realm of power systems, quantum technology is increasingly recognized as a transformative force for addressing numerous computational challenges. For instance, quantum optimization algorithms offer the potential to efficiently address complex power system tasks related to planning, operation, and control. These tasks, such as optimal power flow, unit commitment, economic dispatch, and voltage stability, have traditionally posed difficulties due to their large-scale, nonlinear, and mixed-integer nature. Moreover, quantum machine learning techniques have the ability to substantially enhance data analysis and decision-making in power systems. They help to detect faults, estimate states, forecast loads, respond to demand, and integrate renewable energy by extracting valuable information from large and noisy datasets, facilitating rapid and precise predictions and classifications.

However, the promise of quantum computing, despite its vast potential, remains in its early stages. Present limitations, including constraints on the number of qubits and inherent noise, impede its immediate widespread adoption. Nevertheless, the amalgamation of quantum methodologies with power system challenges has the potential not only to revolutionize the energy sector but also to accelerate the practical application of quantum technologies in real-world scenarios.

Quantum computing offers a distinct advantage in its ability to tackle complex problem-solving tasks. Utilizing the quantum phenomenon of superposition, where qubits can exist in multiple states simultaneously (0, 1, or any combination thereof), quantum computers excel in solving specific problem classes with greater speed and efficiency compared to their classical counterparts.

The field of quantum computing has witnessed significant progress in recent years, particularly in the development of quantum algorithms that surpass classical algorithms in specific tasks. Notable among these is Shor's algorithm [11,12], which excels at factoring large numbers, an endeavor traditionally considered nearly impossible for classical computers. Similarly, Grover's algorithm provides an accelerated method for searching within unordered databases [13,14].

Recent advances in quantum computing have led to the creation of more powerful and sophisticated quantum machines. Unlike the predecessor quantum computers, contemporary systems possess enhanced computational capabilities. For instance, IBM introduced the groundbreaking Osprey processor, which features 433 qubits, a significant step toward applying quantum computers to previously unsolvable problems [15].

With swift progress in quantum processor development, the establishment of a quantum computational paradigm is on the horizon. This marks a pivotal milestone in the journey toward practical quantum computing and ushers in an era of quantum-centric supercomputing. However, despite these advancements, the realization of practical quantum computers and the formulation of quantum algorithms for real-world challenges remain ongoing challenges. Researchers are actively enhancing the stability and scalability of quantum hardware while also crafting new quantum algorithms and software tailored to leverage the unique attributes of quantum systems. In sum, quantum computing is a rapidly evolving field brimming with potential for addressing some of the most formidable computational problems in renewable energy systems.

Quantum computing offers a powerful solution for the study of power systems [16], with the goal of solving computational problems at a pace surpassing classical algorithms on classical computers.

Quantum computers harness quantum-mechanical phenomena like superposition and entanglement, enabling operations within exponentially large Hilbert spaces while incurring only polynomial overheads.

Mathematically, the Dirac notation is frequently employed to denote the state of a quantum bit, commonly known as a qubit. This concept parallels the fundamental unit of classical computation, the bit, which can exist in a state of either 0 or 1. The key distinction between a qubit and a bit lies in the qubit's ability to inhabit states beyond |0⟩ = [1 0] T or |1⟩ = [0 1] T . These states, |0⟩ and |1⟩, serve as the computational basis states for a single qubit, forming an orthonormal basis for the associated vector space.

In quantum computing, it is possible to create linear combinations of |0⟩ and |1⟩ states, known as superpositions. Typically, superpositions of a single qubit are visually represented using a unit three-dimensional Bloch sphere, as depicted in Figure 12.1. Multiple correspondences are given, such as,

which are commonly used for quantum computation. Therefore, according to Figure 12.1, a quantum state |Ψ⟩ can be calculated as (12.2), where 𝜃 and 𝜑 are the angles given in Figure 12.1.

Quantum mechanics reveals that a qubit can exist in a continuum of states spanning between |0⟩ and |1⟩ until it undergoes observation. Upon measurement, a qubit yields either the outcome 0 with a probability of | cos 𝜃 2 | 2 or the outcome 1 with a probability of |e i𝜑 sin 𝜃 2 | 2 . When multiple qubits are involved in computations, the operation of the tensor product becomes essential [16]. As a result, employing N qubits leads to the development of a vector space of 2 N , illustrating the concept of quantum supremacy.

Variational Quantum Computing (VQC) represents a synergy of quantum and classical elements [17,18]. In this framework, quantum computers handle the preparation and measurement of quantum states, while classical computers refine parameters based on these measurements to enhance an objective function. Consequently, variational quantum computing technology naturally offers several advantages.

• Compatibility with the Noisy Intermediate-Scale Quantum (NISQ) Era: We currently find ourselves in the NISQ phase, characterized by advanced yet non-fully fault-tolerant quantum devices prone to errors. Given these limitations, variational quantum algorithms stand as particularly well-suited for NISQ devices. They rely on shorter quantum circuits, which mitigates the impact of noise and errors.

