The 2-RDM of a molecule obeys a set of stringent constraints that are necessary for it to represent an N-electron wave function−constraints known as N-representability conditions.  A 2-RDM that is computed by an approximate method or measured in the presence of noise as on a quantum computer can violate some of these conditions.  

The command Purify2RDM purifies the 2-RDM to obey an important set of N-representability conditions.  Purify2RDM uses an advanced type of optimization, known as semidefinite programming, to project the 2-RDM onto the set of approximately N-representable 2-RDMs.  

The command is especially useful in molecular simulations on quantum computers where noise generates errors in the measured 2-RDMs.  The correction of errors in quantities measured on a quantum computer is known as error mitigation.  

The optional conditions keyword controls the N-representability conditions employed in the calculation.  It can be set to the following strings: "D", "DQ", "DQG" (default), and "DQGT".  

conv_tol = float -- converge threshold. Default is 1*${10}^{-4}\.$

conditions = string -- "D", "DQ", "DQG," or "DQGT". Default is "DQG".

electron_number = posint -- the number of electrons represented by the 2-RDM. 

max_cycle_sdp = posint -- max number of SDP iterations. Default is 50000.

mo_symmetries = Vector -- string labels of the irreducible representations of the molecular orbitals.

spin = nonnegint -- twice the total spin S (= 2S). Default is 0.

symmetry = string/boolean -- is the Schoenflies symbol of the abelian point-group symmetry which can be one of the following:  D2h, C2h, C2v, D2, Cs, Ci, C2, C1. true (default) finds the appropriate symmetry while false does not use symmetry.

verbose = posint -- positive integer between 1 and 5 that controls printing. Default is 1.

J. J. Foley IV and D. A. Mazziotti, Phys. Rev. A 86, 012512 (2012). "Measurement-driven reconstruction of many-particle quantum processes by semidefinite programming with application to photosynthetic light harvesting"

D. A. Mazziotti, Phys. Rev. E 65, 026704 (2002). "Purification of correlated reduced density matrices"

S. E. Smart, J.-N. Boyn, and D. A. Mazziotti, "Resolution of the relative energies of the benzyne isomers on a quantum computer using a contracted Schrödinger equation" (in preparation 2021).

$\mathrm{with}\left(\mathrm{QuantumChemistry}\right)\:$

$\mathrm{hf} ≔ \mathrm{MolecularGeometry}\left(''hydrogen\; fluoride''\right)\;$

$\mathrm{hf}≔\left[\left[''F''\,0\,0\,0\right]\,\left[''H''\,0.43580000\,\mathrm{-0.14770000}\,\mathrm{-0.81880000}\right]\right]$

$\mathrm{data\_hf} ≔ \mathrm{MP2}\left(\mathrm{hf}\, \mathrm{return\_rdm}\=''rdm1\_and\_rdm2''\right)\:$

$$\begin{gathered} \mathrm{D2} ≔ \mathrm{Matrix}\left(36\,36\, \mathrm{shape}\=\mathrm{symmetric}\right)\: \\ \mathrm{D2}\left(..\right) ≔ \mathrm{ArrayTools}:-\mathrm{Permute}\left(\mathrm{data\_hf}\left[\mathrm{rdm2}\right]\, \left[1\,3\,2\,4\right]\right)\; \end{gathered}$$

$\mathrm{eigenvals} ≔ \mathrm{LinearAlgebra}:-\mathrm{Eigenvalues}\left(\mathrm{D2}\right)\;$

$\mathrm{data\_hf\_purified} ≔ \mathrm{Purify2RDM}\left(\mathrm{data\_hf}\left[\mathrm{rdm2}\right]\right)\;$

$$\begin{gathered} \mathrm{D2p} ≔ \mathrm{Matrix}\left(36\,36\, \mathrm{shape}\=\mathrm{symmetric}\right)\: \\ \mathrm{D2p}\left(..\right) ≔ \mathrm{ArrayTools}:-\mathrm{Permute}\left(\mathrm{data\_hf\_purified}\left[\mathrm{rdm2}\right]\, \left[1\,3\,2\,4\right]\right)\; \end{gathered}$$

$\mathrm{eigenvals} ≔ \mathrm{LinearAlgebra}:-\mathrm{Eigenvalues}\left(\mathrm{D2p}\right)\;$