
• Responds efficiently to varying relative pricing between in-house reserved and external on-demand quantum resources.
• Offers resilience to deteriorating communication conditions by prioritizing reliable quantum networks when available and dynamically switching to backup resources if required.

Crucially, it is capable of minimizing the need for expensive “just-in-case” resource reservations, which would otherwise inflate operational costs in networks subject to large fluctuations or partial failures.

6. Implications and Contributions for Distributed Quantum Computing

The outlined resource management and provisioning framework provides several methodological and practical advances for the DQC field:

• Introduces an optimal, risk-aware resource allocation paradigm that encompasses both computational capacity and quantum networking resources under stochasticity.
• Enables practical deployment planning for DQC operators, offering a template for "quantum cloud" providers and research clusters as large-scale DQC deployment matures.
• Demonstrates the integration of detailed network and hardware uncertainty into actionable decision processes, in contrast to prior static or deterministic modeling paradigms.
• The MINLP formulation supports extensibility to account for additional network and operational constraints.

By operationalizing uncertainty-aware and cost-effective allocation, this framework helps lower the barrier to practical, scalable DQC and sets a benchmark for future studies on dynamic scheduling, topology-aware resource allocation, and integration with error correction or higher-level compiler optimizations. The model is particularly relevant for scenarios with variable quantum channel quality, time-varying device participation, and multi-tenant resource sharing.