• Inherent Hybrid Nature: Variational quantum algorithms exhibit a hybrid design, seamlessly integrating quantum and classical functionalities. In this configuration, the quantum component is responsible for state creation and evaluation, while its classical counterpart meticulously adjusts parameters. This collaborative interplay facilitates error mitigation, optimizing the inherent strengths of both classical and quantum computing domains.

To improve the resilience of the renewable energy system, it becomes imperative to efficiently analyze the flow of power in the physical layer and the data traffic in the cyber layer. Some computational tasks, such as optimal power flow and unit commitment, can be mathematically framed as a combinatorial optimization problem, a well-known NP-hard problem [19]. In practical scenarios, classical approximation algorithms are often employed to tackle these problems [20][21][22][23].

The Quantum Approximate Optimization Algorithm (QAOA), a hybrid quantum-classical algorithm, has the promise of delivering better approximate solutions compared to existing classical algorithms [24,25]. QAOA employs classical computation to optimize parameters for a quantum circuit [25]. This parameterized quantum circuit approximates the adiabatic evolution from an initial Hamiltonian, featuring a readily preparable ground energy state, to a final Hamiltonian whose ground energy state encodes the problem solution. An ideal approximation aims to produce an exact solution with high probability [26]. Consequently, the parameters governing the quantum circuit play a pivotal role in achieving high-quality approximations [27][28][29]. However, the efficient determination of the suitable parameters remains an open question.

Mathematically, a renewable energy system can be described as a weighted graph G = (V, E), where |V| = n denotes the number of vertices, |E| = m signifies the number of edges, and 𝑤 ij represents the normalized weight of the edge ⟨i, j⟩ ∈ E, with Max(𝑤 ij ) = 1. The edge weights are derived from power flow calculations for the physical layer and data traffic for the cyber layer.

Many computational tasks within the system can be framed as finding a subset S ⊂ V that maximizes ∑ i∈S,j∉S 𝑤 ij for the cyber or physical layers, respectively. To represent the status of vertices V, we use an n-bit string

where each bit z i equals 1 if the ith vertex is in subset S, otherwise -1. This partition of vertices aids in obtaining the solution. Consequently, the classical cost function is defined as follows:

where C ij (Z) represents the contribution of 𝑤 ij to the cost function.

In quantum computing, we use n qubits

represent the status of n vertices. Each qubit |z i ⟩ can exist in a superposition of quantum states |0⟩ and |1⟩, denoted as |z i ⟩ = a i |0⟩ + b i |1⟩, where |0⟩ and |1⟩ are the eigenstates of the Pauli-Z operator 𝜎 z . Measurement in the computational basis yields outcomes based on the probabilities |a i | 2 and |b i | 2 , introducing variability into measurement results. For deterministic n-bit strings Z k obtained from measurements on 2 n n-qubit eigenstates in the computational basis (|Z k ⟩ with |z k,i ⟩ = |0⟩ or |1⟩ and z k,i = ⟨z k,i |𝜎 z i |z k,i ⟩), we define the quantum cost function as follows:

with the Hamiltonian H C given by, QAOA employs a quantum circuit to approximate the adiabatic evolution from the maximum energy state of an initial Hamiltonian, H B , to the maximum energy state of the final Hamiltonian, H C . The essential idea of QAOA is depicted in Figure 12.2 [29]. According to the adiabatic theorem [30], in an ideal scenario, we expect to obtain the maximum energy state of H C with a high probability, leading to the solution. Implementation on quantum computers involves preparing the initial state |+⟩ ⨂ n , executing the quantum circuit with 2p trainable parameters 𝛾 = ( 𝛾 1 , 𝛾 2 , … , 𝛾 p ) and 𝛽 = ( 𝛽 1 , 𝛽 2 , … , 𝛽 p )

, measuring the output state, and estimating the quantum cost function. A classical-quantum hybrid optimizer is then used to maximize the quantum cost function by iterating over the parameters 2p, resulting in solutions with high approximation ratios even with finite p.

The effectiveness of QAOA heavily relies on the crucial parameters 𝛾 and 𝛽. Previous studies have extensively examined the efficiency and precision of QAOA while keeping the circuit depth consistent [25,[31][32][33][34][35]. Heuristic approaches have shown promise in identifying near-optimal parameters that yield satisfactory solutions, as supported by numerical evidence [29,32,33]. However, a comprehensive exploration of these heuristic approaches in real-world applications remains a challenging endeavor [31].

To address this issue, data-driven methods provide a powerful tool for QAOA. This is to directly provide high-approximation-ratio solutions without the need for parameter optimization, thereby avoiding expensive computational efforts. Multiple metrics can be leveraged to develop the data-driven method, such as the normalized weighted graph density [36,37]. The essential idea of data-driven methods can be outlined below.

1) Formulate the cost function in quantum format.

