Fast Native Three-Qubit Gates and Fault-Tolerant Quantum Error Correction with Trapped Rydberg Ions

TL;DR

The paper addresses the slow, scalability-limited entangling gates in trapped-ion platforms by introducing a native CCZ gate for microwave-dressed Rydberg ions and optimizing a single-pulse protocol that mitigates Rydberg decay. It combines a rigorous Hamiltonian model with pulse optimization to achieve high-fidelity, fast multi-qubit operations and demonstrates their utility through a measurement-free, fault-tolerant Bacon-Shor QEC scheme on a linear ion chain. The results show fidelities surpassing 97% with gate times around 2 μs under cryogenic conditions, and a quadratic scaling of logical error rates in a FC QEC cycle, signaling a viable route toward FT quantum computing with Rydberg ions. However, achieving logical error rates below 10^-6 would require substantial improvements in physical gate fidelities and higher connectivity, motivating exploration of 2D ion-crystal architectures for scalable FT QEC.

Abstract

Trapped ions as one of the most promising quantum-information-processing platforms, yet conventional entangling gates mediated by collective motion remain slow and difficult to scale. Exciting trapped ions to high-lying electronic Rydberg states provides a promising route to overcome these limitations by enabling strong, long-range dipole-dipole interactions that support much faster multi-qubit operations. Here, we introduce the first scheme for implementing a native controlled-controlled-Z gate with microwave-dressed Rydberg ions by optimizing a single-pulse protocol that accounts for the finite Rydberg-state lifetime. The resulting gate outperforms standard decompositions into one- and two-qubit gates by achieving fidelities above 97% under realistic conditions, with execution times of about 2 microseconds at cryogenic temperatures. To explore the potential of trapped Rydberg ions for fault-tolerant quantum error correction, and to illustrate the utility of three-qubit Rydberg-ion gates in this context, we develop and analyze a proposal for fault-tolerant, measurement-free quantum error correction using the nine-qubit Bacon-Shor code. Our simulations confirm that quantum error correction can be performed in a fully fault-tolerant manner on a linear Rydberg-ion chain despite its limited qubit connectivity. These results establish native multiqubit Rydberg-ion gates as a valuable resource for fast, high-fidelity quantum computing and highlight their potential for fault-tolerant quantum error correction.

Figures (9)

