Programmable N-body interactions with trapped ion qubits*

The qubit and gate model of a quantum computer employs a universal set of operations, such as single-qubit rotations and two-qubit controlled-NOT gates. While such few-qubit interactions are sufficient for general computation, and can be used to construct many-body entangled states, many-qubit interactions can dramatically simplify quantum circuit structures, speed up their execution, and extend the power of quantum computer systems facing decoherence. We describe a simple protocol for the single-step generation of N-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 N-body interactions. We show how this N-body gate operation allows the single-step implementation of a family of N-bit gate operations such as the powerful N-Toffoli gate, which flips a single qubit if and only if all other N-1 qubits are in a particular state.