2) Obtain quasi-optimal parameters (𝛾, 𝛽) by developing a proper parameter warm-up strategy and subsequently transmitting these parameters to the quantum processors. 3) Construct the quantum circuit utilizing the adjacency matrix and the parameters (𝛾, 𝛽), and execute it on a quantum processor. Then, measure the output state of the quantum circuit to obtain the probability distribution and compute the value of the cost function. 4) Optimize the parameters when necessary, using a classical optimizer to achieve improved results.

The effective initial parameter guesses (𝛾, 𝛽) play a key role in addressing the problems of barren plateaus [38]. These initial estimates also streamline the interactions between the classical optimizer and the quantum processor, resulting in algorithmic time savings.

The fundamental concept behind the data-driven QAOA method is the parameter warm-up strategy. This strategy typically comprises the following three steps, which are designed to significantly enhance the efficacy of parameter transfer. 1) Establish an initial database for the parameter warm-up strategy. Generate several random seed graphs, each with normalized graph densities ranging from 0 to 1. Given the small size of these graphs, quasi-optimal parameters (𝛾, 𝛽) can be calculated by Brandao et al. [32] and Guerreschi and Smelyanskiy [34]. These parameters offer potential quasi-optimal values for new graphs. 2) Develop a mapping table designed to transfer quasi-optimal parameters from seed graphs to target graphs, assuming that they have the same circuit layer number p. Perform QAOA calculations for each target graph using the parameters obtained from the seed graphs to obtain the cost function values. Based on these values, organize them into a mapping

table. In this table, each column corresponds to one target graph, and each row corresponds to one seed graph. 3) Transfer parameters to new graphs. For a new graph with a normalized graph density, select suitable seed graphs from the mapping table, ensuring that their size is equal to or close to that 10.1002/9781394334599.ch12, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/9781394334599.ch12 by Pennsylvania State University, Wiley Online Library on [10/12/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 12.3 Typical Applications of Quantum Computing 319

of the new graph. On the basis of the obtained entries, identify and transfer the parameters in the pair corresponding to each entry to the new graph.

To improve the precision of the result, we can increase the layer number p accordingly using the parameters obtained from the parameter warm-up strategy.

In summary, the central idea of data-driven QAOA involves multiple key steps. First, we mode the cyber-physical system as two normalized weighted graphs and compute the normalized graph density from the adjacency matrix. Second, in the parameter transfer module, determine the seed graph size and layer number for the QAOA circuit. Then, according to the mapping table, obtain quasi-optimal parameters (𝛾, 𝛽) from seed graphs whose normalized densities are close to the target graph. Third, pass the transferred parameters (𝛾, 𝛽) to the quantum circuit with multiple layers for QAOA. Through measurement, generate the probability distribution, from which we can obtain the solution. Fourth, if improved performance is desired, further optimize (𝛾, 𝛽) for C(|Z⟩) max . This step is optional. In addition, the obtained parameters can also be used to develop an expandable quasi-optimal parameter database for providing quasi-optimal parameters for new target graphs.

Taking into account the advantages of quantum computing mentioned above, it has the potential to bring about significant advances in the field of power systems by leveraging the capabilities of quantum computers [39][40][41][42][43][44]. Potential applications include but are not limited to: 1) Integration of renewable energy: Quantum computing can help optimize the integration of renewable energy sources into the grid by forecasting renewable generation, managing energy storage systems, and balancing supply and demand. 2) Optimal power flow optimization: Quantum computing can be used to solve complex optimization problems, helping the real-time distribution and allocation of power within a grid to minimize losses and improve efficiency. Quantum algorithms have the potential to provide faster solutions to large-scale problems. 3) Energy market optimization: Quantum computing can assist in optimizing energy market operations by modeling supply and demand dynamics, pricing mechanisms, and market clearing processes, ultimately leading to more efficient energy trading. 4) Cybersecurity: Quantum computing can enhance the cybersecurity of power systems by developing quantum-resistant encryption methods and algorithms to protect critical infrastructure from cyberattacks. 5) Real-time monitoring and control: Quantum-enhanced sensors and control systems can provide real-time monitoring and control of power grid components, helping to maintain system stability and prevent cascading failures. 6) Resource allocation for microgrids: Quantum computing can optimize resource allocation in microgrids, ensuring reliable and cost-effective energy supply to local communities, even during grid outages. 7) Grid resilience analysis: Quantum computing can be employed to analyze the resilience of the power grid by simulating various failure scenarios and assessing their impact on the grid. This helps to design more robust and reliable power systems. 8) Load forecasting: Quantum machine learning algorithms, supported by VQC, have the potential to improve load forecasting accuracy by analyzing historical data and complex patterns, leading to better demand-side management and resource allocation. 9) Fault detection and diagnostics: Quantum computing can analyze data from sensors and smart grid devices to detect faults and diagnose issues in real time, allowing proactive maintenance and minimizing downtime. 10) Carbon emission reduction: Quantum computing can aid in the development of strategies to reduce carbon emissions in power systems by optimizing energy generation from cleaner sources and minimizing the use of fossil fuels.