Comparison of spontaneous emission in trapped-ion multiqubit gates at high magnetic fields

Abstract

Penning traps have been used for performing quantum simulations and sensing with hundreds of ions and provide a promising route toward scaling up trapped ion quantum platforms because of the ability to trap and control hundreds or thousands of ions in two- and three-dimensional crystals. In both Penning traps and the more common radiofrequency Paul traps, lasers are often used to drive multiqubit entangling operations. A leading source of decoherence in these operations is off-resonant spontaneous emission. While many trapped ion quantum computers or simulators utilize clock qubits, other systems, especially those with high magnetic fields such as Penning traps, rely on Zeeman qubits, which require a more complex calculation of this decoherence. We therefore examine theoretically the impacts of spontaneous emission on quantum gates performed with trapped-ion ground-state Zeeman qubits in a high magnetic field. In particular, we consider two types of gates---light-shift $({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{i}^{z}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{j}^{z})$ gates and M\o{}lmer-S\o{}rensen $({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{i}^{x}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{j}^{x})$ gates---obtained with laser beams directed approximately perpendicular to the magnetic field (the quantization axis) and compare the decoherence errors in each. Within each gate type, we also compare different operating points with regard to the detunings, polarizations, and required intensity of the laser beams used to drive the gates. We show that both gates can have similar performance at their optimal operating conditions at high magnetic fields and examine the experimental feasibility of various operating points. By examining the magnetic field dependence of each gate, we demonstrate that, when the $P$ state fine-structure splitting is large compared to the Zeeman splittings, the theoretical performance of the M\o{}lmer-S\o{}rensen gate is significantly better than that of the light-shift gate. Additionally, for the light-shift gate, we make an approximate comparison between the fidelities that can be achieved at high fields with the fidelities of state-of-the-art two-qubit trapped ion quantum gates. We show that, with regard to spontaneous emission, the achievable infidelity with our current configuration is about an order of magnitude larger than that of the best low-field gates, but we also discuss several alternative configurations with potential error rates that are comparable with those for state-of-the-art trapped ion gates.