Pontryagin Maximum Principle for Rydberg-blockaded state-to-state transfers: A semi-analytic approach

TL;DR

This work develops a semi-analytic approach to time-optimal state-to-state control for ensembles of Rydberg-blockaded qubits by applying the Pontryagin Maximum Principle (PMP) to the block-diagonalized, independent two-level subsystems. For two qubits ($N=2$), abnormal extremals are shown to be either nonexistent or suboptimal, while normal extremals produce a detuning dynamics governed by an ordinary differential equation that mirrors the motion of a classical particle in a quartic potential. The detuning dynamics $rac{1}{2}\\dot{\Delta}^2 + V(\Delta)=0$ with a quartic potential $V(\Delta)$ reduces the time-optimal control to solving an ODE with coefficients determined by conserved quantities, and the resulting pulses agree with GRAPE solutions, providing a robust semi-analytic route to high-fidelity, time-minimal gates. The framework blends analytic PMP structure with numerical optimization to offer both physical insight and practical control prescriptions for multi-qubit operations in Rydberg-based quantum processors.

Abstract

We study time-optimal state-to-state control for two- and multi-qubit operations motivated by neutral-atom quantum processors within the Rydberg blockade regime. Block-diagonalization of the Hamiltonian simplifies the dynamics and enables the application of a semi-analytic approach to the Pontryagin Maximum Principle to derive optimal laser controls. We provide a general formalism for $N$ qubits. For $N=2$ qubits, we classify normal and abnormal extremals, showcasing examples where abnormal solutions are either absent or suboptimal. For normal extremals, we establish a correspondence between the laser detuning from atomic transitions and the motion of a classical particle in a quartic potential, yielding a reduced, semi-analytic formulation of the control problem. Combining PMP-based insights with numerical optimization, our approach bridges analytic and computational methods for high-fidelity, time-optimal control.

Figures (4)

• Figure 1: Excitation processes for different Rydberg settings. (a) Scheme for the excitation process in the gate setting, for $N=2$ Rydberg atoms. The state $\ket{00}$ is not coupled to any excited state by the laser. The states $\ket{01}$ and $\ket{10}$ undergo a similar excitation process, by reaching the states $\ket{r0}$ and $\ket{0r}$ respectively. We remark that the laser amplitude $\Omega(t)$ has no coefficient in front, accounting for the fact that only one atoms is in the state $\ket{1}$. Finally, we see the excitation $\ket{11}\mapsto \ket{W}:=(\ket{1r}+\ket{r1})/\sqrt{2}$, that comes with amplitude $\sqrt{2}\Omega(t)$. (b) Scheme for the excitation process in the cluster setting, for $N=4$ clusters. We can see four different collections of Rydberg atoms, having an increasing number of atoms. Each one of the clusters behaves as a TLS, where the laser amplitude $\Omega(t)$ has to be rescaled by a factor $\sqrt{k}$, for $k=1,2,3,4.$
• Figure 2: Schematic approach to OC problems. A strategy to obtain a solution is to follow the solid lines in the scheme (cfr. Sec. \ref{['sec:semianalytic']}). Therefore, the optimal candidate can be found either by solving the problem analytically or by leveraging numerical methods. An alternative is to follow the dashed lines in the scheme, i.e. to employ the GRAPE algorithm to devise approximated candidates Sugny1.
• Figure 3: Trajectories on the Bloch sphere corresponding to different normal extremals obtained with the PMP. Panels (a) and (b) show the trajectories associated with Case (i) and Case (ii), respectively. Panel (c) displays an extremal that drives the first TLS from the ground to the excited state while keeping the second TLS in its ground state at the final time. In contrast to the CZ gate of Case (ii), which introduces a phase $\pi$ on the second TLS, panels (d) and (f) show state-to-state transfers generating smaller conditional phase shifts of $\pi/3$ and $\pi/50$, respectively. Panel (e) illustrates an extremal in which the second TLS is steered to the equator of the Bloch sphere while the first TLS is driven back to its ground state. Panels (g) and (h) display trajectories obtained by varying the parameters $\Delta_0$ and $V_0$, respectively, highlighting the effect of these parameters on the extremal evolution. The starting point of each trajectory is marked by an orange dot, and the final point is indicated by a star; if the final point coincides with the starting point, only the dot is visible.
• Figure 4: Depicted are Case (i) and Case (ii), shown respectively in panels (a) and (b). The first column displays the laser pulse $\varphi(t)$, the second one the detuning $\Delta(t)$, both plotted as functions of time, while the third column shows the potential $V(\Delta)$ as a function of the detuning. All quantities are normalized by $\Omega_{\text{max}}$. The starred points mark the inversion points of the motion inside the potential, whereas the pink points indicate the starting and final points of the detuning which correspond to the starting and final point of the motion inside the potential. The dashed blue curves represent the numerical GRAPE solutions, while the solid light-blue curves correspond to the semi-analytic results.

Theorems & Definitions (1)

• Theorem 1: Pontryagin Maximum Principle