Trapped-ion two-qubit gates with >99.99% fidelity without ground-state cooling (2510.17286v1)

Abstract: We introduce the 'smooth gate', an entangling method for trapped-ion qubits where residual spin-motion entanglement errors are adiabatically eliminated by ramping the gate detuning. We demonstrate electronically controlled two-qubit gates with an estimated error of $8.4(7)\times10^{-5}$ without ground-state cooling. We further show that the error remains $\lesssim 5\times10^{-4}$ for ions with average phonon occupation up to $\bar{n}=9.4(3)$ on the gate mode. These results indicate that trapped-ion quantum computation can achieve high fidelity at temperatures above the Doppler limit, which enables faster and simpler device operation.

• The paper introduces a smooth adiabatic gate protocol that achieves >99.99% fidelity without the need for ground-state cooling.
• The protocol employs a detuning ramp with constant Rabi frequency to suppress residual spin-motion entanglement and mitigate noise.
• Experimental results demonstrate robust gate performance at Doppler temperatures with simplified calibration and scalability in QCCD architectures.

Trapped-Ion Two-Qubit Gates with >99.99% Fidelity without Ground-State Cooling

Introduction and Motivation

High-fidelity two-qubit entangling gates are essential for scalable, fault-tolerant quantum computation with trapped ions. Historically, achieving error rates below the quantum error correction (QEC) threshold has required ground-state cooling of the ions' motion, which imposes significant overhead in both time and experimental complexity. The work under review introduces the "smooth gate," a novel adiabatic entangling gate protocol that achieves error rates below $10^{-4}$ without ground-state cooling, thus enabling high-fidelity quantum logic at temperatures above the Doppler limit. This result is particularly significant for architectures where cooling and transport dominate the total circuit runtime, and it has direct implications for the scalability and practicality of trapped-ion quantum computers.

Theoretical Framework: Diabatic vs. Adiabatic Gates

The standard approach to trapped-ion entangling gates is the geometric phase gate, which relies on spin-dependent forces detuned from a motional mode. The conventional "diabatic" approach (DESE) uses a constant detuning and rapidly ramps the force on and off, requiring precise timing to ensure the ions' motion returns to its origin in phase space at the end of the gate. This method is highly sensitive to detuning errors and motional frequency drifts, leading to residual spin-motion entanglement that scales with the thermal occupation number $\bar{n}$.

In contrast, the "adiabatic" approach (AESE) slowly ramps the equilibrium position of the forced states, allowing the ions' motion to adiabatically follow the instantaneous equilibrium. This suppresses residual spin-motion entanglement, making the gate robust to both static and dynamic detuning errors and largely insensitive to the initial motional state.

Figure 1: Schematic comparison of diabatic (top) and adiabatic (bottom) geometric phase gates, and the frequency dynamics during a smooth gate.

The smooth gate protocol leverages AESE by ramping the gate detuning $\delta(t)$ while keeping the Rabi frequency $\Omega_g$ constant, in contrast to previous adiabatic proposals that ramped $\Omega_g$. This choice maintains a high entanglement rate and simplifies calibration, as the entangling angle $\theta_g$ can be tuned by a single parameter ($\delta_\text{min}$).

Filter Function Analysis and Motional Robustness

A central contribution of the paper is the quantitative analysis of the smooth gate's robustness to motional frequency fluctuations, using the filter function formalism. The filter function $F(\omega_f)$ quantifies the gate's sensitivity to noise at frequency $\omega_f$. The smooth gate exhibits a filter function that is significantly suppressed at low frequencies compared to Walsh-modulated diabatic gates, indicating superior robustness to both static and slow time-dependent detuning errors.

Figure 2: Filter function comparison for smooth, Walsh-3, and Walsh-1 gates at $\bar{n}=0$ and $\bar{n}=10$.

