Gradient-based QHD, a quantum Hamiltonian descent variant that inserts the gradient into the kinetic term, is claimed to converge at O(t^-2) in theory and to outperform QHD and classical methods in 2D tests.
Quantitative Convergences of Lie Group Momentum Optimizers
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abstract
Explicit, momentum-based dynamics that optimize functions defined on Lie groups can be constructed via variational optimization and momentum trivialization. Structure preserving time discretizations can then turn this dynamics into optimization algorithms. This article investigates two types of discretization, Lie Heavy-Ball, which is a known splitting scheme, and Lie NAG-SC, which is newly proposed. Their convergence rates are explicitly quantified under $L$-smoothness and local strong convexity assumptions. Lie NAG-SC provides acceleration over the momentumless case, i.e. Riemannian gradient descent, but Lie Heavy-Ball does not. When compared to existing accelerated optimizers for general manifolds, both Lie Heavy-Ball and Lie NAG-SC are computationally cheaper and easier to implement, thanks to their utilization of group structure. Only gradient oracle and exponential map are required, but not logarithm map or parallel transport which are computational costly.
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Quantum Optimization via Gradient-Based Hamiltonian Descent
Gradient-based QHD, a quantum Hamiltonian descent variant that inserts the gradient into the kinetic term, is claimed to converge at O(t^-2) in theory and to outperform QHD and classical methods in 2D tests.