  Hamiltonian encoding is a methodology for revealing the mechanism behind the
dynamics governing controlled quantum systems. In this paper, following Mitra
and Rabitz [Phys. Rev. A 67, 033407 (2003)], we define mechanism via pathways
of eigenstates that describe the evolution of the system, where each pathway is
associated with a complex-valued amplitude corresponding to a term in the Dyson
series. The evolution of the system is determined by the constructive and
destructive interference of these pathway amplitudes. Pathways with similar
attributes can be grouped together into pathway classes. The amplitudes of
pathway classes are computed by modulating the Hamiltonian matrix elements and
decoding the subsequent evolution of the system rather than by direct
computation of the individual terms in the Dyson series. The original
implementation of Hamiltonian encoding was computationally intensive and became
prohibitively expensive in large quantum systems. This paper presents two new
encoding algorithms that calculate the amplitudes of pathway classes by using
techniques from graph theory and algebraic topology to exploit patterns in the
set of allowed transitions, greatly reducing the number of matrix elements that
need to be modulated. These new algorithms provide an exponential decrease in
both computation time and memory utilization with respect to the Hilbert space
dimension of the system. To demonstrate the use of these techniques, they are
applied to two illustrative state-to-state transition problems.
