Q-CHOP: Quantum constrained Hamiltonian optimization (2403.05653v1)

Abstract: Combinatorial optimization problems that arise in science and industry typically have constraints. Yet the presence of constraints makes them challenging to tackle using both classical and quantum optimization algorithms. We propose a new quantum algorithm for constrained optimization, which we call quantum constrained Hamiltonian optimization (Q-CHOP). Our algorithm leverages the observation that for many problems, while the best solution is difficult to find, the worst feasible (constraint-satisfying) solution is known. The basic idea is to to enforce a Hamiltonian constraint at all times, thereby restricting evolution to the subspace of feasible states, and slowly "rotate" an objective Hamiltonian to trace an adiabatic path from the worst feasible state to the best feasible state. We additionally propose a version of Q-CHOP that can start in any feasible state. Finally, we benchmark Q-CHOP against the commonly-used adiabatic algorithm with constraints enforced using a penalty term and find that Q-CHOP performs consistently better on a wide range of problems, including textbook problems on graphs, knapsack, combinatorial auction, as well as a real-world financial use case, namely bond exchange-traded fund basket optimization.

• The paper introduces the Q-CHOP algorithm, which enforces constraints during adiabatic quantum evolution to reliably target optimal solutions.
• It methodically rotates the objective Hamiltonian from a worst feasible state to a solution state, eliminating the need for extraneous driver Hamiltonians.
• Numerical tests on problems like Maximum Independent Set and Knapsack demonstrate Q-CHOP’s superior success probability and scalability compared to penalty-based methods.

Quantum Constrained Hamiltonian Optimization: A New Algorithm for Quantum Optimization

Quantum Optimization in Constrained Problem Spaces

Advances in quantum computing have positioned quantum optimization algorithms as promising tools for solving complex combinatorial problems, which are widespread in fields ranging from finance to logistics and beyond. One critical challenge in leveraging quantum computing for these applications is handling constraints that naturally arise in real-world problems. This paper introduces a novel quantum algorithm, Quantum Constrained Hamiltonian Optimization (Q-CHOP), designed to navigate the challenges posed by constrained optimization landscapes effectively.

The Q-CHOP Algorithm: An Overview

Q-CHOP builds upon the foundation of adiabatic quantum computation, where the aim is to evolve a quantum system's initial ground state into the ground state of a final Hamiltonian encoding the solution to the optimization problem. Central to Q-CHOP is the methodical enforcement of constraints throughout the adiabatic process, ensuring that the evolution is restricted to plausible, constraint-satisfying states. The algorithm identifies the worst feasible solution for a given problem as easily obtainable, then constructs an adiabatic path from this initial state to the optimal feasible solution by "rotating" an objective Hamiltonian. This strategy eliminates the necessity for non-problem-related initial states and driver Hamiltonians, distinct from typical approaches in the field.

Numerical Evidence and Performance

Numerical simulations assessing the performance of Q-CHOP against the standard adiabatic algorithm (SAA), which incorporates constraints through penalty terms, reveal Q-CHOP's superior efficacy. Benchmarks include various problem sets, from Maximum Independent Set (MIS) and Directed Minimum Dominating Set (DMDS) to Knapsack and Combinatorial Auction problems. Across these diverse scenarios, Q-CHOP consistently outperforms the SAA, achieving higher-quality solutions and demonstrating an increased probability of finding optimal solutions with scalability in problem size and quantum runtime.

Practical Implications and Theoretical Insights

The algorithm’s inherent design, particularly its focused search within the feasible solution space and the systematic treatment of constraints, underscores potential enhancements in quantum optimization's applicability and performance. Furthermore, Q-CHOP's framework suggests new directions for imposing constraints in quantum algorithms and invites further exploration into efficient Hamiltonian rotation strategies and counterdiabatic driving techniques for quicker convergence to solutions.

Future Directions in Quantum Constrained Optimization

Q-CHOP opens avenues for future work, including the exploration of variational and discretized versions for gate-based quantum computing, akin to Quantum Approximate Optimization Algorithm (QAOA) methodologies. The pursuit of quantitative performance guarantees, ideally tied to the algorithm’s spectral properties, also presents a promising research direction. Furthermore, the methodology to accommodate inequality constraints warrants additional investigation, aiming to extend Q-CHOP’s applicability across an even broader spectrum of constrained optimization problems.

In conclusion, Q-CHOP represents a significant advancement in quantum optimization, tailored to address the intrinsic complexity of constrained problem spaces. Its development not only highlights the growing sophistication of quantum algorithms but also aligns with the broader goal of harnessing quantum computing's potential to tackle pressing computational challenges across various industries and scientific disciplines.