• Figure 1: Electronic level structure. (a) Considered energy levels of a single $^{88}\mathrm{Sr}^+$-ion. The computational basis states are $\ket{0}$ and $\ket{1}$, with the latter coupled to the Rydberg state $\ket{ns}$ with Rabi frequency $\Omega_\mathrm{L}$ and detuning $\Delta_\mathrm{L}$. The Rydberg states $\ket{ns}$ and $\ket{np}$ are resonantly coupled by a microwave driving field with Rabi frequency $\Omega_\mathrm{MW}$, which yields the dressed states $\ket{\mathcal{D}_\pm}$. (b) Three Rydberg ions confined in a linear Paul trap interacting via dipole--dipole interactions between nearest neighbors, $V_{12}$ and $V_{23}$, and next-nearest neighbors, $V_{13}$. (c) Complete energy-level diagram of the interacting three-ion system. Both dressed states $\ket{\mathcal{D}_\pm}$ are coupled to $\ket{1}$ with Rabi frequency $\frac{\Omega_\mathrm{L}}{\sqrt{2}}$ and effective laser detuning $\frac{\Omega_\mathrm{MW}}{2}\pm\Delta_\mathrm{L}$ (cf. Hamiltonian \ref{['eqn:Hamiltonian']}). The decay rate $\gamma_\mathrm{R}$ accounts for the finite lifetime of the Rydberg states.
• Figure 2: CCZ-gate optimization results. (a) Fidelity of the optimized CCZ-gate protocols \ref{['eqn:pulses']} vs. the dimensionless gate time $\tau V$ for different decay rates $\gamma_\mathrm{R}$. Crosses ($\times$) represent approximate fidelities considering only next-nearest neighbor entangling-phase errors $\bar{\varphi}_{101}^\mathrm{ent}$ and decay-induced population errors $\bar{p}_\mathrm{dec}$. (b) Gate dynamics with $\mathcal{F}\approx 1$ and $\gamma_\mathrm{R}=0$. Top: Modulation of the Rabi frequency $\Omega_\mathrm{L}$ and the detuning $\Delta_\mathrm{L}$ of the excitation laser. The solid gray lines at $\Delta_\mathrm{L}=\mp\frac{\Omega_\mathrm{MW}}{2}$ indicate resonance with the dressed Rydberg states $\ket{\mathcal{D}_\pm}$. Center: Population dynamics of the computational basis states for the initial state $\ket{+++}$. Bottom: Accumulated entangling phases $\varphi_{abc}^\mathrm{ent}$. (c) Same as in (b) for a gate with reduced fidelity $\mathcal{F}\approx 96.75\,\%$ due to shorter gate time $\tau V$. (d) Gate fidelity for various gate times vs. the interaction ratio $\alpha = \frac{V_{13}}{V}$ for $\gamma_\mathrm{R}=1.64\cdot10^{-3}\,V/2\pi$. The dashed gray line indicates the linear Coulomb crystal configuration ($V_{13} = \frac{V}{8}$). The optimized parameters, gate fidelities, population and phase errors of the gates can be found in App. \ref{['app:parameter']}.
• Figure 3: Comparison of CCZ-gate optimization with and without decay. Depicted are the population dynamics for initial state $\ket{+++}$ (top) and accumulated entangling phases $\varphi_{abc}^\mathrm{ent}$ (bottom) both simulated with a decay rate ${\gamma_\mathrm{R}=1.64\cdot10^{-3}\,V/2\pi}$. (a) Gate optimization without accounting for decay results in significant population loss and a reduced fidelity of $\mathcal{F}=89.65\,\%$. (b) The same gate optimized while accounting for decay. The population loss is decreased by reducing the time spent in the Rydberg states, yielding a substantially improved fidelity of $\mathcal{F}=97.29\,\%$. The optimized parameters, gate fidelities, population loss and phase errors of the gates can be found in App. \ref{['app:parameter']}.
• Figure 4: Measurement-free FT QEC protocol for the nine-qubit Bacon-Shor code and its implementation with the available Rydberg-ion gate set. (a) Measurement-free FT protocol of the nine-qubit Bacon-Shor code without connectivity restrictions Veroni2024. Fault tolerance is ensured by extracting redundant syndrome information and by an appropriate ordering of the stabilizer readouts, as discussed in the main text. The illustrated error-propagation examples show that hook errors map only onto gauge operators, possibly accompanied by at most one correctable weight-one data error. (b) Nine-qubit Bacon-Shor code on a $3\times3$ lattice including the stabilizers $S_X^{1,2}$ and $S_Z^{1,2}$ together with the logical operators $X_\mathrm{L}$ and $Z_\mathrm{L}$. (c) Realization of the $S_X^{1}$-readout with the available Rydberg-ion gate set using additional SWAP operations. Dashed lines indicate SWAP ancillas required for FT SWAPs (blue). (d) Realization of the coherent correction with the available Rydberg-ion gate set using additional SWAP operations and native CCZ gates. (e) Fault-tolerant SWAP operation in which the exchange between two qubits is mediated via an ancilla (dashed line), preventing correlated errors on the two data qubits (cf. red error propagation). (f) Logical error rate $p_\mathrm{L}$ as a function of the noise-scaling parameter $\lambda$ for logical states initialized in $\ket{0}_\mathrm{L}$ and $\ket{+}_\mathrm{L}$. The dashed line indicates a $\lambda^2$ dependence, illustrating the expected quadratic scaling of $p_\mathrm{L}$ characteristic of an FT circuit. A power-law fit of the data, excluding the two largest values of $\lambda$, yields $p_\mathrm{L}(\lambda) = C \lambda^\alpha$ with $\alpha \approx 1.99$ for both logical states.
• Figure 5: $\mathrm{C}_1\mathrm{Z}_3$-gate optimization results. Comparison of the fidelity of optimized $\mathrm{C}_1\mathrm{Z}_3$- and CCZ-gate protocols \ref{['eqn:pulses']} vs. gate duration $\tau$ for different decay rates $\gamma_\mathrm{R}$. The grey vertical dashed line indicates the three gates plotted in Fig. \ref{['fig:C1Z3_gates']}.