Abstract
We describe a simple protocol for the single-step generation of -body entangling interactions between trapped atomic ion qubits. We show that qubit state-dependent squeezing operations and displacement forces on the collective atomic motion can generate full -body interactions. Similar to the M{\o}lmer-S{\o}rensen two-body Ising interaction at the core of most trapped ion quantum computers and simulators, the proposed operation is relatively insensitive to the state of motion. We show how this -body gate operation allows the single-step implementation of a family of -bit gate operations such as the powerful -Toffoli gate, which flips a single qubit if and only if all other - qubits are in a particular state.