Applying 4-site gates to PEPS with ITensorNetworks

I’m trying to implement a code performing TEBD for a toric code Hamiltonian in a 2D square lattice using ITensorNetworks. I’m facing an issue related to applying a 4-site gate to a PEPS and retrieving the state as a PEPS again. While in the case of 2 site gates it is clear what to after applying the gate to decompose the resulting tensor back to the PEPS structure, if I decompose the 4-site tensor (resulting from applying the gate) naively, I will end up with a missing bond between two of the sites in the final PEPS, which breaks the entanglement structure of my state.

What is the correct way of applying 4-site gates to a PEPS? If someone could point me to some reference where I can check the algorithm for this, it would be very helpful.

Thank you in advance!

The best description I can find of applying 4-site gates is Section 4.2.2 of [1110.3632] Breakdown of a perturbed Z_N topological phase, though it is lacking in details.

The optimization of the 4 PEPS tensors where the plaquette term is being applied can be carried out by the same optimization procedure that is used in the so-called “full update” PEPS gate application algorithm, described in Section 6.4 of [0907.2796] Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems. Another description of that optimization procedure that includes tensor network diagrams can be found in Section II. B. of [1503.05345] The iPEPS algorithm, improved: fast full update and gauge fixing and starting at Slide 44 of That procedure should generalize in a straightforward way to 4-site gates, but of course at higher cost.

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Of course it would be best if you can break down the 4-site gates into a series of 2-site gates, though maybe there isn’t a straightforward way to do that in your case.