



01891nas a2200145   4500008004100000245003200041210003200073260001300105520152500118100001601643700001901659700001501678700001501693856003701708       2023                        eng d00aQuantum Hamiltonian Descent0 aQuantum Hamiltonian Descent  c3/2/20233 a<p>Gradient descent is a fundamental algorithm in both theory and practice for continuous optimization. Identifying its quantum counterpart would be appealing to both theoretical and practical quantum applications. A conventional approach to quantum speedups in optimization relies on the quantum acceleration of intermediate steps of classical algorithms, while keeping the overall algorithmic trajectory and solution quality unchanged. We propose Quantum Hamiltonian Descent (QHD), which is derived from the path integral of dynamical systems referring to the continuous-time limit of classical gradient descent algorithms, as a truly quantum counterpart of classical gradient methods where the contribution from classically-prohibited trajectories can significantly boost QHD&#39;s performance for non-convex optimization. Moreover, QHD is described as a Hamiltonian evolution efficiently simulatable on both digital and analog quantum computers. By embedding the dynamics of QHD into the evolution of the so-called Quantum Ising Machine (including D-Wave and others), we empirically observe that the D-Wave-implemented QHD outperforms a selection of state-of-the-art gradient-based classical solvers and the standard quantum adiabatic algorithm, based on the time-to-solution metric, on non-convex constrained quadratic programming instances up to 75 dimensions. Finally, we propose a &quot;three-phase picture&quot; to explain the behavior of QHD, especially its difference from the quantum adiabatic algorithm.</p>1 aLeng, Jiaqi1 aHickman, Ethan1 aLi, Joseph1 aWu, Xiaodi  uhttps://arxiv.org/abs/2303.01471