Title: Efficient Formulation and Quantum Optimization of  Combinatorial Problems through Parametrized Hamiltonians
Primary area: optimization
Keywords: Quantum optimization, Parameterized Hamiltonians, Combinatorial optimization, Implicit differentiation, Quantum Approximate Optimization Algorithm (QAOA), Optimization, Physics-inspired machine learning, Software frameworks for quantum optimization

Abstract: Combinatorial optimization problems (COPs) represent a promising application domain for quantum computing, yet current quantum optimization approaches treat each problem instance independently, requiring expensive re-optimization for every configuration. In this paper we propose a different paradigm inspired by quantum many-body physics, where parameterized Hamiltonians naturally encode system variations under changing global conditions. Our parametrized COPs formulation, where a global parameter changes the problem configuration, allows to model parameterized problems and opens access to problem classes that were previously difficult and inefficient to formulate. Second, we provide a concrete algorithmic framework, using implicit differentiation to solve these parameterized COPs classes efficiently. Drawing from techniques used in quantum susceptibility calculations, our method propagates optimal circuit parameters across different Hamiltonian configurations without expensive re-optimization. We demonstrate this approach by finding globally optimal configurations in Max-Cut problems, where the Hamiltonian parameter controls edge weight distributions. Our implementation systematically generates parameterized problem families from Max-Cut, Knapsack, and Portfolio Optimization domains and translates them into quantum formulations suitable for variational algorithms. Experiments on simulated quantum hardware demonstrate substantial computational speedups compared to independent optimization approaches.

TL;DR: We introduce parameterized Hamiltonians as a framework for combinatorial optimization, enabling new problem types and efficient global optimization via QAOA with implicit differentiation.