Session: Quantum Linear Algebra and Optimization (Part 1))
Chair: Mohammadhossein Mohammadisiahroudi
Cluster: Optimization for Emerging Technologies (LLMs, Quantum Computing, ...)
Talk 1: Quantum Computing-based Sensitivity Analysis for PDE-constrained Optimization
Speaker: Sicheng He
Abstract: Quantum computing is an emerging paradigm offering significant speed-ups for solving specific mathematical problems. In recent years, optimization and scientific computing researchers have developed quantum algorithms that demonstrate complexity advantage for large-scale problems. A key area of focus has been to leverage quantum linear algebra techniques to solve linear systems that arise in optimization and scientific computing applications. We propose quantum computing-based direct and adjoint methods for implicit sensitivity analysis in PDE-constrained optimization. The proposed quantum approaches achieve exponential speed-up in complexity with respect to the problem dimension, i.e., the number of state variables, compared to classical methods. Notably, in the quantum computing framework, both the direct and adjoint methods exhibit similar computational complexity, a departure from their classical counterparts. We demonstrate the proposed method using a simple heat transfer problem implemented with the IBM Qiskit simulators
Talk 2: Recent Advances in Quantum Interior Point Methods
Speaker: Tamas Terlaky
Abstract: Quantum Interior Point Methods (QIPMs) have recently emerged as a potential approach to accelerating the solution of large-scale conic optimization problems by leveraging quantum linear system algorithms for solving the Newton systems in IPMs. However, the one of significant challenges of QIPMs is the inexact and noisy nature of quantum solvers. In this talk, we discuss recent advancements in the design of efficient QIPMs that effectively manage errors. We introduce novel reformulations of the Newton system that enable maintaining feasibility despite inexact Newton directions. Additionally, we employ iterative refinement techniques to enhance solution accuracy while operating under limited precision. Our proposed QIPMs achieve the best-known iteration complexity, offering a significant step forward in the practical realization of quantum-accelerated optimization.
Talk 3: Quantum Approaches to Mixed Integer PDE-Constrained Optimization
Speaker: Adrian Harkness
Abstract: Mixed-integer PDE-constrained optimization (MIPDECO) problems arise in applications like gas and power networks or turbine placement. These problems combine the combinatorial complexity of integer programming with the large-scale linear systems of PDE-constrained optimization. This work investigates quantum computing methods for solving MIPDECO problems, specifically with binary control variables and a knapsack constraint. By using a first-discretize-then-optimize approach, we derive a binary quadratic optimization (BQO) formulation. We then explore two quantum algorithmic strategies based on the Quantum Approximate Optimization Algorithm (QAOA): a constraint-enforcing approach using custom mixer Hamiltonians, and a penalty-based method embedding constraints into the objective function. We discuss penalty parameter selection for feasibility and compare simulations of both quantum approaches. Our results illustrate the potential advantages of quantum methods for solving PDE-constrained optimization problems
