  In this note we consider optimised circuits for implementing Shor's quantum
factoring algorithm. First I give a circuit for which none of the about 2n
qubits need to be initialised (though we still have to make the usual 2n
measurements later on). Then I show how the modular additions in the algorithm
can be carried out with a superposition of an arithmetic sequence. This makes
parallelisation of Shor's algorithm easier. Finally I show how one can factor
with only about 1.5n qubits, and maybe even fewer.
