Fast design and scaling of multi-qubit gates in large-scale trapped-ion quantum computers

TL;DR

This work tackles scaling multiqubit entangling gates in trapped-ion crystals by introducing the Large-Scale Fast (LSF) method, which converts a hard quadratic gate-design problem into a practical, polynomial-time pipeline based on seed zero-phase solutions and a two-stage conversion-plus-minimization process. The approach reveals a scaling law where the total entangling power grows as $\mathcal{O}(N^2)$ while the minimal gate time scales linearly with the number of ions, $T_{\min}\propto N$, and provides a nuclear-norm-based bound $\Omega_{\text{nuc}}$ for drive power. Through error analysis of motional-frequency drifts, drive-amplitude jitter, and heating, LSF shows how robustness constraints can mitigate adverse scaling and demonstrates feasibility with a 49-ion surface-code stabilizer example achieving infidelity $<10^{-4}$. Benchmarks indicate an order-of-magnitude speedup over traditional optimization methods, enabling offline compilation for large ion-crystal circuits and offering a path toward hundreds of qubits in ion-trap quantum processors. The framework is extensible to additional robustness constraints and different drive configurations, positioning trapped-ion architectures for scalable quantum computation.

Abstract

Quantum computers based on crystals of trapped ions are a prominent technology for quantum computation. A unique feature of trapped ions is their long-range Coulomb interactions, which can be exploited to realize large-scale multiqubit entanglement gates. However, scaling up the number of qubits, $N$, in these systems, while retaining high-fidelity and high-speed operations, is challenging. Specifically, designing multiqubit entanglement gates in long ion crystals of hundreds of ions involves an NP-hard optimization problem, rendering scale-up not only a technological challenge, but also a conceptual challenge. Here we introduce a method that mitigates this challenge, effectively allowing for a polynomial-time design of fast, robust, and programmable entanglement gates, acting on the entire ion-crystal. We show that while the number of simultaneous entanglement operations scales as $N^2$, the gate duration scales as $N$, leading to a scaling advantage. We use our methods to investigate the drive-power requirements and susceptibility to noise and errors of these multiqubit gates. Our method delineates a path towards scaling up quantum computers based on ion-crystals with hundreds of qubits.

Figures (11)

• Figure 1: Correlation between the solution drive amplitude and its estimator $\Omega_{\text{nuc}}$ for 150 different phase maps on a single $N = 50$ ion crystal. Color indicates the category of the multiqubit coupling maps. These include structured maps (e.g., all-to-all), where $N$ or a subset of ions follow a defined entangling pattern, and unstructured maps, where phases are randomly assigned to ion pairs. Phases are either restricted to $\{0, \pi/4\}$ or drawn from the continuous range $[-\pi/4, \pi/4]$ for random-phase maps. The dashed line indicates perfect correlation, $|r| = \Omega_{\text{nuc}}$.
• Figure 2: Summary of results obtained by our method, shown as total required Rabi frequency vs. gate time for various lengths of ion crystals (color). Each point corresponds to a different entanglement operation. The gate time (horizontal) is normalized by $T_\nu$, and the total Rabi frequency, $\left|r\right|$, (vertical, log scale), normalized according to the nuclear-norm based estimation, $\Omega_\text{nuc}$ and $T_g/T_\nu$. As seen with this scaling, all our solutions collapse on a single line (black dashed), signifying an inverse proportion between required power and gate time.
• Figure 3: Gate fidelity, $F$, due to a drift in the trap RF voltage, estimated for gates generating GHZ states for ion-crystals of 10 to 60 ions. Gate times of $T_g=2\cdot T_{\text{min}}$ (light) and $3\cdot T_{\text{min}}$ (dark) collapse to similar trends when rescaling the frequency error by $T_g$ (horizontal). Dashed colored-brown lines show gates on 60 ions where only a subset (of size coded by the color) of the ions are entangled, forming a sub- crystal size GHZ state. Inset: A quadratic approximation of the infidelity, with Filled circles showing the quadratic coefficient for varying crystal size, with (blue) and without (red) additional linear constraints aiming at reducing the gate sensitivity to such drifts. Both coefficients scale almost linearly with the number of bi-partite couplings in the gate, $N_\text{c}\sim N^2$, yielding a constant slope of approximately 2.
• Figure 4: Gate infidelity due to fluctuating laser amplitude. For each crystal size, the drive amplitude was rescaled by $1+\epsilon$ with $\epsilon\sim\mathcal{N}(0,\sigma^2)$. Filled circles correspond to numerical averages (mean of 300 samples), while solid lines show the expectation value of Eq. \ref{['eqErrorLaserAmplitude']}.
• Figure 5: Infidelity due to motional heating. Main frame: Heating-induced errors for GHZ gates in crystals of 10–60 ions. To remove the trivial gate-time scaling, the COM-mode heating rate is normalized to $0.02$ quanta per gate time. Results are shown for correlated noise ($\zeta=150~\mu$m, spanning most of the crystal) and uncorrelated noise ($\zeta=1~\mu$m). Solutions including motional error minimization linear constraints (blue) are compared with unconstrained solutions (Red). Inset: Error growth with gate time in a 20-ion crystal. Errors increase linearly with $T_g$, as expected from Eq. \ref{['eqHeatingRateBound']}, with robust gates (blue) showing roughly half the error rate.