Simple update for PEPO (3-site version) (#262)

* Add 3-site simple update for PEPO

* fix InfinitePEPO rotation

* Fix physical axis fusing

* Add J1-J2 example test

* Modify _apply_gate(mpo!)

Author: Yue-Zhengyuan

src/algorithms/time_evolution/evoltools.jl

@@ -189,7 +189,7 @@ Apply 2-site `gate` on the reduced matrices `a`, `b`
 ```
 """
 function _apply_gate(
-        a::AbstractTensorMap{T, S}, b::AbstractTensorMap{T, S},
+        a::AbstractTensorMap, b::AbstractTensorMap,
         gate::AbstractTensorMap{T, S, 2, 2}, trscheme::TruncationScheme
     ) where {T <: Number, S <: ElementarySpace}
     V = space(b, 1)

src/algorithms/time_evolution/simpleupdate.jl

@@ -200,7 +200,7 @@ using the Hamiltonian `ham`, which can contain up to next-nearest-neighbor inter
 ## Keyword Arguments
 
 - `bipartite::Bool=false`: If `true`, enforces the bipartite structure of the PEPS. 
-- `gate_bothsides::Bool=true`: (Effective only for PEPO evolution)
+- `gate_bothsides::Bool=true`: (Effective only for PEPO evolution) If true, apply the Trotter gate to both side of the PEPO. Otherwise, the gate is only applied to the physical codomain legs. 
 - `force_3site::Bool=false`: Forces the use of the 3-site simple update algorithm, even if the Hamiltonian contains only nearest-neighbor terms.
 - `check_interval::Int=500`: Specifies the number of evolution steps between printing progress information.
 
@@ -220,7 +220,7 @@ function simpleupdate(
     @assert !(bipartite && use_3site) "3-site simple update is incompatible with bipartite lattice."
     # TODO: check SiteDependentTruncation is compatible with bipartite structure
     if use_3site
-        return _simpleupdate3site(state, ham, alg, env; check_interval)
+        return _simpleupdate3site(state, ham, alg, env; gate_bothsides, check_interval)
     else
         return _simpleupdate2site(state, ham, alg, env; bipartite, gate_bothsides, check_interval)
     end