Trotter Gates with PBC?

Question

Hi All,

Is it possible to time-evolve a MPS with trotter gates including a gate connecting the 1st and the last sites? I think the main difficulty is how to effectively compress MPS each time after applying a gate. Because there's no orthogonality center for MPS with PBC. Is there a way to overcome this difficulty? Thanks very much.

Jin

Answer 1

Hi Jin,
Someone else may answer who knows more about this than I do, but I don't think there is a nice MPS algorithm that does what you want. There is an older paper by Verstraete and Cirac that discusses optimizing periodic MPS; you could look at that for some ideas.

Also you could do the "poor man's periodic" approach where you actually use an open-boundary MPS (the regular kind used in ITensor) and just apply a two-site gate that acts on sites 1 and N, using a sequence of swap gates to swap site N all the way to site 2 and then back every time. This would be pretty slow of course. But it might work ok.

There is a paper by Zaletel and Pollmann about constructing MPOs that are the exponential of certain local Hamiltonians. Their construction might straightforwardly work for periodic Hamiltonians. Then if you can find an algorithm to apply a periodic MPO to a periodic MPS that is another possible route. (But there may not be such a nice algorithm for that either.)

Finally, please think hard about whether you really need periodic boundary conditions. One of the strengths of MPS techniques is their scalability to very long 1d systems. So often it's better to just study really long open systems than to use PBC to attempt to reduce finite-size effects. And for some quantities and systems, PBC actually gives worse finite-size scaling than OBC!

Miles

Comments

One other quick comment: if by time evolve you actually mean *imaginary* time evolve, then methods such as TNR (Evenbly and Vidal) and the earlier TRG can be used in principle to imaginary time evolve some simple product state with a periodic Hamiltonian, with the result being represented as either a tree-tensor-network (TRG) or as a MERA (TNR). I don't think these algorithms would work well at all for real-time evolution though.