Decompositions of n-qubit Toffoli Gates with Linear Circuit Complexity

Toffoli gates are natural elements for the circuit model based quantum computation. We investigate general resource requirements for arbitrary n-qubit Toffoli gate. These resources consist of the nontrivial Clifford gate (CNOT), non-Clifford gate (T gate), ancillary qubits, and circuit depth. To implement n-qubit Toffoli gates, we consider two cases: only one auxiliary qubit and unlimited auxiliary qubits. The key of the first case is to decompose an n-qubit Toffoli gate into the reduced Toffoli gate modulo phase shift using the Clifford gates and one ancillary qubit. With this construction, it only requires O(n) number of general resources for an n-qubit Toffoli gate. For the second case, an approximate Toffoli gate is constructed to obtain efficient decomposition of a Toffoli gate. The new decomposition can further reduce general resources except auxiliary qubits.