Implemetation of a shooting technique for quantum optimal control on spin qudits

Abstract

High-fidelity control of quantum systems is essential for scalable quantum technologies. We introduce a shooting-based quantum optimal control algorithm for systems of finite-dimensional Hilbert spaces, and demonstrate its performances through numerical simulations on systems inspired from single molecule magnets. Our method efficiently decomposes quantum gates into selective electromagnetic pulses, outperforming the standard GRAPE method against which it is benchmarked, especially in larger Hilbert spaces

Figures (12)

• Figure 1: (a) A double decker, $\text{Tb}\text{Pc}_2$. The blue atoms are some nitrogen and the grey ones are carbons. Image from hartm2024. (b) A triple decker, $\text{Tb}_2\text{Pc}_3$. The two dark blue atoms are still some $\text{Tb}^{3+}$ ions. Image from hartm2024.
• Figure 2: (a) The energy levels graph of a double decker $\text{Tb}\text{Pc}_2$. The spacing is unequal thanks to a non linear term in the Hamiltonian. Otherwise energy levels would be positioned on the dashed lines. The resulting graph is a linearly coupled one. Values from godfr2017. (b) The energy levels graph of a triple decker $\text{Tb}_2\text{Pc}_3$. Each node represent an energy level, ranked by their energy. The edges represent adressable rabi transitions between nodes, for a certain energy difference in Kelvin. The energy levels are denoted by "lvl" with the first digit denoting the nuclear energy level for the first molecular magnet and the second digit the other nuclear energy level. This figure is done using parameters that differ slightly from Tb2Pc3_expe, and at a high magnetic field to suppress level mixing.
• Figure 3: (a) Figure for the controls evolution evolutions for the GRD for a QFT gate runned on the double decker graph.. The controls correspond to addressable energy difference between levels visible in \ref{['fig:double_levels']}, each couple corresponding to those 4 levels numbered from 0 to 3. Each blue activation, delimited by black lines, correspond to a different $(\theta, \phi)$ for a GR in \ref{['eq:GRD']}. The length of each blue activation is $\phi$, in units of the inverse of the maximum allowed Rabi amplitude $\Omega$. One can notice that we start by the z-rotations, that are in block of 3 operations. (b) Figure for the populations evolution. The initial state is $\ket{\psi}=\frac{1}{\sqrt{N}}\sum_{i=0}^{N-1}\ket{i}$, a superposition of all the states, so it ends in the state $\ket{0}$.
• Figure 4: Controls and population evolutions for the GRAPE method for a QFT gate on the double-decker graph. Each color in (a) corresponds to a control amplitude; for 3 links with $\sigma_x$ and $\sigma_y$ controls per link, there are 6 controls. (b) Populations illustrating the gate dynamics starting from a completely superposed initial state, ending in the void $\ket{0}$. The evolution shown corresponds to the GRAPE control minimizing implementation time among 1000 random initializations. The execution time is in units of the inverse of the maximum allowed Rabi amplitude $\Omega$ according to the \ref{['sec:GRDvsGRAPE']}.
• Figure 5: Controls and population evolutions for MAGICARP for a QFT gate on the double-decker graph. (a) Each color corresponds to a control amplitude, for 3 links with $\sigma_x$ and $\sigma_y$ controls per link, there are 6 controls. The colors correspond to the same controls as in Fig. \ref{['subfig:GRAPE_amps_H_4']}. (b) Populations illustrating the gate dynamics from a completely superposed initial state, ending in the void $\ket{0}$. The evolution shown corresponds to the MAGICARP control minimizing implementation time among 1000 random initializations. The execution time is in units of the inverse of the maximum allowed Rabi amplitude $\Omega$ according to the \ref{['sec:GRDvsGRAPE']}.

Theorems & Definitions (5)

• Definition 2.1
• proof
• Lemma C.4: Jacobian-vector product Magicarp
• Lemma C.5: Vector-jacobian product Magicarp
• Remark 1