How to create a particular initial state as a MPS?

Question

Given an initial state written, in the full Hilber Space, as

$$
|\psi\rangle = \otimes_{j=1}^{L/2}|\leftarrow\rangle \otimes_{j=L/2+1}^L |\rightarrow\rangle,
$$

such that,

$$
\hat{\sigma}_x |\leftarrow\rangle = +|\leftarrow\rangle\
$$

$$
\hat{\sigma}_x |\rightarrow\rangle = -|\rightarrow\rangle.
$$

How can I write it in the form of an MPS?

Answer 1

One way is to set all qubits in this form (left half of the chain spin up, right half with spin down):

SpinHalf sites = SpinHalf(N);

auto init = InitState(sites);
for(auto n : range1(N)) {
    if (n <= N/2) { 
        init.set(n, "Up"); 
    } else { 
        init.set(n, "Dn");
    }
}
auto psi = MPS(init);

and then applying the Hadamard gate in all the qubits:

for(int n = 1; n <= N; ++n)
{
    auto ind = sites(n);
    auto indP = prime(sites(n));
    auto Had = ITensor(ind,indP);
    Had.set(ind(1),indP(1), ISqrt2);
    Had.set(ind(1),indP(2), ISqrt2);
    Had.set(ind(2),indP(1), ISqrt2);
    Had.set(ind(2),indP(2), -ISqrt2);
    psi.setA(n,psi.A(n)*Had);
}
psi.mapprime(1,0,Site);

You should get the desired product state.

Comments

Thank you very much.
I implemented it, but something strange happened: iIn order to control if the states is the desired one, I calculate @@\langle \hat{sigma}\_x (j) \rangle@@ with @@j=1,2,...,L@@. I obtain the right values from @@[1,L-1]@@, but not for @@L@@, for which @@\langle \sigma\_x(L) \rangle = 0@@.

Does it happen also to you? If you could check it would be great.

---

I just tested and it work as desired, including the last site. Here is my code:

    for(int j = 1; j <= N; ++j)
    {
    psi.position(j);
    Real Sxj = (psi.A(j)
               * sites.op("Sx",j)
               * dag(prime(psi.A(j),Site))).real();
    println("Sx_",j," = ",Sxj);
    }

and the result for 10 sites:

Sx_1 = 0.5
Sx_2 = 0.5
Sx_3 = 0.5
Sx_4 = 0.5
Sx_5 = 0.5
Sx_6 = -0.5
Sx_7 = -0.5
Sx_8 = -0.5
Sx_9 = -0.5
Sx_10 = -0.5

---

Okok, it works fine.  Thank you very much!

---

how can we do the same for julia version?

Answer 2

Diego's answer is a good one.

Another way to do what you want, setting the tensor elements directly, is to use similar code to what olabaz provides in his question here:
http://itensor.org/support/1357/setting-random-initial-product-state