In this paper, we establish a direct quantum-classical duality based on the degree-2 Sum-of-Squares (SoS) semidefinite programming cone: the matrix of two-qubit Pauli-Z correlation functions obtained from any quantum state ( \rho ) is automatically a feasible point of the classical Goemans-Williamson (GW) relaxation. This observation provides a universal 'safety net' for quantum optimization algorithms: applying GW random hyperplane rounding to the quantum-driven moment matrix yields a certified expected cut value ( \mathbb{E}[\mathrm{Cut}] \geq \alpha{\mathrm{GW}}\langle\mathcal{H}\rangle\rho ), valid for every state produced by variational algorithms such as QAOA or the Variational Quantum Power Method (VQPM), regardless of convergence quality.
We further show that the same moment matrix reveals the tensor-product structure of the underlying unitary circuit, enabling a polynomial-time, correlation-based circuit cutting procedure with rigorous error bounds. The framework is validated numerically on Max-Cut instances for variational quantum algorithms and on random states for circuit cutting, demonstrating that the cheap two-point correlation data are sufficient to locate near-optimal bipartitions and that the theoretical error bounds hold in practice.
Blogger's Review: This paper offers a fresh perspective on quantum optimization algorithms through the quantum-classical moment duality, particularly showcasing its practicality in circuit cutting. The combination of theory and real-world application, especially the rigor of error bounds, is noteworthy and deserves further attention and exploration. The universality of this method provides an important theoretical foundation for future quantum algorithm designs.