**Introduction **

Optimization is a way of finding the most favorable solution or condition for any given system. Engineering applications of Optimization are set to finding out the optimal designs with respect to initially given criteria or constraints. Efficient optimization not only leads to better designs and resource management, but can minimize costs, maximize profits, and improve efficiency in various applications like product design, reliability, safety and scheduling time to market. For instance, engineers are often required to identify a few appropriate design solutions, and then decide which one best meets the needs of the client.

As technology and systems get more complex, the number of variables to be optimized also goes up. Complex optimization is a time-consuming and iterative process, mainly due to the computational challenges associated with classical computing hardware and classical algorithms.

Quantum algorithms can speed up optimization process and solve many unsolvable problems due to their nature and working principles. BosonQ Psi’s Simulation of Things takes advantage of quantum algorithms to advance optimization and simulation in engineering. This article talks about optimization and optimization algorithms such as classical and quantum for better results.

Engineers can minimize costs, maximize profits, and improve efficiency in various applications like product design, reliability, safety, time-to-market, and scheduling. Making optimal decisions in the early stages of a project is crucial as changes become more costly later. Efficient optimization leads to better designs and resource management.

**Steps to formulate a model for Problem Solving:**

To optimize a real-world problem, it is essential to formulate a model as per the given Problem based on decision variables, design constraints, and design objectives.

Fig: Formulation of Optimization Model

The step-by-step procedure for formulating an optimization model is as follows:

**Decision Variables:**Identify the variables in the problem that can be adjusted such as material usage, design alteration, etc. to improve performance. Engineers represent these decision variables using algebraic symbols.**Design Constraints:**Identify the restrictions or constraints that bind the problem. In Optimization problems, these constraints such as costs are expressed as mathematical equations.**Design Objectives:**Express the measures of system effectiveness as one or more objective functions. These functions represent what we want to minimize or maximize such as weight and volume.

The design constraints and objectives are often mathematical functions of the decision variables.

**Constrained Optimization Problem: **A constrained optimization problem involves finding the best solution that meets a specific set of constraints. It requires optimizing an objective function while satisfying inequality, equality, and/or functional constraints (limitations). The majority of the industry's problems encountered are from the vast area of constrained optimization.

In many optimization problems, the space of feasible design vectors is not entirely free but bounded by constraints. Constraints can arise from limited resources, physical quantities that must satisfy certain conditions, and other limitations.

The total number of constraints in real-world engineering applications can be substantial, even in the thousands. This makes the problems even harder for classical techniques to optimize.

**Challenges faced in traditional optimization techniques**

In a constrained optimization problem, the search for solutions is restricted by the conventional and functional constraints. Solutions are found by combining the constraints with the objective function(s) and utilizing appropriate search algorithms. This is why classical algorithms find it challenging to achieve the best-suited solution faster while using lesser computational resources.

Secondly, classical algorithms such as Gradient Descend algorithms have largely been used in optimization in various industries. It searches for a single solution at a time increases time to reach the most optimized solution. These classical algorithms on classical computers are deterministic in nature that limits its parallelism, i.e., the ability to simultaneously calculate multiple mathematical functions.

In turn, these solutions are poor in quality, and the time-to-solution increases with the problem size. To combat this, many parameters are often omitted to reach a solution within a reasonable amount of computation time, making the results inconclusive. Even the most powerful supercomputers, whose performance exceeds a hundred quadrillion FLOPS, can reach their computation limit.

In contrast, search-based algorithms such as quantum algorithms are showing promise in solving Optimization problems across industries. It can reduce the time of computation and errors in calculations. It works on completely different principles of Quantum Information processing which is probabilistic in nature.

Advantages of Quantum Algorithm over classical for real-world complex problems

**Introduction to quantum computing and its potential for optimization**

Quantum computing exploits quantum mechanics to process data using qubits. Qubit is a unit of quantum bits which is a quantum mechanical analogue to classical bits such as 0’s and 1’s. A quantum bit can be in the ‘0’ and ‘1’ state at the same time and allows superpositions and entanglement between qubits. It enables parallel processing, enhancing optimization tasks' speed and complexity. Quantum Information processing uses quantum phenomena such as Superposition and Entanglement. These are the basis of quantum parallelism, the fundamental quantum–mechanical property enabling the simultaneous evaluation of mathematical functions. This behavior differentiates quantum from classical parallel computing, where multiple Boolean circuits can only evaluate parts of the input simultaneously.

