Quantum Hamiltonian Descent
This work addresses the challenge of quantum optimization for non-convex problems, offering a novel method that could benefit quantum computing applications, though it appears incremental as it builds on existing quantum and classical techniques.
The authors tackled the problem of developing a quantum counterpart to classical gradient descent for continuous optimization, proposing Quantum Hamiltonian Descent (QHD) and empirically showing that it outperforms state-of-the-art classical solvers and the quantum adiabatic algorithm on non-convex constrained quadratic programming up to 75 dimensions based on time-to-solution metrics.
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'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 "three-phase picture" to explain the behavior of QHD, especially its difference from the quantum adiabatic algorithm.