Huge SDP problems arising from quantum chemistry

I uploaded some SDP problems arising from quantum chemistry
http://accc.riken.jp/maho/rdmsdp/sdp_rdm.html

mathematical challenges from theoretical / computational chemistry

In mathematical challenges from theoretical / computational chemistry by Committee on Mathematical Challenges from Computational Chemistry, "The N-and V-Representability Problems" is listed.
http://www.nap.edu/openbook.php?record_id=4886&page=48

> Unfortunately, only an incomplete set of necessary conditions are known, but these are already so complex that further work in this area has been abandoned by chemists.

For N-representability:
I think this is solved with negative result: computational complexity is QMA-complete.

For V-representability:
If I remembered correctly Ayers et al. have already published proof.

Excited state 2-RDM

The ground state of 2-RDM should lie on the boundary of N-representable cone. So what about the excited state 2-RDM? Yasuda-sensei showed that there are many wavefunctions that reduces to the same 2-RDM which correspond to an excited state wavefunction in 2002.

Why 1-RDM doesn't matter?

Usually molecules have even number of electrons, the ground state is not degenrated, and occupy same orbital but different spin.
In this case, Darwin Smith's theorem applies, thus problem can be very simplified.
http://link.aps.org/doi/10.1103/PhysRev.147.896
This is very important theorem that we do not obtain 1-RDM from fake wave function.

However, in doublet state, we don't have such symmetry, thus we have degenerated ground state. The total energy can be calculated very accurately, but fails to calculate the dipole moment which should be non-zero but zero.

Quantum Marginals and Density Matrices Workshop in Toronto 2009

After a long thought, I decided to go to Canada to attend this conference.
I have a presentation in 7/25, so I'll be Canada on 7/27 16:00. I'll miss some important
presentations. If possible I'll have a talk.

Quantum Marginals and Density Matrices Workshop in Toronto 2009

I'm really interested in following talk by Stefano Pironio. Unforutunately
I cannot attend this conference... though ... what a pity... Japan is located
too far from Canada. I'll go to US in this August. Toooooo hard schedule.
There will be a session by Erdahl....
(At CSC2009, Garnet and Ayers kindly told me about this conference, thanks!)

http://atlas-conferences.com/cgi-bin/abstract/cazc-03

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Non-commutative polynomial optimization and the varianional RDM method by Stefano Pironio

http://atlas-conferences.com/cgi-bin/abstract/cazc-03

A standard problem in optimization theory is to find the minimum of a polynomial function subject to polynomial inequality constraints. We introduce a generalization of this problem where the optimization variables are not real numbers, but non-commutative variables, i.e., operators acting on Hilbert spaces of arbitrary dimension. We show how semidefinite programming (SDP) can be used to solve this problem. Specifically, we introduce a sequence of SDP relaxations of the original problem, whose optima converge monotically to the global optimum.

Our method can find applications to compute the ground state energy of quantum many-body systems. In particular, it gives a new interpretation to and should strengthens the RDM method used in quantum chemistry to compute electronic energies. Our method provides a computation technique for many-body systems that is not based on states (and thus directly linked to entanglement) but that is rather based on the algebraic structure of quantum operators.

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I asked him to send me a preprint!
Maybe I cannot sleep until their preprint is sent to me.

What they are doing? Some magic? Yes I'm sure!

SDP now become very popular indeed!!!

Mpack: multiple precision arithmetic version of BLAS and LAPACK

This is a byproduct of SDPA-GMP. I implemented a part of LAPACK/BLAS but apparently
only for SDPA-GMP. It is not a constructive. So I decided to implement whole part of BLAS
and possible part of LAPACK. It took very long and still ongoing.

It's very portable and my motivation is providing a reference library, and not optimized one.
(of course this is in my TODO list)

SDPA-QD, DD version were done in a day or so.

I was invited to have a presentation at Shinjyuku, in 2009/2. I don't know which society
is interested in this, so your advice is really appreciated.