We propose and study ways speeding up of the entangling operations in the trapped ions system with high fidelity. First, we find a scheme to increase the speed of a two-qubit gate without the limitation of trap frequency, which was considered as the fundamental limit. Second, we study the fast gate scheme for entangling more than two qubits simultaneously. We apply the method of applying multiple frequency components on laser beams for the gate operations. In particular, in order to avoid infinite terms from the coupling to carrier transition, we focus on the phase-insensitive gate scheme here. We carefully study the effect of large excitation of motional mode beyond the limit of Lamb-Dicke approximation by including up to second order terms of the Lamb-Dicke parameter. We study the speed limit of multi-qubit global entangling gates without individual addressing requirements. Furthermore, our gates can be made insensitive to the fluctuation of initial motional phases which are difficult to stabilise in the phase-insensitive gate scheme.

We propose and study how to increase the speed of the entangling gates with trapped ions while maintaining high fidelities from two aspects. First, we find a scheme to increase the speed of a two-qubit gate without the limitation of the trap frequency, which was considered the fundamental limit. Second, we simultaneously study the possibility and speed limit in applying entangling gates for more than two qubits, the global entangling gates. For the speedup, we explore modulating various control parameters such as phases and intensities, and applying multiple-frequency laser beams for the gate operations. In particular, in order to avoid infinite terms from carrier transitions, we focus on phase-insensitive gates, equivalent to light shift gates. We find that if phase modulation is included, the gate speed can be increased without the problem of infidelity caused by excessively large excitation of motional mode beyond the limit of Lamb-Dicke approximation. We also find that, by combining phase and intensity modulations, global entangling gates can be performed close to the limit of the trap frequency without individual addressing requirements. Furthermore, our gates are robust to the fluctuation of initial motional phases which are considered uncontrollable in the phase-insensitive gate scheme.