Krylov-subspace approach for the efficient control of quantum many-body dynamics

The gradient ascent pulse engineering (GRAPE) algorithm is a celebrated control algorithm with excellent converging rates, owing to a piecewise-constant ansatz for the control function that allows for cheap objective gradients. However, the computational effort involved in the exact simulation of quantum dynamics quickly becomes a bottleneck limiting the control of large systems. In this paper, we experiment with a modified version of GRAPE that uses Krylov approximations (K-GRAPE) to deal efficiently with high-dimensional state spaces. Even though the number of parameters required by an arbitrary control task scale linearly with the dimension of the system, we find a constant elementary computational effort (the effort per parameter). Since the elementary effort of GRAPE is superquadratic, this speed up allows us to reach dimensions far beyond. The performance of the K-GRAPE algorithm is benchmarked in the paradigmatic $XXZ$ spin-chain model.

Figure 1

Artist's impression. (a) Krylov's approximation: An initial state with an arbitrary distribution on the full $D$-dimensional state space is mapped into the ground state of an effective $N$-level system. This ground state evolves under the action of $T_N$, the Hamiltonian in the reduced Krylov basis. Finally, the evolved state on the full Hilbert space is recovered with the inverse mapping. (b) GRAPE's ansatz: A piecewise-constant control field. (c) The zeroth-order approximation of the gradient is equivalent to the bracket of the control Hamiltonian between the forward evolved initial state and the backward evolved target-projected final state at each time slot. At each iteration, the control field $\epsilon$ is updated using the gradient $\nabla I$, the inverse of an approximated Hessian $B$, and a backtracked step length $\alpha$.

Figure 2

Algorithm benchmarking: (a) Optimization runtime $\mathcal{R}$ and (b) elementary runtime $\tilde{\mathcal{R}}$ (runtime per iteration per time step) for the state transfer task in Eq. (12) using GRAPE (dotted line and red squares) and K-GRAPE (dashed line and green squares) as a function of $D$, the dimension of the subspace holding the dynamics. The data points corresponding to GRAPE and K-GRAPE are fitted with linear and a cubic functions, respectively. In the first low-dimensional regime GRAPE outperforms (see the inset), while in the large-dimensional one, K-GRAPE does.

Figure 3

Time-step study: (a) Minimum infidelity, (b) mean runtime $\mathcal{R}$, (c) mean iterations, and (d) $\tilde{\mathcal{R}}$ runtime per iteration per time step, as a function of $\mathrm{\Delta }t$. The different curves correspond to different values of the truncation parameter, $N=2$, 10, and 18, marked with blue squares, green circles, and red diamonds, respectively. The yellow pluses correspond to the centered zeroth-order gradient, evaluated using an exact evolution. See text for details.

Figure 4

“Death” of a control window: Minimum infidelity as a function of time step $\mathrm{\Delta }t$ for a fixed truncation of $N=6$ and different dimensions $D=10$, 19, 44, and 146 (blue squares, green circles, red diamonds, and yellow pluses).

Figure 5

Truncation study: (a) Runtime $\mathcal{R}$, (b) iterations, and (c) elementary runtime $\tilde{\mathcal{R}}$ involved in the achievement of controls as a function of dimension $D$. The different curves correspond to different values of the truncation parameter $N=8$, 10, and 12, marked with blue squares, green circles, and red diamonds, respectively.