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).