🤖 AI Summary
This work addresses the lack of general performance guarantees in existing quantum optimization algorithms and the difficulty of balancing efficiency with error control in circuit partitioning. It introduces, for the first time, a quantum–classical moment duality framework applicable to arbitrary quantum states. By leveraging a second-order sum-of-squares (SoS) semidefinite program, the approach establishes a duality between two-qubit Pauli-Z correlation matrices and the Goemans–Williamson relaxation, yielding certifiable lower bounds on Max-Cut values. Simultaneously, this correlation matrix reveals the tensor structure of quantum circuits, enabling efficient partitioning with rigorous error bounds. Experimental results demonstrate that near-optimal lower bounds can be achieved using only two-point correlation data, and the theoretical error bounds are empirically validated to hold tightly in practice.
📝 Abstract
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 \emph{any} quantum state $ρ$ 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 $\mathbb{E}[\mathrm{Cut}] \ge α_{\mathrm{GW}}\langle\mathcal{H}\rangle_ρ$, 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.