The smooth gate's filter function remains nearly unchanged as the thermal occupation increases, demonstrating that the protocol's error rate is nearly independent of $\bar{n}$ over a wide range. This is in stark contrast to diabatic gates, where the error rate increases linearly with $2\bar{n}+1$.

Experimental Implementation and Calibration

The experimental realization uses a pair of $^{40}$Ca$^+$ ions in a cryogenic trap, with qubits encoded in Zeeman sublevels and entangling gates implemented via bichromatic near-field microwave currents. The smooth gate sequence consists of:

1. Ramping $\Omega_g$ up at large detuning ($\delta_\text{max}$).
2. Adiabatically ramping $\delta(t)$ from $\delta_\text{max}$ to $\delta_\text{min}$.
3. Holding at $\delta_\text{min}$ for a calibrated duration.
4. Reversing the detuning ramp.
5. Ramping $\Omega_g$ down.

The calibration is straightforward: the entangling angle $\theta_g$ is set by scanning $\delta_\text{min}$ and identifying the point where the populations of $\ket{\uparrow\uparrow}$ and $\ket{\downarrow\downarrow}$ cross at 0.5, corresponding to a maximally entangling gate.

Figure 3: Measured populations as a function of $\delta_\text{min}$, showing the calibration of the entangling angle.

Benchmarking and Error Analysis

The fidelity of the smooth gate is characterized using Subspace Leakage Error Randomized Benchmarking (SLERB), which isolates two-qubit gate errors from single-qubit and SPAM errors. The protocol applies sequences of Clifford operations built from two-qubit gates and measures the decay of survival and leakage probabilities.

Figure 4: SLERB results showing population decay and extracted error rates for smooth gates at Doppler temperature.

The key results are:

• Gate error per two-qubit gate: $8.4(7) \times 10^{-5}$ for sequences up to 432 gates.
• Error remains below $5 \times 10^{-4}$ for $\bar{n}$ up to 9.4, i.e., well above the Doppler limit.
• The dominant error is geometric phase error, with spin-motion entanglement (leakage) suppressed to $<10^{-4}$.

  Figure 5: Survival and leakage probabilities for smooth and Walsh-1 gates as a function of static detuning offset, demonstrating the superior robustness of the smooth gate.

  

  Figure 6: Two-qubit gate error as a function of initial motional occupation number $\bar{n}$, showing insensitivity to temperature.

Implications for Quantum Computing Architectures

The smooth gate protocol has several architectural implications:

• Elimination of ground-state cooling: Enables high-fidelity operation at Doppler or higher temperatures, reducing circuit time and experimental complexity.
• Simplified calibration: Single-parameter tuning of the entangling angle, with no need for precise amplitude control or multi-parameter searches.
• Robustness to motional noise: Tolerance to both static and dynamic detuning errors, facilitating operation in large-scale, multi-zone QCCD architectures.
• Scalability: The protocol is compatible with global entangling operations and local frequency shimming, supporting parallel gate execution and flexible addressing.

The reduction in cooling and transport overheads can yield an order-of-magnitude speedup in circuit execution, and the relaxed requirements on laser power and beam geometry simplify the engineering of large-scale devices.

Future Directions

Potential avenues for further development include:

• Optimization of detuning ramp profiles to further minimize gate duration and residual errors.
• Extension to multi-qubit and multi-mode gates in larger ion crystals.
• Integration with advanced QEC protocols (e.g., qLDPC codes) to exploit the low error rates and high throughput enabled by the smooth gate.
• Application to other physical platforms where adiabatic elimination of unwanted couplings can enhance robustness.

Conclusion

The smooth gate protocol represents a significant advance in the practical implementation of high-fidelity entangling gates for trapped-ion quantum computing. By adiabatically eliminating spin-motion entanglement through detuning ramps, the protocol achieves error rates below $10^{-4}$ without ground-state cooling, with robustness to motional noise and simplified calibration. These features directly address key bottlenecks in current QCCD architectures and pave the way for faster, simpler, and more scalable trapped-ion quantum computers.