Q-CHOP: Quantum constrained Hamiltonian optimization

TL;DR

Q-CHOP tackles constrained optimization on quantum hardware by enforcing the feasible subspace with a constraint Hamiltonian $H_\text{con}$ and steering from the worst to the best feasible state along a rotated objective $H_\text{obj}(\theta)$. It contrasts with penalty-based adiabatic methods by separating constraint and objective roles and providing a principled choice of the penalty factor $\lambda$ tied to problem data. Across MIS, DMDS, knapsack, combinatorial auction, and ETF basket problems, Q-CHOP consistently achieves higher in-constraint performance and higher probability of near-optimal solutions than the standard SQAA at comparable runtimes. The results indicate potential practical benefits for constrained quantum optimization and motivate further work on counterdiabatic driving, inequality constraints, and discretized variants for near-term devices. Overall, Q-CHOP offers a general, parameter-light framework that improves constrained adiabatic optimization and opens several avenues for theoretical and experimental refinement.

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 of Q-CHOP is 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. Q-CHOP thereby assigns qualitatively distinct roles to the constraint and objective functions of a constrained optimization problem. 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 of quantum annealing with an objective function that penalizes constraint violation, and find that Q-CHOP consistently performs significantly better on a wide range of problems, including textbook graph problems, knapsack problems, combinatorial auctions, and a real-world financial use case of bond exchange-traded fund basket optimization.

Figures (8)

• Figure 1: Summary of MIS simulation results with Q-CHOP and the SQAA. (Top row) In-constraint approximation ratio $\braket{r}$ throughout simulations. (Top row) Expectation value of the optimal-state projector $P_\text{opt}$ throughout simulations. (Left-most panels) Time-series data in simulations of 10 random graphs with $N=10$ vertices and quantum runtime $T=2\pi N^2$. The inset shows the in-constraint probability $\braket{P_\text{feas}}$ over time for the same simulations. (Middle panels) Values at the end of simulations with the same random graphs, but different quantum runtimes $T$. (Right-most panels) Values at the end of simulations with different qubit numbers (vertices). Lines in the right-most panels track average values for each qubit number $N$.
• Figure 2: Summary of DMDS simulation results with random directed graphs, presented in a format identical to Figure \ref{['fig:MIS-summary']}.
• Figure 3: Summary of knapsack simulation results, presented in a format identical to Figure \ref{['fig:MIS-summary']}. Here $N$ is the number of items in a knapsack instance.
• Figure 4: (Top panels) Histograms of final in-constraint approximation ratio $\braket{r}$ and optimal-state probability $\braket{P_\text{opt}}$ from knapsack simulations with $N=8$ items and quantum runtime $T=2\pi N^2$. The data in the top panels here is a coarse-grained slice of the data at time $t=T$ in the left-most panels of Figure \ref{['fig:KS-summary']}. Darker red bars indicate that Q-CHOP and SQAA values lie within the same histogram bin. (Bottom panels) Histograms of the performance differences (as measured by $\braket{r}$ and $\braket{P_\text{opt}}$) between Q-CHOP and the SQAA in the same simulations as the top panels. Q-CHOP outperforms the SQAA in every simulated knapsack problem instance with $N=8$ items.
• Figure 5: Summary of combinatorial auction simulation results, presented in a format identical to Figure \ref{['fig:MIS-summary']}. Here $N$ is the number of bids in the combinatorial auction.