As been proposed by @Adwait-Naravane in #101, it may be interesting to consider two-site update in Loop-TNR. I am still thinking whether it can be achieved in some way.
To my understanding, it may have the following two advantanges:
- In the final coarse-graining step, to make the coarse grained tensor to be low-rank, it is sufficient to require only diagonal legs to be low rank. The horizontal and vertical legs can be arbitrarily high rank. If we consider two-site updates, this additional rank can be used to further lower the cost
- Usually in DMRG, two-site update can help get rid of local minimum, especially when there is quantum number locking.
The two-site update Loop-TNR, based on my previous understanding, has the following main problem, that the $\langle \psi_B|\psi_A\rangle$ part of the cost function involves the following contraction:
---A'---B'---A'---B'---
-C---D---C----D----C-
whose constraction cost may rises to $O(\chi^9)$ or $O(\chi^7)$ if we contract the transfer matrix tensor-by-tensor. This can be seen from the middle term of equation (B5) of TNR+.
Maybe we can try the following strategy, by performing a two-step approximation. Firstly, we use a MPS $|\psi_{intermediate}\rangle$ with $k\chi$ to approximate $|\psi_A\rangle$, using the usual LoopTNR. This $k$ can be larger than 1. Then we use a two-site MPS $|\psi_B\rangle$ to approximate $|\psi_{intermediate}\rangle$. Then the overall contraction cost becomes $O(k^3\chi^6)$.
As been proposed by @Adwait-Naravane in #101, it may be interesting to consider two-site update in Loop-TNR. I am still thinking whether it can be achieved in some way.
To my understanding, it may have the following two advantanges:
The two-site update Loop-TNR, based on my previous understanding, has the following main problem, that the$\langle \psi_B|\psi_A\rangle$ part of the cost function involves the following contraction:
whose constraction cost may rises to$O(\chi^9)$ or $O(\chi^7)$ if we contract the transfer matrix tensor-by-tensor. This can be seen from the middle term of equation (B5) of TNR+.
Maybe we can try the following strategy, by performing a two-step approximation. Firstly, we use a MPS$|\psi_{intermediate}\rangle$ with $k\chi$ to approximate $|\psi_A\rangle$ , using the usual LoopTNR. This $k$ can be larger than 1. Then we use a two-site MPS $|\psi_B\rangle$ to approximate $|\psi_{intermediate}\rangle$ . Then the overall contraction cost becomes $O(k^3\chi^6)$ .