Finite elements and moving asymptotes accelerate quantum optimal control - FEMMA

TL;DR

The paper presents FEMMA, a method that accelerates quantum optimal control by solving the Liouville–von Neumann evolution as a finite-element linear system, with gradients obtained efficiently via an adjoint formulation. By discretizing time and employing a Helmholtz-based regularization plus hyperbolic tangent scaling, the approach enables fast, constrained optimization of spin pulses using MMA, outperforming L-BFGS and approaching Newton-level efficiency. In extensive tests on single-spin problems, FEMMA delivers over an order-of-magnitude speedup compared with GRAPE while maintaining sub-percent accuracy, and demonstrates strong performance in both broadband excitation and universal-rotation tasks, including ensemble settings with parallelization. The work suggests substantial practical impact for rapid design of RF pulses in MRS/MRI and related quantum-control applications, with potential extensions to larger spin networks and diffusion-like spatial variables.

Abstract

Quantum optimal control is central to designing spin manipulation pulses. While GRAPE efficiently computes gradients, realistic ensemble models make optimization time-consuming. In this work, we accelerated single-spin optimal control by combining the finite element method with the method of moving asymptotes. By treating discretized time as spatial coordinates, the Liouville-von Neumann equation was reformulated as a linear system, yielding gradients solving over an order of magnitude faster than GRAPE with less than one percent relative-accuracy loss. The moving asymptotes further improves convergence, outperforming L-BFGS and approaching Newton-level efficiency.

Figures (6)

• Figure 1: View of the one-dimensional linear Lagrange global shape functions. The time interval between two nodes is uniformly set to $\Delta t$. The labels $\mathbf{L}_1, \mathbf{L}_2, \ldots, \mathbf{L}_N$ represent the discretized Liouvillian for a piecewise-constant waveform.
• Figure 2: FEM performance versus GRAPE for a single-spin system. (a) Relative error using Lagrange elements. (b) Relative error using cubic Hermite elements, $R$ denotes the Helmholtz-filter radius, the solid and dashed lines represent the spin-trajectory and gradient error, respectively. Panels (a) and (b) use the pulse duration $T=0.5$ ms, $\|\mathbf{L}\|=2\times10^{4}\,\,\mathrm{rad}\cdot\mathrm{s}^{-1}$, and sweep $N$ from 100 to 1000. (c) Speedup of using three elements over GRAPE versus the number of time steps, with $\|\mathbf{L}\|\Delta t=0.063$ and $T=N\Delta t$.
• Figure 3: Performance of the excitation pulse optimization for an ensemble single-spin system. (a) Comparison of convergence rates with the gradient calculated by FEM and GRAPE, each method was repeated 15 times using different random initial guesses, and the MMA algorithm was used for optimization. (b) Histogram showing the time consumption of the two methods. The shaped pulse steers $I_{\text{z}}$ to $I_{\text{x}}$ with RF amplitude $10~\unit{kHz}$ and $\pm20\%$ scaling ($n_{\text{rf}}=5$). A $15~\unit{kHz}$ bandwidth was discretized into $n_{\text{off}}=40$ offsets, giving $N_{\text{ens}}=200$. The $500~\unit{\mu s}$ pulse was piecewise constant with 500 segments, fixed amplitude 1, and optimized phases. The target ensemble fidelity was 0.995.
• Figure 4: Performance of the universal rotation pulse optimization for an ensemble single-spin system. (a) Comparison of convergence rates for the MMA, L-BFGS, and Newton methods, each method was repeated 15 times using different random initial guesses, GRAPE was used to compute the gradient and Hessian. (b) Histogram showing the time consumption of the three methods. The shaped pulse implements a $90_{\text{x}}^{\text{o}}$ universal rotation with nominal RF amplitude $10~\unit{kHz}$ and $\pm10\%$ scaling ($n_{\text{rf}}=5$). A $20~\unit{kHz}$ bandwidth was discretized into $n_{\text{off}}=40$ offsets, yielding $N_{\text{ens}}=200$. The $500~\unit{\mu s}$ pulse was piecewise constant with 500 slices, fixed amplitude 1, and optimized phases. The target ensemble fidelity was 0.995 with a limit of 100 iterations.
• Figure S1: View of the piecewise-linear waveform, the Liouvillians are defined on the nodes.