solving a linear problem with alternating_update

DMRG and Numerical Methods
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1

Hi Matt and Miles,
I have been looking into the linsolve(A,b,x_0;…) function in ITensorTDVP that solves the A|x>=|b> problem using alternating_update (which basically replaces the eigsolve_updater in DMRG with a linsolve_updater) with @srwhite. If we understand correctly based on reducedlinearproblem.jl and reducedconstantterm.jl, the algorithm projects both the operator A and the constant term |b> under the basis of |x> that is being optimized. Graphically the reduced problem is defined as:

*--*--*-      -*--*--*--*--*--* <x|
|  |  |  |  |  |  |  |  |  |  |
o--o--o--o--o--o--o--o--o--o--o  A      (A_reduced)
|  |  |  |  |  |  |  |  |  |  |
*--*--*-      -*--*--*--*--*--* |x>

o--o--o--o--o--o--o--o--o--o--o <b|  
|  |  |  |  |  |  |  |  |  |  |         (|b_reduced>)
*--*--*-      -*--*--*--*--*--* |x>

Then A_reduced and |b_reduced> is fed into the KrylovKit.linsolve() together with an initial guess |x_0_reduced>. It seems to us this algorithm requires that |b> and the final |x> share the same MPS basis to some degree. For example, if A changes the quantum number by a definite amount, say being a C^+, then either the basis of |x> needs to be expanded to include a different quantum number sector (which is not necessary since |x> also has a definite quantum number), or A_reduced will be zero.
It seems to us more natural to set up the problem in the framework of reducedcontractproblem.jl, such that the reduced problem is:

o--o--o-      -o--o--o--o--o--o <b|
|  |  |  |  |  |  |  |  |  |  |
o--o--o--o--o--o--o--o--o--o--o  A     (A_reduced)
|  |  |  |  |  |  |  |  |  |  |
*--*--*-      -*--*--*--*--*--* |x>

-o--o-   (|b_reduced> that is taken out of the sandwich above)
 |  |

In this way, one seems to avoid the above problem.

Best,
Shengtao

2

I believe I tried that and couldn’t get it to work but maybe there was something I was missing.

3

Hi Matt,
Thanks for the quick response and I will experiment with it myself and will let you know if I find anything.
Could you share a little more about why it couldn’t work? Was it an ITensor issue or that the CG/GMRES in KrylovKit.linsolve() doesn’t converge?

Best,
Shengtao

4

I found that it didn’t converge to the correct state |x>.

5

I think if the bases |x> and |b> were similar, it was ok, but if they were different enough from each other it failed very badly. One limit to think about is if the basis of |b> is very small, say a product state. In that limit, my intuition was that <b|A|x> was projecting out too much of A|x>, but that was just a rough guess about why it was failing.

1 Like
6

If you have success with using the |b> basis (or some other variation) please let us know!

7

Of course! Steve has suggested to improve the current version by using density-matrix mixing to incorporate some |b> into the basis of |x>, maybe in a way like running DMRG with noise on for some initial sweeps. I will implement that and let you know how that works.

8

I think I tried something like that but didn’t push that idea very far, I’m curious to hear if you get something like that working.