Practical Guide to Quantum Computing – Variational Algorithms: Cost Functions (Based on Materials from IBM Q) # 4

Abstract This practical guide focuses on the evaluation and implementation of cost functions in variational quantum algorithms (VQAs), based on IBM Q learning resources. The exercises were performed by the author on May 2, 2024, using IBM Q cloud resources. Cost functions are central to variational algorithms, defining the objective that the optimizer seeks to minimize or maximize, and linking the parameterized quantum state to the computational task. In this lesson, participants first learn to utilize Qiskit Runtime primitives to implement efficient and accurate evaluation routines. The cost function C(θ⃗)C(\vec{\theta})C(θ) is defined according to the specific problem, encoding the target metric that guides the variational optimization. The guide also introduces strategies for measurement selection and execution to optimize computational speed and reduce statistical errors during evaluation, leveraging cloud-based quantum resources. Hands-on exercises in Qiskit provide experience in constructing cost function evaluations, applying them to parameterized ansätze, and integrating them with classical optimization routines. Participants explore the trade-offs between measurement overhead, convergence speed, and accuracy, reinforcing the understanding of how cost functions drive the performance of hybrid quantum-classical algorithms on noisy intermediate-scale quantum (NISQ) devices. This lesson equips participants with practical skills in defining, implementing, and measuring cost functions, forming the foundation for efficient variational algorithm design in quantum chemistry, optimization, and machine learning applications.