Quantum Riemannian Hamiltonian Descent

Abstract

We propose Quantum Riemannian Hamiltonian Descent (QRHD), a quantum algorithm for continuous optimization on Riemannian manifolds that extends Quantum Hamiltonian Descent (QHD) by incorporating geometric structure of the parameter space via a position-dependent metric in the kinetic term. We formulate QRHD at both operator and path integral formalisms and derive the corresponding quantum equations of motion, showing that quantum corrections appear in the action integral but they are suppressed at late times by the time-dependent dissipation factor. This implies that convergence near optimal points is controlled by the classical potential while quantum effects influence early-time dynamics. By analyzing the semiclassical equation, we estimate a lower bound on the convergence time and numerically demonstrate whether QRHD work as a quantum optimization algorithm in some examples. A quantum circuit implementation based on time-dependent Hamiltonian simulation is also discussed and the query complexity is estimated.

Figures (7)

• Figure 1: (Left) Schematic illustration of quantum optimization algorithms using quantum dynamics. The blue point and black curve represent the parameters to be optimized and the landscape of the loss function, respectively. The light blue region denotes the probability density of the parameters, which evolves in time according to the Schrödinger equation. Quantum tunneling enables the parameters to escape from local minima. (Right) Comparison between the conventional idea, QHD, and our proposal, QRHD. While QHD performs parameter search in a Cartesian coordinate system on a flat space, QRHD enables parameter search on a specified Riemannian manifold.
• Figure 2: Comparison of convergence behavior for two-dimensional quadratic convex optimization. The upper panels show the QHD case and the lower panels show the QRHD case. In both figures, time evolves from left to right, and each panel shows a heat map of the squared magnitude of the wave function. The initial state is taken to be a random distribution in both cases.
• Figure 3: A schematic illustration of a conformally flat coordinate systems on the sphere. (Left) The $N$-dimensional vector $\bm{x}$ constrained to line on the sphere $\mathbb{S}^{N-1}$ by $\bm{x}^2 = R^2$. (Middle) The correspondence between $\bm{x}$ and the stereographic projection $\bm{u}$ from the north pole $\bm{x}_N$. $\bm{u}$ denotes the $(N-1)$-dimensional coordinates. (Right) The correspondence between $\bm{x}$ and the stereographic projection $\bm{v}$ from the north pole $\bm{x}_N$. $\bm{v}$ denotes the $(N-1)$-dimensional coordinates.
• Figure 4: Numerical demonstrations of quadratic convex optimization on a two-dimensional sphere. (Upper) The time evolution in the coordinate system ${\bm{u}}$. (Lower) The time evolution in the coordinate system ${\bm{v}}$. The time evolves from left to right, and each panel shows a heat map of the squared magnitude of the wave function. The optimal point of this problem is indicated by a star. The initial state is taken to be a random distribution in both cases. At late times, we can see that the wave function in QRHD for ${\bm{v}}$ converges to the optimal point.
• Figure 5: Schematic quantum circuit diagrams for time-dependent Hamiltonian simulation. The left panel is based on the Dyson series expansion 2018-Dyson, and the right panel further adopts the interaction picture 2018-Dyson-IntPic. In both panels, $\ket{\mathtt{clock}}$ is a register that manages time ordering, and $\ket{\mathtt{order}}$ is a register that manages the order in the Dyson series expansion.