# Improvement of Quantum Approximate Optimization Algorithm for Max-Cut Problems
## 1 Introduction  
Value chains linked to Industry 4.0 (I4.0) involve complex cyber-physical networks in which information is processed efficiently by humans and machines to deliver the desired product to a customer [1, 2, 3]. I4.0 and the Industrial Internet of Things (IIoT) both describe further emerging landscapes for an integrated human-machine interaction [4, 5]. Together, the two concepts are grounded in intelligent, interconnected cyber-physical manufacturing systems that are fully equipped and capable of controlling the process flow of industrial production. Given that many decisions are made independently by machines interoperating with production planning and fabrication systems, the integration of human users requires new paradigms [6]. In the realm of IIoT I4.0 manufacturing, I4.0 vision has advanced the notions of smart fabrication and smart factory by augmenting all assets with sensor-based connectivity [7]. These intelligent sensors generate a large amount of manufacturing data that helps to create digital twins as support for a live mirror of physical processes [8, 9]. The ambition is to capture process variability within this approach, with the capability to process all relevant information by analyzing big data in cloud computation so that manufacturers are able to find bottlenecks in manufacturing processes, identify the causes and impacts of problems in such a way that the effective application of measures is useful for both product design and manufacturing engineering, including maintenance, repair and overhaul [10]. Quantum near-term simulations in classical computers have been recently used to solve different applications [6, 11], including Industry 4.0 challenges such as the modelling of organizational decision networks as quantum circuits [12]. In this work, with the help of quantum simulations, a new solution for the combinatorial optimization problem is proposed, which can be applied to a wide range of applications including in the Industry4.0 environment. It consists in finding the \"optimal\" partitioning of a value chain into two classes, such that the number of connections between them is maximized. Direct applications are linked to introduce flexibility in value chain models by enabling extra resilience to it, no matter whether the related processes are logistics or production related ones. Solving this industrial process design problem potentially allows maximizing the interaction between the elements of the value chain and thus maximizes its productivity [13; 14]. Other applications can be connected to the Narrowband Internet of Things (NB-IoT) technology. NB-IoT is a cellular radio-based access protocol specified by 3 GPP to tackle the quickly growing market for low-power wide-area connectivity significantly targeting mobile use cases. To realize the global outreach and broad adoption of NB-IoT services, mobile network operators (MNOs) need to guarantee end-to-end devices and services across several vendors connected to the deployed NB-IoT systems, and that the data transport capacity and connection modes are well understood. In this context, efficient dynamic partitions depending on the low power network available are relevant for providing a robust integrative configuration with limited transport overhead. A solution to these problems can be formulated in terms of a combinatorial optimization approach, which involves finding an optimal object out of a finite set of objects. In this particular case, it involves finding \"optimal\" bitstrings composed of 0's and 1's among a finite set of bitstrings. Such bitstring represents a partition of nodes of a graph into two sets, such that the number of edges between the sets is maximum. Each of the sets represents the allocation of nodes in the value stream network or nodes in the IoT system to specific managerial structures giving a maximal flexibility by providing the highest degree of connectivity.