Abid A. Khan

The Sherrington-Kirkpatrick (SK) model serves as a foundational framework for understanding disordered systems. The Quantum Approximate Optimization Algorithm (QAOA) is a quantum optimization algorithm whose performance monotonically improves with its depth $p$. In this work, we introduce a new equivalence between the task of evaluating the energy of QAOA applied to the SK model in the infinite-size limit and the task of simulating a spin-boson system, which we show can be done with modest cost using matrix product states. Using this equivalence, we optimize QAOA parameters and provide numerical evidence that QAOA obtains a $(1-\epsilon)$ approximation to the optimal energy with circuit depth $\mathcal{O}(n/\epsilon^{\infiniteSizeLimitOneOverEta})$ in the average case, with $\varepsilon\lesssim\infiniteSizeLastpError\%$ at $p=\infiniteSizeLastp$. We then use these optimized QAOA parameters to evaluate the QAOA energy for finite-sized instances with up to $30$ qubits and find convergence to the ground state consistent with the infinite-size limit prediction. Our results provide strong numerical evidence that QAOA can efficiently approximate the ground state of the SK model in the average case.