Toffoli gate based on a three-body fine-structure-state-changing Förster resonance in Rydberg atoms

TL;DR

This work presents an improved scheme for implementing a three-qubit Toffoli gate using a fine-structure state-changing three-body Förster resonance in a line of Rubidium Rydberg atoms. By exploiting Stark-tuned three-body interactions and avoiding the need for magnetic tuning, the approach yields simplified population and phase dynamics with identical initial states across qubits. Numerical simulations indicate gate fidelities exceeding 99% with relatively short operation times, and a reduced sensitivity to electric-field fluctuations when interatomic spacing is optimized. The results point to a practical pathway for high-fidelity multi-qubit gates in large-scale neutral-atom registers, while highlighting lifetimes as a limiting factor and suggesting cryogenic environments as a possible improvement.

Abstract

We have developed an improved scheme of a three-qubit Toffoli gate based on fine structure state changing three-body Stark-tuned Rydberg interaction. This scheme is a substantial improvement of our previous proposal [I.I.Beterov et al., Physical Review A 98, 042704 (2018)]. Due to the use of a different type of three-body Förster resonance we substantially simplified the scheme of laser excitation and phase dynamics of collective three-body states. This type of Förster resonance exists only in systems with more than two atoms, while the two-body resonance is absent. We reduced the sensitivity of the gate fidelity to fluctuations of external electric field and eliminated the necessity to use external magnetic field for fine tuning of the resonant electric field value, compared to the previous scheme of Toffoli gate based on Rydberg atoms. A gate fidelity of >99% was demonstrated in the calculations.

Figures (4)

• Figure 1: (a) Numerically calculated Stark structure of the collective energy levels, involved in three-body Förster resonance $|70P_{3/2}\rangle^{\otimes 3} \to |70S_{1/2};71S_{1/2};70P_{1/2}\rangle$. The intersections 1-4 mark the positions of three-body resonances. (b) Numerically calculated dependence of the fraction $\rho$ of atoms in the final $|71S_{1/2}\rangle$ state after three-body interaction on the external dc electric field for the initial state $|70P_{3/2}\left(m=1/2\right)\rangle^{\otimes 3}$ (marked as 1 in Fig. 1(a)). The atoms are located along the Z axis at interatomic distance $R=10$ microns.
• Figure 2: (a) General scheme of the three-qubit Toffoli gate. (b) Scheme of the Toffoli gate based on three-body Rydberg interactions. Three atoms are located in the individual optical dipole traps aligned along the Z axis, which is co-directed with the controlling dc electric field. Laser Raman (or microwave) pulses 1 and 8 drive transitions between the logical states $|0\rangle$ and $|1\rangle$ of the target qubit. Laser pulses 2-7 excite and de-excite the chosen Rydberg states of the three atoms. The $\pi$ phase shift due to the three-body interaction appears only if all three atoms are excited into Rydberg states. The green and blue arrows here indicate $|70P_{3/2}\rangle^{\otimes 3} \to |70S_{1/2};70P_{3/2};71S_{1/2}\rangle$ and $|70S_{1/2};70P_{3/2};71S_{1/2}\rangle \to |70S_{1/2};71S_{1/2};70P_{1/2}\rangle$ intermediate two-body transitions, respectively. (c) Timing diagram of the pulses in the proposed gate scheme. The whole gate scheme includes the following 5 steps: application of pulse 1, simultaneous application of pulses 2-4, application of a constant external electric field, simultaneous application of pulses 5-7, application of pulse 8.
• Figure 3: Numerically calculated time dependences of the populations and phases of the initially excited collective states of three interacting atoms. The upper row (a, b, g, h) depicts the multiparticle state population and phase evolution when all three atoms are excited into Rydberg states ($|rrr\rangle$). The middle (c, d, i, j) and lower (e, f, k, l) rows belong to configurations $|rgr\rangle$ and $|grr\rangle$ ($|rrg\rangle$), respectively. Here $|g\rangle$ is the ground state which can be either $|0\rangle$ or $|1\rangle$, $|r\rangle$ is the Rydberg state $|70P_{3/2}\left(m=1/2\right)\rangle$ The phase values are presented in ordinary units in the range ($-\pi$, $\pi$). System parameters: (a - f) $R = 10$ µ m; $E=0.14235$ V/cm; $T = 1.15$ µ s; (g - l) $R = 8.5$ µ m;$E=0.1469$ V/cm; $T= 0.42$ µ s.
• Figure 4: Dependence of the fidelity of the Toffoli gate on the dc electric field for two different interatomic distances: $R=10$ µ m (red curve) and $R=8.5$ µ m (blue curve). The maximum fidelity of $99.05\%$ is achieved with an electric field of $0.14232$ V/cm. The interaction times coincide with those indicated in the description of Fig. \ref{['Phase_diag']} for both cases.(a) For a wide range of electric field values (0.1-0.2 V/cm). (b) Near the fidelity maxima.