  We show that under the matrix product state formalism the states produced in
Shor's algorithm can be represented using O(max($4lr^2$, $2^{2l}$)) space,
where l is the number of bits in the number to factorise, and r is the order
and the solution to the related order-finding problem. The reduction in space
compared to an amplitude formalism approach is significant, allowing
simulations as large as 42 qubits to be run on a single processor with 32GB
RAM. This approach is readily adapted to a distributed memory environment, and
we have simulated a 45 qubit case using 8 cores with 16GB RAM in approximately
one hour.
