Haomu Yuan

Quadratic Unconstrained Binary Optimization (QUBO) problems are prevalent in various applications and are known to be NP-hard. The seminal work of Goemans and Williamson introduced a semidefinite programming (SDP) relaxation for such problems, solvable in polynomial time that upper bounds the optimal value. Their approach also enables randomized rounding techniques to obtain feasible solutions with provable performance guarantees. In this work, we identify instances of QUBO problems where matrix multiplicative weight methods lead to quantum and quantum-inspired algorithms that approximate the Goemans-Williamson SDP exponentially faster than existing methods, achieving polylogarithmic time complexity relative to the problem dimension. This speedup is attainable under the assumption that the QUBO cost matrix is sparse when expressed as a linear combination of Pauli strings satisfying certain algebraic constraints, and leverages efficient quantum and classical simulation results for quantum Gibbs states. We demonstrate how to verify these conditions efficiently given the decomposition. Additionally, we explore heuristic methods for randomized rounding procedures and extract the energy of a feasible point of the QUBO in polylogarithmic time. While the practical relevance of instances where our methods excel remains to be fully established, we propose heuristic algorithms with broader applicability and identify Kronecker graphs as a promising class for applying our techniques. We conduct numerical experiments to benchmark our methods. Notably, by utilizing tensor network methods, we solve an SDP with $D = 2^{50}$ variables and extract a feasible point which is certifiably within $0.15\%$ of the optimum of the QUBO through our approach on a desktop, reaching dimensions millions of times larger than those handled by existing SDP or QUBO solvers, whether heuristic or rigorous.

Cardinality-constrained binary optimization is a fundamental computational primitive with broad applications in machine learning, finance, and scientific computing. In this work, we introduce a Grover-based quantum algorithm that exploits the structure of the fixed-cardinality feasible subspace under a natural promise on solution existence. For quadratic objectives, our approach achieves $\mathcal{O}\left(\sqrt{\frac{\binom{n}{k}}{{M}}}\right)$ Grover rotations for any fixed cardinality $k$ and degeneracy of the optima $M$, yielding an exponential reduction in the number of Grover iterations compared with unstructured search over $\{0,1\}^n$. Building on this result, we develop a hybrid classical--quantum framework based on the alternating direction method of multipliers (ADMM) algorithm. The proposed framework is guaranteed to output an $\epsilon$-approximate solution with a consistency tolerance $\epsilon + \delta$ using at most $ \mathcal{O}\left(\sqrt{\binom{n}{k}}\frac{n^{6}k^{3/2} }{ \sqrt{M}\epsilon^2 \delta }\right)$ queries to a quadratic oracle, together with $\mathcal{O}\left(\frac{n^{6}k^{3/2}}{\epsilon^2\delta}\right)$ classical overhead. Overall, our method suggests a practical use of quantum resources and demonstrates an exponential improvements over existing Grover-based approaches in certain parameter regimes, thereby paving the way toward quantum advantage in constrained binary optimization.