Constrained Quantum Optimization via Iterative Warm-Start XY-Mixers

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a leading hybrid heuristic for combinatorial optimization, but efficiently handling hard constraints remains a significant challenge. XY-mixers successfully confine quantum state evolution to a feasible subspace, such as the Hamming-weight-1 sector for one-hot constraints. On the contrary, warm-starting biases the search toward promising regions based on preliminary solutions. Combining these two techniques requires maintaining the essential alignment between the initial state and the mixer Hamiltonian to preserve convergence guarantees. Previous work demonstrated warm-starting with XY-mixers via a biased initial state, but relying only on standard mixer Hamiltonians. Consequently, the initial state is no longer a ground state of the mixer. In this work, we overcome these limitations by formulating a warm-started XY-mixer Hamiltonian for one-hot constraints and proving its ground-state properties. Furthermore, we provide a shallow circuit implementation suitable for NISQ implementations. We embed the warm-starting into a classical heuristic that iteratively updates the bias based on previous samples, called Iterative Warm-Starting (IWS). Extensive numerical simulations on Max-$k$-Cut and Traveling Salesperson Problem instances demonstrate that IWS-QAOA significantly accelerates the solution-finding process, increasing the probability of sampling optimal solutions by orders of magnitude compared to standard XY-QAOA. Finally, we validate our approach on the ibm_boston QPU using hardware-tailored 144-qubit problem instances. By coupling IWS-QAOA with a greedy steepest-descent post-processing strategy to repair infeasible measurements caused by hardware noise, we successfully identify optimal solutions on actual quantum devices.

Figures (11)

• Figure 1: Contour plots showing the energy landscape in terms of approximation ratio of WS-QAOA for $p=5$ on a TSP instance with $N=7$. The leftmost plot applies no warm starting; the following two plots show the landscape with warm starting based on the ideal solution, using regularization with $\epsilon = 0.25$ and $\epsilon = 0.1$. The final two plots show warm-starting using $\ket{W_p}$ and the default XY-mixer, which is not aligned with the initial state.
• Figure 2: Approximation ratio (a) and approximation trace (b) of IWS-QAOA at $p=1$ as a function of the total number of shots ($M_\text{agg}$ from Algorithm \ref{['alg:iws-qaoa']}) for $M \in \{100, 200, 500\}$ and $\overline{M} = 3000$ across four MkC instance classes. Each panel displays the median over five instances, with ten runs per instance. Error bands indicate the interquartile range. The solid black line represents the median performance of IWS using classical random sampling with $M = 100$ (the best-performing sample size among those tested). Vertical dashed lines in panel (b) mark the median number of total shots required to identify the optimal solution.
• Figure 3: Median improvement ratio of the optimal solution probability, $P_\text{opt}$, achieved by IWS-QAOA compared to the baseline without warm-starting (no WS) across four MkC problem classes for various sample sizes $M$. Error bars indicate the interquartile range. The gray bars represent the performance of IWS using classical random sampling with $M=100$. For IWS-QAOA, $P_\text{opt}$ is evaluated directly from the state vector following the final iteration of IWS-QAOA.
• Figure 4: Approximation trace of IWS-QAOA at $p=1$ as a function of the total number of shots for $M\in \{100, 200, 500\}$ and $\overline{M} = 3000$ on TSP instances ranging from 6 to 9 cities. Each panel displays the median over five instances, with ten runs per instance. Error bands indicate the interquartile range. The solid black line represents the median baseline performance of IWS using classical random sampling with $M=200$, which outperformed $M=100$ for these TSP instances.
• Figure 5: Median improvement ratio of $P_\text{opt}$ achieved by IWS-QAOA relative to the baseline without warm-starting (WS) for the four TSP instance sizes across various $M$ values. Error bars indicate the interquartile range. The gray bars represent IWS with classical random sampling at $M=200$. For IWS-QAOA, $P_\text{opt}$ is evaluated directly from the state vector following the final iteration of IWS-QAOA.

Theorems & Definitions (10)

• Proposition 1
• proof
• Corollary 1.1
• Corollary 1.2
• Corollary 1.3
• Proposition 2
• proof : Proof of Corollary \ref{['cor:1']}
• proof : Proof of Corollary \ref{['cor:app']}
• proof : Proof of Corollary \ref{['cor:2']}
• proof