Contracting MPS with different lengths

Question

Dear community,

The problem I have in hand is: I have three MPS, |\psi1>, |\psi2>, |\psi3>, with length 2*N, N, N, respectively. I want to contract the first N index of |\psi1> with |\psi2>, and the second N index with |\psi3>.

I had a solution for the small system, which requires me to change the data type from MPS to ITensor, since I cannot multiply MPS using innerC with different size. Here is my code:

// Change the data type, so that we can perform contraction
 ITensor mpsToTensor(MPS & psi0){

    ITensor psi = psi0.A(1);
    int N = length(psi0);

    for (int j=2; j <=N; ++j){
        psi *= psi0.A(j);
    }

    return psi;

 }

This works for small system. But when I tried N=20, it throws error terminate called after throwing an instance of 'std::bad_alloc' what(): std::bad_alloc.
Do you have a smarter way to perform this task that won't cause this error? I am thinking about making a big MPS by combining |\psi2> and |\psi3>, but haven't found a way to do this.

Many thanks,
Yuhan

Answer 1

Converting an MPS to an ITensor will require an exponential amount of memory in the number of sites, so it is best to avoid that.

I would recommend combining the two MPS @@\psi2@@ and @@\psi3@@ into a single MPS (say @@\phi@@) and then taking the inner product of @@\phi@@ with @@\psi1@@.

Comments

Thanks Matt! Could you also explain a little bit how to combine two MPS into a single MPS?

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You should be able to initialize a new MPS @@\phi@@ of length @@2N@@ and fill the elements in a loop with the tensors of the MPSs @@\psi2@@ and @@\psi3@@. For example something like this:

    auto phi = MPS(2N);
    for(int j=1; j <=2N; ++j)
        {
        if(j <= N) phi.set(j, psi2(j));
        else phi.set(j, psi3(j - N));
        }

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Thank you very much! It is really helpful. I am also a bit curious why this conversion takes exponential amount of memory in sites. From my naive understanding, in the case where I have bond dimension being 64 and spin 1/2 on the site, each MPS stores 64*64*2 complex numbers. In the case where it causes error, the sites N = 20.  I guess 64*64*2*20 shouldn't be a large number for a machine? I assume the required memory scales linearly with N and don't quite understand the exponential scaling.

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When you contract the MPS tensors with each other, you sum over the internal dimensions (the bond dimensions) and the site dimension remain. So for N = 20 and spin-1/2 particles, the full many-body wavefunction has a dimension 2^20.