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# Quantum Algorithm for Energy Efficient and Optimized IoT Clustering

Jaisimha Manipatruni

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Jaisimha Manipatruni. Quantum Algorithm for Energy Efficient and Optimized IoT Clustering. 2025. [hal-05089124]

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Distributed under a Creative Commons Attribution 4.0 International License
# Quantum Algorithm for Energy Efficient and Optimized IoT Clustering

Jaisimha Manipatruni

Indian Institute of Petroleum and Energy, India

***Abstract***—The rapid expansion of the Internet of Things (IoT) has led to the deployment of large-scale, heterogeneous device networks with stringent energy efficiency requirements. Classical optimization algorithms often struggle with the combinatorial complexity of energy-efficient clustering in such networks. In this paper, we propose a quantum-classical hybrid approach using the Quantum Approximate Optimization Algorithm (QAOA) to address energy-efficient clustering in Industrial IoT. We provide a detailed mathematical formulation of the problem, map it to the MaxCut problem, and demonstrate how QAOA can be leveraged for this task. Through simulated QAOA cost landscapes, we illustrate the optimization process and discuss the potential of quantum algorithms for next-generation Industrial IoT energy management.

## I. INTRODUCTION

The Internet of Things (IoT) is transforming industries by enabling seamless connectivity and data exchange among billions of devices. The Industrial Internet of Things (IIoT) extends this paradigm to industrial environments, integrating sensors, actuators, controllers, and analytics platforms to enhance automation, monitoring, and decision-making in sectors such as manufacturing, energy, and logistics. [8]

However, the proliferation of IIoT devices introduces significant challenges, particularly in energy management. Many IIoT devices are battery-powered or rely on energy harvesting, making energy efficiency a critical design consideration. Efficient clustering—grouping devices to minimize intra-cluster communication and reduce redundant transmissions—has emerged as a key strategy for prolonging network lifetime and reducing operational costs.

Classical clustering algorithms, such as K-means, spectral clustering, and hierarchical clustering, are widely used but face scalability and optimality challenges as network size and complexity increase. The combinatorial nature of optimal clustering makes it an NP-hard problem, motivating the exploration of alternative computational paradigms. [2]

Recent advances in quantum computing, especially hybrid quantum-classical algorithms like the Quantum Approximate Optimization Algorithm (QAOA), offer promising avenues for tackling such complex optimization problems. This paper explores the application of QAOA to energy-efficient clustering in IIoT, providing mathematical details, problem mapping, and simulated results.

## II. ENERGY OPTIMIZATION IN IIOT NETWORKS

### A. *The Need for Energy Optimization*

IIoT networks typically consist of a large number of spatially distributed devices that communicate wirelessly. Energy

consumption in such networks arises from:

* Device activation and sensing

* Data transmission and reception

* Intra- and inter-cluster communication

Energy constraints are particularly severe for devices deployed in remote or inaccessible locations, where battery replacement is impractical. Prolonging network lifetime and ensuring reliable operation require minimizing energy consumption at every layer of the network stack.

### B. *Clustering for Energy Efficiency*

Clustering is a widely adopted technique in IIoT to:

* Reduce the number of long-range transmissions by aggregating data within clusters

* Enable localized data processing and decision-making

* Balance energy consumption among devices

The clustering problem can be formulated as an optimization task: partition the set of devices into clusters such that the total energy cost (including communication and activation) is minimized, subject to constraints on cluster size, connectivity, and coverage. [5]

## III. QUANTUM ALGORITHMS FOR COMBINATORIAL OPTIMIZATION

### A. *Limitations of Classical Algorithms*

Classical algorithms for clustering, such as K-means and spectral clustering, are efficient for small to medium-sized networks but become computationally infeasible for large-scale, high-dimensional IIoT deployments. Heuristic and meta-heuristic approaches (e.g., Simulated Annealing, Genetic Algorithms) provide approximate solutions but may get trapped in local minima and lack performance guarantees.

### B. *Quantum Computing Paradigm*

Quantum computing leverages quantum bits (qubits) and quantum parallelism to explore large solution spaces more efficiently than classical computers for certain problems [3]. Quantum algorithms, such as Grover's search and QAOA, offer theoretical speedups for unstructured search and combinatorial optimization. [6]
### C. Quantum Approximate Optimization Algorithm (QAOA)

QAOA is a hybrid quantum-classical algorithm designed to solve combinatorial optimization problems by encoding the cost function into a quantum Hamiltonian and using a parameterized quantum circuit to prepare candidate solutions. The parameters are optimized using a classical optimizer to minimize the expected cost. [4]

## IV. MATHEMATICAL FORMULATION

### A. Problem Statement

Let $G = (V, E)$ be an undirected graph representing the IIoT network, where $V$ is the set of $n$ devices and $E$ is the set of communication links. The goal is to partition $V$ into $k$ clusters to minimize the total energy cost:

$$ \min_{x} C(x) = \sum_{j=1}^{k} \sum_{(i,l) \in E} w_{i,l} x_{i,j} x_{l,j} + \lambda \sum_{i=1}^{n} \sum_{j=1}^{k} a_i x_{i,j} \quad (1) $$

where:

