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Performing a Wigner transformation on this HEOM, the quantum Fokker-Planck equation with low temperature correction terms emerges.<ref name="Tanimura152">{{Citation|last=Tanimura|first=Yoshitaka|title=Real-time and imaginary-time quantum hierarchical Fokker-Planck equations|journal=J. Chem. Phys.|volume=141|issue=14|pages=044114|year=2015|arxiv=1502.04077|doi=10.1063/1.4916647|authorlink=Yoshitaka Tanimura}}</ref><ref>{{Cite journal|last=Tanimura|first=Yoshitaka|last2=Wolynes|first2=Peter G.|date=1991-04-01|title=Quantum and classical Fokker-Planck equations for a Gaussian-Markovian noise bath|url=https://link.aps.org/doi/10.1103/PhysRevA.43.4131|journal=Physical Review A|language=en|volume=43|issue=8|pages=4131–4142|doi=10.1103/PhysRevA.43.4131|issn=1050-2947}}</ref><br />
===Algorithm and Computational Cost===
When the [[open quantum system]] is represented by <math>M</math> levels and <math>M</math> baths with each bath response function represented by <math>K</math> exponentials, a hierarchy with <math>\mathcal{N}</math> layers will contain:
<math>
\frac{\left(MK + \mathcal{N}\right)!}{\left(MK\right)!\mathcal{N}!}
</math>
matrices, each with <math>M^2</math> complex-valued (containing both real and imaginary parts) elements. Therefore, the limiting factor in HEOM calculations is the amount of [[RAM]] required, since if one copy of each matrix is stored, the total RAM required would be:
<math>
16M^2\frac{\left(MK + \mathcal{N}\right)!}{\left(MK\right)!\mathcal{N}!}
</math>
[[bytes]] if using double-precision, and half of that if using single-precision (however, numerically exact open quantum system dynamics has been shown to require double-precision in Fig. 2 of <ref name=Dattani4>{{Citation | last = Dattani | first = Nike | year = 2013 | authorlink = Nike Dattani |title=FeynDyn: A MATLAB program for fast numerical Feynman integral calculations for open quantum system dynamics on GPUs | journal = Computer Physics Communications | volume = 184 | issue = 12 | pages = 2828–2833 | bibcode = 2013CoPhC.184.2828D | doi = 10.1016/j.cpc.2013.07.001 | arxiv = 1205.6872 }}</ref>).
Further RAM may be necessary depending on how the system of differential equations is numerically solved. The algorithm which requires the least amount of memory to store the auxiliary density operators (ADOs), yet is still stable numerically, was presented in 2015 by Wilkins and Dattani,<ref name=Wilkins>{{Citation | last = Wilkins | first = David |author2= Dattani, Nike| year = 2015 | authorlink = David M. Wilkins | author2link = Nike Dattani |title=Why Quantum Coherence Is Not Important in the Fenna–Matthews–Olsen Complex | journal = Journal of Chemical Theory and Computation | volume = 11 | issue = 7 | pages = 3411–3419| doi = 10.1021/ct501066k | pmid = 26575775 | arxiv = 1411.3654 }}</ref> and has been implemented for GPUs by Tsuchimoto and Tanimura.<ref name=Tsuchimoto>{{Citation | last = Tsuchimoto | first = Masashi |author2= Tanimura, Yoshitaka| year = 2015 | authorlink = Masashi Tsuchimoto | author2link = Yoshitaka Tanimura |title=Spins Dynamics in a Dissipative Environment: Hierarchal Equations of Motion Approach Using a Graphics Processing Unit (GPU) | journal = Journal of Chemical Theory and Computation | volume = 11 | issue = 7 | pages = 3859–3865| doi = 10.1021/acs.jctc.5b00488 | pmid = 26574467 }}</ref> An implementation for CPUs is given in the original paper.<ref name="Wilkins" /> An open source parallel CPU implementation is available from the [[Klaus Schulten|Schulten]] group. <ref>url=https://www.ks.uiuc.edu/Research/phi/</ref>
===Applications===
The largest number of levels coupled to the bath, that have been successfully treated with the HEOM using '''''one''''' bath, with the bath response function represented by '''''one''''' exponential is 4096 (equivalent to 12 [[qubits]]) in an application to the [[Ising model]].<ref name="Tsuchimoto" /> The largest number of levels treated using '''''two''''' exponentials and <math>M</math> baths is 24 in an application to the [[Fenna-Matthews-Olson complex]].<ref name="Wilkins" />
==See also==
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