Multiqubit Rydberg Gates for Quantum Error Correction

TL;DR

This work develops a coherent framework for implementing multiqubit gates with global Rydberg pulses in neutral-atom systems and provides analytic, decay-aware pulse shapes via an open-source optimization tool. It demonstrates that symmetric, low-parameter pulses can realize time- and Rydberg-time-optimal CZ and CCZ gates, enabling measurement-free fault-tolerant QEC and Floquet QEC protocols that place break-even thresholds within reach of current hardware. The study shows that native multiqubit gates can reduce circuit depth and shuttling overhead, especially under biased noise, while finite-interaction geometries introduce robust, geometry-dependent tradeoffs. Overall, these results indicate practical pathways to fault-tolerant quantum computation in single-species neutral-atom processors using global three-qubit gates for stabilizer readout and measurement-free operations. The work emphasizes that, despite higher raw error rates, native multiqubit gates can outperform decomposed implementations in realistic, biased-noise environments, with significant implications for near-term quantum error correction experiments.

Abstract

Multiqubit gates that involve three or more qubits are usually thought to be of little significance for fault-tolerant quantum error correction because single gate faults can lead to high-weight correlated errors. However, recent works have shown that multiqubit gates can be beneficial for measurement-free fault-tolerant quantum error correction and for fault-tolerant stabilizer readout in unrotated surface codes. In this work, we investigate multiqubit Rydberg gates that are useful for fault-tolerant quantum error correction in single-species neutral-atom platforms and can be implemented with a single, non-addressed laser pulse. We develop an open-source Python package to generate analytical, few-parameter pulses that implement the desired gates while minimizing gate errors due to Rydberg-state decay. The tool also allows us to identify parameter-optimal pulses, characterized by a minimal parameter count for the pulse ansatz. Measurement-free quantum error correction protocols require CCZ gates, which we analyze for atoms arranged in symmetric and asymmetric configurations. We investigate the performance of these schemes for various single-, two-, and three-qubit gate error rates, showing that break-even performance of measurement-free QEC is within reach of current hardware. Moreover, we study Floquet quantum error correction protocols that comprise two-body stabilizer measurements. Those can be realized using global three-qubit gates, and we show that this can lead to a significant reduction in shuttling operations. Simulations with realistic circuit-level noise indicate that applying three-qubit gates for stabilizer measurements in Floquet codes can yield competitive logical qubit performance in experimentally relevant error regimes.

Figures (16)

