Compressed space quantum approximate optimization algorithm for constrained combinatorial optimization

TL;DR

CS-QAOA addresses constrained COPs by compressing the feasible solution space into $m$ qubits with $2^m \ge |F|$ via a unitary $\\hat{U}_{cs}$, enabling optimized search within a reduced space while leveraging the original Ising formulation. It offers a deterministic path for one-hot and parity constraints and a scalable variational path for general constraints, including a method to construct a compressed-space Hamiltonian $\\hat{H}_{cs}$ and two ansatz families (C-ansatz and D-ansatz) to minimize $E(\\hat{U},\\hat{H}_{cs})$. Coherent simulations across Max-$k$ cut, QAP, and QKP show CS-QAOA can outperform conventional QAOA in many settings, particularly when infeasible states are minimized or eliminated in the compressed space. The approach highlights potential gains in constrained quantum optimization, while identifying scalability and noise-related challenges that motivate future hardware experiments, error-mitigation strategies, and more efficient compression architectures.

Abstract

Combinatorial optimization is a promising area for achieving quantum speedup. Quantum approximate optimization algorithm (QAOA) is designed to search for low-energy states of the Ising model, which correspond to near-optimal solutions of combinatorial optimization problems (COPs). However, effectively dealing with constraints of COPs remains a significant challenge. Existing methods, such as tailoring mixing operators, are typically limited to specific constraint types, like one-hot constraints. To address these limitations, we introduce a method for engineering a compressed space that represents the feasible solution space with fewer qubits than the original. Our approach includes a scalable technique for determining the unitary transformation between the compressed and original spaces on gate-based quantum computers. We then propose compressed space QAOA, which seeks near-optimal solutions within this reduced space, while utilizing the Ising model formulated in the original Hilbert space. Experimental results on a quantum simulator demonstrate the effectiveness of our method in solving various constrained COPs.

Figures (13)

• Figure 1: Schematic picture of a compressed space $\mathcal{H}_\mathrm{cs}$ for a simple constrained COP with two binary variables $x_A, x_B \in \{0, 1\}$. The feasible solution space $\mathcal{H}_\mathrm{F}$ is mapped to $\mathcal{H}_\mathrm{cs}$ by the unitary transformation $\hat{U}_\mathrm{cs}$. Local minimum disappears in the compressed space.
• Figure 2:
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• Figure 5: One-hot constraint
• Figure 6: Parity constraint