Algorithms like Quantum approximate optimization algorithm (QAOA) and Quantum Inspired Evolutionary algorithms (QIEA), show promise in optimization by efficiently exploring solution spaces. Based on that, heuristic and stochastic search methods have greater prospects with quantum algorithms. It speeds up the computation process involved in optimization. By utilizing these search methods with quantum algorithms, optimization problems can be effectively solved, leading to improved performance and efficiency in engineering applications.

**Here are some benefits of using quantum algorithms for these categories of problems:**

Explores larger search (solution) space to deliver the most optimized results

Speeds up the search process due to its quantum mechanical principles

Reduces the use of computational resources used in searching

**BQP's Optimization approach and its simulation platform based on quantum algorithms **

**BosonQ Psi (BQP)** has integrated quantum algorithms into its simulation platform that can run on HPCs (high-performance computers) to deliver more accurate, faster, and efficient results for complex and intractable optimization problems.

BQP’s platform enables solving different classes of optimization problems more efficiently than traditional methods. Design optimization and Combinatorial Optimization are some of the classes of problems that BosonQ Psi has been solving using its quantum algorithms.

BQP's simulation platform and capabilities

Engineering applications of BQP’s solver in different classes of problems are as follows:** **

**Topology Optimization:**Engineering applications include designing lightweight structures by optimizing material distribution to achieve maximum stiffness or strength.**Shape Optimization:**This involves optimizing the shape of a component to improve its performance or meet specific design requirements, such as reducing drag in aerodynamics or enhancing fuel efficiency in automotive engineering.**Sizing Optimization:**This optimization technique determines the optimal size of components or systems to achieve desired performance while considering constraints like weight, cost, or space limitations.**Structural-thermal Optimization**: It focuses on optimizing the design of structures to achieve the desired thermal performance, such as minimizing heat generation or improving heat dissipation.**Thermo-mechanical Optimization:**This application optimizes systems considering both thermal and mechanical factors, such as designing components that can withstand high temperatures while maintaining structural integrity.**Machine Learning Training Optimization**: Optimization techniques are used to train machine learning models more efficiently, improving accuracy and reducing training time.**Portfolio Optimization:**It is a well-defined computational task that appears in many fields, but its utility is applied for financial operations and decision-making. It can help companies to invest smartly, maximizing the expected return.**Supply chain Optimization**: Supply chains are complicated and supply chain optimization uses technology and resources like blockchain, artificial intelligence (AI), and the Internet of things (IoT) to maximize efficiency and performance in a supply network using analytical models.

**Some real-world cases of BQP’s quantum-based optimization**

Previously, BQP has achieved better optimization results through its quantum-inspired approach which shows the efficiency of its methods than traditional.

Here are few examples:

BQP’s simulation platform reduced automotive control arm weight 3.2 times more than the classical approach while using 8x less computational resources.

In a second case, it was able to achieve 18% more optimized shape of an airfoil (racecar) leading to 40% cost saving.

Airfoil optimization for volume and compliance minimization: in this use case, BQP bettered the volume and compliance minimization of an aircraft wing airfoil while limiting the usage of computational resources.

**Broader Impact of BQP's Approach in Optimization (Industries)**

BQP's integration of quantum algorithms into its simulation platform has a significant impact across various industries. By leveraging quantum-inspired techniques, BQP enables more efficient and accurate optimization solutions for complex problems in engineering and beyond.

In the automotive industry, BQP's methods can have impacts on developing lighter, more fuel-efficient vehicles with improved performance and lower manufacturing costs. The aerospace sector benefits from optimized aerodynamic shapes, resulting in increased efficiency and reduced emissions.

Manufacturers can optimize their production processes, minimizing material usage and improving overall product quality. Additionally, a more diverse applications impact can impact other industries such as the finance industry, and manufacturing industry by improving decision-making and forecasting with Quantum-based solutions. BQP's quantum-based approach has the potential to transform these industries by providing more efficient, cost-effective, and sustainable solutions to complex optimization challenges.

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