Read a lot of papers. One of the most understandable
is the original paper by Shor on
algorithms for prime
factorization and discrete logarithms,
but I could not figure out how to reproduce the Figure 5.1 probability plot
using equation (5.7).
However, using an alternative equation (3.50) from
the lecture notes of N. David Mermin, prob.c reproduces the
plot.
This probability distribution could be used to
simulate the quantum algorithm on a conventional computer,
but the distribution is a function of the parameter we are trying to
find, so such a simulation would be rather pointless.
Further conceptual insight is provided in Appendix A, page 27, of
Shor's discrete logarithm quantum algorithm for elliptic curves
by John Proos and Christof Zalka, using an eigenstate representation.
The toy theory
of Robert Spekkens is helpful in understanding quantum states.
Nitin Jain's timing attacks
paper also provides a simplified magical ball model
for quantum cryptography.
QBism
by Christopher Fuchs is an alternative approach to quantum mechanics
using Baysian probability theory.
Quantum annealing by
Boixo, et. al. describes experiments using the
D-Wave quantum computer.
Also see pqcrypto.org
for information on post-quantum cryptography.