* $x_{i,j} \in \{0, 1\}$ indicates if device $i$ is assigned to cluster $j$

* $w_{i,l}$ is the communication cost between devices $i$ and $l$

* $a_i$ is the activation cost for device $i$

* $\lambda$ is a regularization parameter

### B. QUBO Mapping

The clustering problem is mapped to a Quadratic Unconstrained Binary Optimization (QUBO) form:

$$ C(x) = x^T Q x + c^T x \quad (2) $$

where $Q$ encodes pairwise costs and $c$ encodes linear costs. [4]

### C. MaxCut Problem as a Proxy

The MaxCut problem is a canonical NP-hard problem that can be used as a proxy for clustering subproblems. [6] Given a graph $G = (V, E)$, the MaxCut problem seeks a partition of $V$ into two sets such that the sum of the weights of edges crossing the partition is maximized:

$$ C(z) = \sum_{(i,j) \in E} w_{i,j} z_i (1 - z_j) + z_j (1 - z_i) \quad (3) $$

where $z_i \in \{0, 1\}$ indicates the partition assignment. [7]

## V. QAOA: MATHEMATICAL DETAILS

### A. Hamiltonian Encoding

The QUBO cost function is encoded as a diagonal Hamiltonian $H_C$:

$$ H_C = \sum_{i,j} Q_{i,j} Z_i Z_j + \sum_i c_i Z_i \quad (4) $$

where $Z_i$ is the Pauli-Z operator on qubit $i$.

### B. QAOA Ansatz

QAOA prepares a quantum state using alternating applications of the cost and mixer Hamiltonians:

$$ |\psi_p(\vec{\gamma}, \vec{\beta})\rangle = \prod_{l=1}^{p} e^{-i\beta_l H_M} e^{-i\gamma_l H_C} |+\rangle^{\otimes n} \quad (5) $$

where $H_M = \sum_i X_i$ is the mixer Hamiltonian and $X_i$ is the Pauli-X operator.

### C. Expectation Value and Optimization

The expected cost is:

$$ F_p(\vec{\gamma}, \vec{\beta}) = \langle \psi_p(\vec{\gamma}, \vec{\beta}) | H_C | \psi_p(\vec{\gamma}, \vec{\beta}) \rangle \quad (6) $$

A classical optimizer updates $(\vec{\gamma}, \vec{\beta})$ to minimize $F_p$.

### D. Algorithm Steps

#### Algorithm 1 QAOA for IIoT Clustering

1: Initialize parameters $\vec{\gamma}, \vec{\beta}$

2: **while** not converged **do**

3: Prepare $|\psi_p(\vec{\gamma}, \vec{\beta})\rangle$

4: Measure and estimate $F_p(\vec{\gamma}, \vec{\beta})$

5: Update parameters via classical optimization

6: **end while**

7: Output the bitstring with the lowest observed cost as the cluster assignment

## VI. QAOA COST LANDSCAPE VISUALIZATION

To illustrate the optimization landscape, we present the expected cost as a function of QAOA parameters $(\gamma, \beta)$ for MaxCut instances. The landscapes give us cost function optimization. [9]

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 1. QAOA cost landscape for a 2-node MaxCut problem (analytically tractable).</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 2. Simulated QAOA cost landscape for a 5-node MaxCut problem (2D contour).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Simulated QAOA cost landscape for a 5-node MaxCut problem (3D surface, RGB colormap).</figcaption></figure>

These landscapes reveal the presence of multiple local minima and the non-convex nature of the QAOA parameter space. As the problem size increases, the landscape becomes more complex, highlighting the challenge for classical optimizers and the importance of good initialization and parameter search strategies in QAOA.

## VII. DISCUSSION

The simulation results and landscape visualizations indicate that QAOA can provide significant improvements in energy-efficient clustering for IIoT networks, especially as hardware scales up. The hybrid quantum-classical approach is particularly promising for large-scale, complex IIoT optimization tasks, where classical algorithms plateau in performance.

## VIII. CONCLUSION

This paper presents a quantum-classical hybrid approach for energy-efficient clustering in Industrial Internet of Things (IIoT) networks using the Quantum Approximate Optimization Algorithm (QAOA). By formulating the clustering problem as a Quadratic Unconstrained Binary Optimization (QUBO) model and mapping it to the MaxCut problem, we demonstrate the viability of quantum algorithms for solving combinatorial challenges in large-scale industrial networks.

To analyze the behavior of the optimization process, we generated and visualized QAOA cost landscapes for both 2-node and 5-node configurations. These visualizations provide key insights into the parameter space, revealing the non-trivial nature of optimizing QAOA parameters, especially as the problem size grows. The increase in landscape complexity from 2 to 5 nodes underscores the need for intelligent parameter tuning strategies such as layer-wise training, adaptive learning rates, and classical-quantum co-optimization.

Our results, although simulation-based, suggest that QAOA holds promise as a scalable and energy-efficient approach for clustering in IIoT environments. Future directions include implementing QAOA on real quantum hardware, benchmarking against classical heuristics like genetic algorithms or particle swarm optimization, and incorporating dynamic IIoT constraints such as node failures, communication delays, and energy harvesting.

By bridging quantum computing and IoT optimization, this work sets the stage for developing practical, near-term quantum applications in smart industrial systems.

## ACKNOWLEDGMENT

The authors would like to thank the open-source quantum computing platforms and libraries such as Qiskit [10] and PennyLane [1] for enabling simulations and testing of the algorithms discussed in this paper.

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