Factoring with n+2 clean qubits and n-1 dirty qubits

Abstract: We present reversible classical circuits for performing various arithmetic operations aided by dirty ancillae (i.e. extra qubits in an unknown state that must be restored before the circuit ends). We improve the number of clean qubits needed to factor an n-bit number with Shor's algorithm from 1.5n+O(1) to n+2, assisted by n-1 dirty qubits, without increasing the asymptotic size or depth of the circuit.