It is commonly assumed that Shor's quantum algorithm for the efficient factorization of a large number $N$ requires a pure initial state. Here we demonstrate that a single pure qubit, together with a collection of ${log}_{2}N$ qubits in an arbitrary mixed state, is sufficient to implement Shor's factorization algorithm efficiently.

• Received 19 January 2000

DOI:https://doi.org/10.1103/PhysRevLett.85.3049