Optimal Control for Rydberg multi-qubit operations

TL;DR

The paper applies quantum optimal control (via the Krotov method) to design a single continuous laser pulse that implements multi-qubit gates on Rydberg atom processors, exploiting Rydberg blockade and state-dependent transition rates to realize native controlled-phase, -NOT, and Fredkin gates. This approach reduces circuit depth, suppresses population in short-lived Rydberg states, and enhances robustness against environmental noise and Doppler effects, achieving a Fredkin gate fidelity of 99.74% under realistic imperfections. The method provides a framework for scalable, high-fidelity multi-qubit operations with fewer operational steps than circuit-based decompositions, and suggests extensions to larger CKZ gates and cat-state generation. It also outlines the Krotov-based optimization framework, including cost function design and parameter tuning, as a general tool for hardware-specific quantum gate design.

Abstract

Quantum computing algorithms can be decomposed into a universal set of elementary one- and two-qubit gates. Different physical implementations of quantum computing, however, employ interactions that permit direct conditional dynamics on multiple qubits in a single step. In this work, we leverage quantum optimal control techniques to design single continuous laser pulses that implement multi-qubit controlled-phase, -NOT and -swap (Fredkin) gates on Rydberg atom quantum processors. The identification of robust multi-qubit operations leads to reduced operation time and less decoherence, and the control field provides continuous protection of the atoms from environmental noise. Notably, we find that the controlled-swap (Fredkin) gate, implemented using this approach achieves 99.74\% fidelity while accounting for imperfections such as spontaneous emission, laser fluctuations, and Doppler dephasing.

Figures (8)

• Figure 1: Fredkin (controlled-swap) gate (a) Level scheme. The qubit basis is the Hyperfine states of Cs, and $|r\rangle$ is the highly excited Rydberg state $|100S_{1/2}\rangle$. Target qubit rotation is achieved via the $\Omega_p$ laser. To make the swap conditional, all control and target qubits in state $|1\rangle$ are coupled to the Rydberg level. The three atoms are within the blockade region $\Omega_r \ll V$ in a triangular spatial geometry. (b) Optimal Rabi frequencies that lead to Fredkin gate with 0.0078 infidelity. Here only amplitudes are modulated while the lasers' phase are constant. Alternatively (c) optimizes both (c1) amplitude and (c2) phase of the Rabi frequencies allowing for further reduction of the infidelity to 0.0026. Reported infidelities include spontaneous emission as well as rotation errors.
• Figure 2: Controlled-swap Mechanism - The two closed subspaces governing the dynamics for (a) $|0_c\rangle$ and (b) $|1_c\rangle$ are illustrated. The two Lambda transitions, shown in red and blue, mediate the swapping operations, controlled by quantum interference Kha24Bou02Sor00 through the Rydberg transition Kha24. Under the blockade effect, the enhanced Rabi oscillations $\sqrt{m}\Omega_2$ control the swapping rate in the Lambda transitions, enabling controlled swap operations. The orange side transition regulates the acquired phase. (c, d) Display the conditional swapping process and the total population within the closed subspaces defined in (a, b), as shown by the blue lines. (e, f) Show the population dynamics of intermediate states in the upper and lower Lambda transitions, peaking at the moments of population swap observed in (c, d).
• Figure 3: Truth tables of the Fredkin gate operated with the set of parameters of Fig 1c.
• Figure 4: Robustness against fluctuations. (a) The gate infidelity as a function of the laser intensity fluctuations for the set of optimized pulses in fig. 1c. The data are generated by Gaussian random sampling with different half-width at half maximum of relative intensity noise (RIN). (b) Effects of the laser noise of Eq. \ref{['Eq2']} on the fidelity. The parameters are extracted from Jia23, see the text. The effects of the servo bumps are negligible. The sensitivity to the white noise is plotted in (b). (c) Doppler dephasing. The scattered points show infidelity due to the Doppler effect as a function of atomic temperature. Each point is averaged over 50 runs of gate simulation with the pulses of Fig. \ref{['Fig1']}c. The results are compared with the Doppler sensitivity of the Rydberg antiblockade Fredkin gate (dotted-dashed line) extracted from Wu21.
• Figure S1: Controlled-Phase multi-qubit gate (a) The level scheme for performing a C$_2$-Z gate in the Cs atomic lattice. To exploit maximum control, we control the intensity and phase of 459.4 and 1040 nm lasers acting on a 2-photon Rydberg excitation. (b) The optimal profile of Rabi frequencies with real and imaginary parts is plotted by red and blue lines, respectively. The simulation considers exciting atoms to the Rydberg state $|r\rangle=|70S_{1/2}\rangle$ via the intermediate state $|p\rangle=|7P_{1/2}\rangle$ with laser detuning of 1GHz. Here a triangular lattice with 3$\mu$m interatomic distance is considered.