## Learn to Use ITensor

# Time Evolving an MPS with an MPO

First set up the MPS you would like to time evolve. An easy way to do this is to make a product state using the InitState helper class:

```
#include "itensor/all.h"
//...
int N = 100;
auto sites = SpinHalf(N);
auto state = InitState(sites);
for(int i = 1; i <= N; ++i)
{
//Neel state
state.set(i,i%2==1 ? "Up" : "Dn");
}
auto psi = MPS(state);
```

Next we can use AutoMPO to specify a Hamiltonian, and use the "toExpH" function to exponentiate this Hamiltonian:

```
auto ampo = AutoMPO(sites);
//Make the Heisenberg Hamiltonian
for(int b = 1; b < N; ++b)
{
ampo += 0.5,"S+",b,"S-",b+1;
ampo += 0.5,"S-",b,"S+",b+1;
ampo += "Sz",b,"Sz",b+1;
}
auto H = toMPO(ampo);
auto tau = 0.1;
auto expH = toExpH(ampo,tau);
```

In the above example, expH will be approximately equal to @@exp(-\tau H)@@ up to terms of order @@\tau^2@@ . For more details on this construction, see the following article: Phys. Rev. B 91, 165112. (Alternatively arxiv:1407.1832.) As mentioned in the article, you can also combine two complex time steps to further reduce the scaling of the errors with @@\tau@@ .

To do real-time evolution instead, change the time step to be imaginary:

```
auto expH = toExpH(ampo,tau*Cplx_i);
```

which gives @@exp(-i\tau H)@@ . Fully complex time steps @@\tau=a+ib@@ are fine too.

To carry out the actual time evolution, repeatedly apply the MPO to the MPS using the applyMPO method.

The recommended practice is to use `"Method=","DensityMatrix"`

to make sure your code is working.
Then consider switching to `"Method=","Fit"`

if you need more efficiency, especially if the MPO you
are applying has a large bond dimension.
But verify that the results you obtain with "Fit" are very similar to the ones you obtain with "DensityMatrix".

Here is some sample time evolution code using `applyMPO`

:

```
auto args = Args("Method=","DensityMatrix","Cutoff=",1E-9,"MaxDim=",3000);
auto ttotal = 3.0;
auto nt = int(ttotal/tau+(1e-9*(ttotal/tau)));
for(int n = 1; n <= nt; ++n)
{
psi = applyMPO(expH,psi,args);
psi.noPrime().normalize();
}
```

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