Multibit Gates for Quantum Computing

We present a general technique to implement products of many qubit operators communicating via a joint harmonic oscillator degree of freedom in a quantum computer. By conditional displacements and rotations we can implement Hamiltonians which are trigonometric functions of qubit operators. With such operators we can effectively implement higher order gates such as Toffoli gates and ${\mathrm{C}}^n-\mathrm{NOT}$ gates, and we show that the entire Grover search algorithm can be implemented in a direct way.