Figure 10. Computing the absorption spectra: a) a code
snippet illustrating a Fourier transform of the ACF computed above; b)
the computed absorption line shapes for the two temperatures considered.
3.2.3. Some complexity
considerations
In this section, we discuss the computational requirements for running
HEOM calculations and we assess the runtimes expected for problems of
various sizes. The main parameters that affect the complexity and
runtime of HEOM simulations are the maximal hierarchy level (\(L\),
defined by the parameter “LL”), the number of Matsubara frequencies
(\(K\), defined by the parameter “KK”), and the number of quantum
states (\(M\), defined by the dimensionality of the input Hamiltonian
matrix). The latter two define the length of the multi-dimensional index
vectors enumerating ADMs, \(d=M*(K+1)\). The total number of
distinct ADMs is given by:
\(N_{\text{ADM}}=\frac{\left(L+d\right)!}{L!d!}=\prod_{i=1}^{d}\frac{L+i}{i}\).
(19)
This number is returned by the “compute_nn_tot” function of our
module. The factorial growth of the number of ADMs w.r.t. any of the M,
K or L parameters is the main reason for the method’s complexity.
Considering that the convergence w.r.t. to both K and L numbers should
be achieved, the number of ADMs can be quite large. Here, we considered
a number of calculations with the systematically-varying “KK” and
“LL” parameters for a 2-level system, Eq. 17. The calculations are
executed on Intel(R) Xeon(R) CPU E5-2620 v3 @ 2.40GHz processors using
only one thread. We separately measure the time to initialize the
calculations (Figure 11a) and integrate HEOM (Figure 11b).