# Combined re-orthogonalization of an MPS and repositioning of a ProjMPO

**URL:** <https://itensor.discourse.group/t/combined-re-orthogonalization-of-an-mps-and-repositioning-of-a-projmpo/704>\
**Category:** ITensor Julia Questions\
**Tags:** julia, mps, mpo\
**Created:** [March 16, 2023, 3:39pm UTC](https://itensor.discourse.group/t/combined-re-orthogonalization-of-an-mps-and-repositioning-of-a-projmpo/704 "2023-03-16T15:39:15Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![phx](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/phx/32/673_2.png) [@phx](https://itensor.discourse.group/u/phx)\
**Post date:** [March 16, 2023, 3:39pm UTC](https://itensor.discourse.group/t/combined-re-orthogonalization-of-an-mps-and-repositioning-of-a-projmpo/704/1 "2023-03-16T15:39:15Z")

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Hi all!

I have an MPS called `state` and an MPO called `H`; I’m trying to calculate the product of the projection, using `ProjMPO`, of `H` with some tensors obtained from `state`.  
This is partly like a typical TDVP1 sweep, and in fact I based my code on that algorithm.  
I need to compute the following.

1. The product of `H`’s one-site projection on the first site with the first tensor of the MPS (with its orthogonality center on the first site):

```julia
PH = ProjMPO(H)
PH.nsite = 1
orthogonalize!(state, 1)
ITensors.position!(PH, state, 1)
H1 = PH(state[1])

```

1. then, I need to decompose the first site of the MPS in a left-orthogonal tensor `A_1` and a “residue” `C`, and multiply this by the zero-site projection of `H`:

```julia
_, S, V = svd(state[1], uniqueinds(state[1], state[2]))
C = S * V
PH.nsite = 0
ITensors.position!(PH, state, 2)
K = PH(C)

```

1. The same as point 1, but on the second bond, i.e. the product of `H`’s one-site projection on the _second_ site with the _second_ tensor of `state` with the orthogonality center moved on the _second_ bond as well:

```julia
PH.nsite = 1
orthogonalize!(state, 2)
ITensors.position!(PH, state, 2)
H2 = PH(state[2]) # <---- error

```

When I run this code (in sequence) I get the following error, referring to the line I pointed out:

```plaintext
ERROR: LoadError: The order of the ProjMPO-ITensor product P*v is not equal to the order of the ITensor v, this is probably due to an index mismatch.
Common reasons for this error: 
(1) You are trying to multiply the ProjMPO with the 1-site wave-function at the wrong position.
(2) `orthogonalize!` was called, changing the MPS without updating the ProjMPO.

P*v inds: ((dim=1|id=245|"Link,l=1"), (dim=1|id=875|"Link,l=1"), (dim=1|id=875|"Link,l=1")', (dim=4|id=788|"Site,n=2,vS=1/2")', (dim=1|id=898|"Link,l=2")') 

v inds: ((dim=4|id=788|"Site,n=2,vS=1/2"), (dim=1|id=898|"Link,l=2"), (dim=1|id=245|"Link,l=1"))

```

I can see that there are two spurious indices on `P*v`, `id=875` and its primed version, that weren’t present in `v`, but I don’t get where they come from.  
I did call `orthogonalize!` on the state, but also I updated the ProjMPO as well. Or so I think. Does my code not do what I think it does?

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**Author:** ![phx](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/phx/32/673_2.png) [@phx](https://itensor.discourse.group/u/phx)\
**Post date:** [March 16, 2023, 5:56pm UTC](https://itensor.discourse.group/t/combined-re-orthogonalization-of-an-mps-and-repositioning-of-a-projmpo/704/2 "2023-03-16T17:56:46Z")

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By the way, here’s a minimal (not) working example:

```julia
using ITensors

let
    s = siteinds("S=1/2", 6)
    z = randomMPS(s; linkdims=4)
    h = randomMPO(s)

    ph = ProjMPO(h)
    
    ph.nsite = 1
    orthogonalize!(z, 1)
    position!(ph, z, 1)
    ph(z[1])

    ph.nsite = 0
    _, S, V = svd(z[1], uniqueinds(z[1], z[2]))
    position!(ph, z, 2)
    ph(S * V)

    ph.nsite = 1
    orthogonalize!(z, 2)
    position!(ph, z, 2)
    ph(z[2])
end

```

I find that if I remove the `position!(ph, z, 2)` then it doesn’t give any error, but I don’t understand why. I thought that I needed to re-project the ProjMPO object on the newly-orthogonalized state.

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**Author:** ![phx](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/phx/32/673_2.png) [@phx](https://itensor.discourse.group/u/phx)\
**Post date:** [April 11, 2023, 4:07pm UTC](https://itensor.discourse.group/t/combined-re-orthogonalization-of-an-mps-and-repositioning-of-a-projmpo/704/3 "2023-04-11T16:07:37Z")

</div>

After some trial and error I managed to figure out the culprit. I’ll post my solution.

Clearly the SVD factorization doesn’t affect the MPS `z` so it can be left out. Actually, if I remove the whole block containing the factorization I encounter no issues, i.e. the following code works:

```julia
using ITensors

let
    s = siteinds("S=1/2", 6)
    z = randomMPS(s; linkdims=4)
    h = randomMPO(s)

    ph = ProjMPO(h)
    
    ph.nsite = 1
    orthogonalize!(z, 1)
    position!(ph, z, 1)
    ph(z[1])

    orthogonalize!(z, 2)
    position!(ph, z, 2)
    ph(z[2])
end

```

The following, however, does not:

```julia
using ITensors

let
    s = siteinds("S=1/2", 6)
    z = randomMPS(s; linkdims=4)
    h = randomMPO(s)

    ph = ProjMPO(h)
    
    ph.nsite = 1
    orthogonalize!(z, 1)
    position!(ph, z, 1)
    ph(z[1])

    position!(ph, z, 2) # <-------- !!

    orthogonalize!(z, 2)
    position!(ph, z, 2)
    ph(z[2])
end

```

By looking at the code for `position!` I see that there are some checks on the site parameter, such that if it hasn’t changed since the last time the function was called (and `ph.nsite` is not changed either) then the function does nothing. This means that the second `position!(ph, z, 2)` call doesn’t change `ph` at all, even if the MPS `z` has been changed in the meantime.  
As a result, when `ph(z[2])` is called, `z` has a new Index object (i.e. with a different id) on the (1,2) link due to the re-orthogonalization process, while `ph` is still using the old set of indices from `z`, causing an index mismatch.

My previous example works if I switch the last two blocks—then `position!` uses the new information on `ph.nsite` and the projections are correctly updated after the re-orthogonalization of the MPS.