• Figure 1: Multiqubit Rydberg gates on three atoms. (a) Three atoms arranged in an isosceles triangle are illuminated by a global Rydberg laser beam. This is not the most general arrangement of three atoms; however, it encompasses many geometries one might encounter in a lattice (e.g. equilateral triangle, right triangle, or straight line). The interaction between the first atom and the third one may be different from the interaction between the second atom and each of the other ones. This allows, in principle, for the realization of multiqubit gates that are not symmetric with respect to the exchange of any two qubits. (b) A parameterization of all multiqubit gates that the global Rydberg pulse can realize. The constituent gates are defined as $\mathrm{C}_{n}\mathrm{Z}(\alpha) := \mathrm{diag}(1, ..., 1, e^{i\alpha})$ on $n+1$ qubits, and $\mathrm{Z}(\alpha) := \mathrm{C}_{0}\mathrm{Z}(\alpha) = \mathrm{diag}(1, e^{i\alpha})$.
• Figure 2: Time-optimal and Rydberg time-optimal $\bm{\mathrm{CZ}}$ gate pulses in the perfect Rydberg blockade regime. The time-optimal pulse jandura2022timeoptimalpagano2022error requires only 4 parameters using the antisymmetric ansatz \ref{['eq:ansatz_sin']} ($\Omega_0 T = 7.611, \, \Omega_0 T_R = 2.958$). The Rydberg time $T_R$ can be reduced marginally when the pulse optimization is performed in the presence of decay from Rydberg states, which we model by adding a non-Hermitian term to the Hamiltonian (see Eq. \ref{['eq:H_decay']}). A 6-parameter pulse with ansatz \ref{['eq:ansatz_sin']} yields $\Omega_0 T = 7.725, \, \Omega_0 T_R = 2.936$. All pulse parameters are provided in Appendix \ref{['app:pulse_parameters']}, Table \ref{['tab:timeoptimal_CZ']}. Here and in all following plots of pulse profiles, a constant detuning $\Delta_0$ is translated into a linear contribution $\Delta_0 t$ to the phase profile $\xi(t)$, and a constant is added such that $\xi(0)=0$.
• Figure 3: Fast implementation of a Rydberg CCZ gate. A CCZ gate can be realized by physically applying the gate $G_{3}(\pi,\pi,\pi)$$=: \overline{\mathrm{CCZ}}$ preceded and succeeded by global single-qubit rotations. As described in Ref. evered2023highfidelity, the $\overline{\mathrm{CCZ}}$ gate can be implemented with a faster global Rydberg pulse than the traditional CCZ gate. While the minimal pulse duration of the former gate is $\Omega_0 T = 10.8$, the latter one takes at least a time $\Omega_0 T = 16.4$ in the perfect blockade regime.
• Figure 4: Time-optimal and $\bm{T_R}$-optimal $\bm{\overline{\mathrm{CCZ}}}$ gates in the perfect Rydberg blockade regime. For an increasing number of pulse parameters, and thus increasing pulse complexity, we optimize $\overline{\mathrm{CCZ}}$ gate pulses using the general ansatz \ref{['eq:ansatz_sin_cos']} and the antisymmetric ansatz \ref{['eq:ansatz_sin']}. Panel (a) shows the minimal pulse durations $\Omega_0 T$ found among $4\times10^4$ optimization runs for each data point. Every such pulse realizes the target gate with infidelity $<\!10^{-7}$. The time-optimal pulse evered2023highfidelity, marked with a grey number 1 in the plot, requires 14 parameters (1: $\Omega_0 T = 10.83, \, \Omega_0 T_R = 4.91$). A nearly time-optimal gate can be realized with a pulse described by 8 parameters (2: $\Omega_0 T = 10.97, \, \Omega_0 T_R = 4.18$). The minimal number of pulse parameters required to realize the gate is 6 (3: $\Omega_0 T = 12.24, \, \Omega_0 T_R = 4.40$). In panel (b), pulses are optimized in the presence of Rydberg state decay with rate $\gamma/\Omega_0 = 10^{-4}$. Again, each data point corresponds to the best out of $4\times10^4$ optimization runs. In order to achieve the smallest possible gate error, the Rydberg time $T_R$ is minimized. We calculate both quantities separately, confirming $1-F=\gamma T_R$ for $\gamma T_R \ll 1$. A 10-parameter pulse can reduce $T_R$ considerably compared to the time-optimal gates (4: $\Omega_0 T = 12.73, \, \Omega_0 T_R = 3.95$). Note that a smaller gate duration does not imply smaller Rydberg times $T_R$. Panel (c) shows the pulse profiles for the gates 1--4 discussed in panels (a, b). The respective pulse parameters are provided in Appendix \ref{['app:pulse_parameters']}, Table \ref{['tab:timeoptimal_CCZ']}.
• Figure 5: $\bm{\overline{\mathrm{CCZ}}}$ gate at finite interaction strengths and asymmetric atomic geometries. (a) A CCZ gate followed by reset of two qubits can be realized by a global multiatom Rydberg gate $G_3(0, \theta', \pi)$$=: \theta'$-$\mathrm{CCZ}$, where the precise value of the gate parameter $\theta'$ does not matter. In Appendix \ref{['app:more_multiqubit_gates']}, Fig. \ref{['fig:circuit_CCZ_fast_eps']}, we show that the faster Rydberg gate $G_3(\pi, \theta', \pi)$$=: \theta'$-$\overline{\mathrm{CCZ}}$ can also implement a CCZ gate followed by a reset of two qubits. (b) Minimal Rydberg times achievable for $\overline{\mathrm{CCZ}}$ and $\theta'$-$\overline{\mathrm{CCZ}}$ gates on three atoms arranged in a right triangle ($V_{\mathrm{nn}}/(\hbar \Omega_0) = 32$, $V_{\mathrm{nnn}}/(\hbar \Omega_0) = 4$) and on a line ($V_{\mathrm{nn}}/(\hbar \Omega_0) = 32$, $V_{\mathrm{nnn}}/(\hbar \Omega_0) = 0.5$). We use the antisymmetric pulse ansatz \ref{['eq:ansatz_sin']}. (c) Sensitivity of the gates mentioned above to variations in the interatomic distances, i.e., variations in the interaction strengths. The left plot considers a 10-parameter pulse calibrated for $V_{\mathrm{nn}}/(\hbar \Omega_0) = V_{\mathrm{nnn}}/(\hbar \Omega_0) = 32$. The center panel investigates two 14-parameter pulses calibrated for atoms arranged in a right triangle. The right panel considers two 12-parameter pulses calibrated for atoms arranged on a line. In all panels in this figure, the decay strength is set to $\gamma/\Omega_0=10^{-4}$.