Jeffrey Larson

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.

Achieving computational advantage using NISQ devices on practically useful problems is a long standing challenge. We report two papers that experimentally demonstrate a concrete application, namely certified randomness generation, which could be useful for multi-party cryptographic protocols and improving imperfect physical sources of randomness. Both papers involve substantial theoretical contributions to the protocol. We devise a realistic protocol that maximizes practical hardness. The verifier first asks the server to prepare a quantum state using a random circuit and then sends a random measurement basis right before the result must be received. This is repeated for many rounds. We show complexity theoretic evidence for entropy generation and provide improved entropy bounds against adversaries with oracle access to the random circuits. We also construct an end-to-end application of randomness amplification of imperfect sources into nearly perfect randomness, notably achieving everlasting security which uplifts computational security to information theoretic security.