# contract a gate with a MPO

**URL:** <https://itensor.discourse.group/t/contract-a-gate-with-a-mpo/73>\
**Category:** ITensor Julia Questions\
**Tags:** mpo\
**Created:** [April 23, 2022, 5:43pm UTC](https://itensor.discourse.group/t/contract-a-gate-with-a-mpo/73 "2022-04-23T17:43:38Z")\
**Posts on this page:** 7\
**Page:** 1

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**Author:** ![jisutich](https://avatars.discourse-cdn.com/v4/letter/j/e79b87/32.png) [@jisutich](https://itensor.discourse.group/u/jisutich)\
**Post date:** [April 23, 2022, 5:43pm UTC](https://itensor.discourse.group/t/contract-a-gate-with-a-mpo/73/1 "2022-04-23T17:43:38Z")

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Hi,

I have a question about contracting gates with MPO. Suppose I have a MPO `A` defined on sites `s`. Then for each site `i`, `A[i]` has two site indexes. How to create a gate and contract it with a specific gate?

I tried

```auto
n = 4
s = siteinds("Qubit",n)
A = randomMPO(s)
g = op("X",s[1])

```

Now I want to act this operator on `A`. I find that `g*A` doesn’t work but `apply(g,A)` seems to work. But I am not sure which site index of A[1] is contracted with operator `g`.

Thanks

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**Author:** ![miles](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/miles/32/6_2.png) [@miles](https://itensor.discourse.group/u/miles)\
**Post date:** [April 23, 2022, 10:45pm UTC](https://itensor.discourse.group/t/contract-a-gate-with-a-mpo/73/2 "2022-04-23T22:45:25Z")

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`apply` is a good way to apply the ITensor `g` to the MPO `A`, and it will apply `g` to the first MPO tensor `A[1]` and update this tensor for you. It is able to figure out that this is the right tensor since it has a matching index `s[1]`.

To clarify your question: is it whether the ITensor `g` will be contracted with the `s[1]'` Index of A[1] or with the `s[1]` Index of A[1]? I can see why this isn’t obvious.

The answer is that it will be contracted with the primed index `s[1]'` of A[1] and then that tensor of the MPO will be updated. You can think of what `apply` is doing as being like a matrix product where each operator (`g` and `A`) are mappings from the unprimed index space to the primed index space. Then `apply` composes these two operators, conceptually multiplying them like matrices.

Here is a diagram of what happens when apply is called for this case:

 ![IMG_6825FE0C0812-1](https://global.discourse-cdn.com/free1/uploads/itensor/original/1X/55d129f60c74df5b50e4889c634c0d068b7bf0fd.jpeg)

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<div class="post-metadata">

**Author:** ![jisutich](https://avatars.discourse-cdn.com/v4/letter/j/e79b87/32.png) [@jisutich](https://itensor.discourse.group/u/jisutich)\
**Post date:** [April 23, 2022, 11:14pm UTC](https://itensor.discourse.group/t/contract-a-gate-with-a-mpo/73/3 "2022-04-23T23:14:33Z")

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Hi Miles, thanks a lot for the answer. I have a further question, how to contract operator `g` with the index `s[i]` rather than `s[i]'`?. Basically what I am thinking is acting operator `g`on the two sides of `A` like g A g^{\dagger}.

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<div class="post-metadata">

**Author:** ![jisutich](https://avatars.discourse-cdn.com/v4/letter/j/e79b87/32.png) [@jisutich](https://itensor.discourse.group/u/jisutich)\
**Post date:** [April 24, 2022, 12:54am UTC](https://itensor.discourse.group/t/contract-a-gate-with-a-mpo/73/4 "2022-04-24T00:54:03Z")

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Oh I guess the answer is simply apply(A,dag(g)) if I want to get Ag^{\dagger}.

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<div class="post-metadata">

**Author:** ![miles](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/miles/32/6_2.png) [@miles](https://itensor.discourse.group/u/miles)\
**Post date:** [April 24, 2022, 1:54am UTC](https://itensor.discourse.group/t/contract-a-gate-with-a-mpo/73/5 "2022-04-24T01:54:20Z")

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It’s a good guess, but actually `apply(::MPO,::ITensor)` isn’t defined so that won’t work.

We actually have a keyword argument to `apply` that will do exactly what you are looking for, which is to compute gAg^\dagger. This argument is called `apply_dag` and you can use it like this:

```julia
B = apply(g,A; apply_dag=true)

```

The result will be B = gAg^\dagger.

One reason that computing Ag^\dagger isn’t as simple as `apply(A,dag(g))` is that the `dag(g)` function, perhaps confusingly, doesn’t compute g^\dagger in the operator sense. The `dag` function does conjugate all of the elements of an ITensor and reverse all of the Index arrows, but it has no effect on the prime levels of the indices. So to compute the Hermitian conjugate of a single-site operator ITensor, you need to do `swapprime(dag(g),0,1)`. Perhaps that’s something we should improve in the future, like adding a `hermitian_conj(::ITensor)` function or similar.

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<div class="post-metadata">

**Author:** ![jisutich](https://avatars.discourse-cdn.com/v4/letter/j/e79b87/32.png) [@jisutich](https://itensor.discourse.group/u/jisutich)\
**Post date:** [April 24, 2022, 2:04am UTC](https://itensor.discourse.group/t/contract-a-gate-with-a-mpo/73/6 "2022-04-24T02:04:21Z")

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Thanks! This is exactly what I need. All these work for double sites operators right?

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<div class="post-metadata">

**Author:** ![miles](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/miles/32/6_2.png) [@miles](https://itensor.discourse.group/u/miles)\
**Post date:** [April 24, 2022, 4:42am UTC](https://itensor.discourse.group/t/contract-a-gate-with-a-mpo/73/7 "2022-04-24T04:42:43Z")

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Yes they should work for two-site or multi-site operators or gates (ITensors) as well.