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

Quantum computers based on trapped-ion crystals offer intrinsic all-to-all connectivity via long-range Coulomb interactions, enabling the efficient generation of large-scale entanglement. Designing fast, high-fidelity, and programmable entangling gates in long ion chains becomes increasingly challenging as the number of ions, and with it the density of the motional spectrum, grows. Moreover, finding power-optimal solutions to the multiqubit gate-design problem is NP-hard.

We present a numerical framework, termed large-scale fast (LSF), that enables polynomial-time design of fully programmable multiqubit entangling gates in large ion crystals. LSF constructs exact gate solutions in a single step, followed by numerical power minimization via a quadratically constrained quadratic programming (QCQP) algorithm tailored to the algebraic structure of the problem.

We exploit the efficient construction of gate solutions to study the scaling of resources, constraints, and errors as both crystal size and entanglement complexity increase. In particular, we find that the minimal gate duration, below which high-fidelity solutions cannot be obtained, scales linearly with system size, despite the gate implementing pairwise entangling operations. We further derive an estimate for the optimal total drive amplitude and analyze its dependence on gate and crystal parameters. Finally, we investigate the sensitivity of these gates to common experimental error sources, such as motional mode frequency drifts, and characterize their scaling with gate time and system size.