# Time Evolving an MPS with an MPO

<p style="color:red;">
Note: this method of time evolution is no longer recommended
as a first option, but should be used only to check results. A much better
choice with all of the advantages of the method below is to use the 
<a href="https://github.com/ITensor/TDVP">time-dependent variational principle (TDVP)</a> 
method which you can download at <a href="https://github.com/ITensor/TDVP">this link</a>,
including a sample (imaginary time evolution) code.
</p>

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: <a href="http://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.165112" target="_blank">Phys. Rev. B 91, 165112</a>. (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|classes/mps_mpo_algs]].

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();
        }