diff --git a/src/PEPSKit.jl b/src/PEPSKit.jl
index a8476abb3..cd4cfae62 100644
--- a/src/PEPSKit.jl
+++ b/src/PEPSKit.jl
@@ -5,7 +5,11 @@ using Compat
 using Accessors: @set, @reset
 using VectorInterface
 import VectorInterface as VI
-using TensorKit, KrylovKit, OptimKit, TensorOperations
+
+using TensorKit
+using TensorKit: TruncationScheme
+
+using KrylovKit, OptimKit, TensorOperations
 using ChainRulesCore, Zygote
 using LoggingExtras
 
@@ -83,7 +87,8 @@ using .Defaults: set_scheduler!
 export set_scheduler!
 export SVDAdjoint, FullSVDReverseRule, IterSVD
 export CTMRGEnv, SequentialCTMRG, SimultaneousCTMRG
-export FixedSpaceTruncation, HalfInfiniteProjector, FullInfiniteProjector
+export FixedSpaceTruncation, SiteDependentTruncation
+export HalfInfiniteProjector, FullInfiniteProjector
 export LocalOperator, physicalspace
 export expectation_value, cost_function, product_peps, correlation_length, network_value
 export correlator
diff --git a/src/algorithms/time_evolution/simpleupdate.jl b/src/algorithms/time_evolution/simpleupdate.jl
index 01eb070d0..db6e0aa49 100644
--- a/src/algorithms/time_evolution/simpleupdate.jl
+++ b/src/algorithms/time_evolution/simpleupdate.jl
@@ -12,7 +12,7 @@ struct SimpleUpdate
     dt::Number
     tol::Float64
     maxiter::Int
-    trscheme::TensorKit.TruncationScheme
+    trscheme::TruncationScheme
 end
 # TODO: add kwarg constructor and SU Defaults
 
@@ -34,7 +34,7 @@ function _su_xbond!(
     col::Int,
     gate::AbstractTensorMap{T,S,2,2},
     peps::InfiniteWeightPEPS,
-    alg::SimpleUpdate,
+    trscheme::TruncationScheme,
 ) where {T<:Number,S<:ElementarySpace}
     Nr, Nc = size(peps)
     @assert 1 <= row <= Nr && 1 <= col <= Nc
@@ -47,7 +47,7 @@ function _su_xbond!(
     B = _absorb_weights(B, peps.weights, row, cp1, Tuple(1:4), sqrtsB, false)
     # apply gate
     X, a, b, Y = _qr_bond(A, B)
-    a, s, b, ϵ = _apply_gate(a, b, gate, alg.trscheme)
+    a, s, b, ϵ = _apply_gate(a, b, gate, trscheme)
     A, B = _qr_bond_undo(X, a, b, Y)
     # remove environment weights
     _allfalse = ntuple(Returns(false), 3)
@@ -86,6 +86,9 @@ function su_iter(
         # to update them using code for x-weights
         if direction == 2
             peps2 = mirror_antidiag(peps2)
+            trscheme = mirror_antidiag(alg.trscheme)
+        else
+            trscheme = alg.trscheme
         end
         if bipartite
             for r in 1:2
@@ -94,7 +97,7 @@ function su_iter(
                     direction == 1 ? gate : gate_mirrored,
                     (CartesianIndex(r, 1), CartesianIndex(r, 2)),
                 )
-                ϵ = _su_xbond!(r, 1, term, peps2, alg)
+                ϵ = _su_xbond!(r, 1, term, peps2, truncation_scheme(trscheme, 1, r, 1))
                 peps2.vertices[rp1, 2] = deepcopy(peps2.vertices[r, 1])
                 peps2.vertices[rp1, 1] = deepcopy(peps2.vertices[r, 2])
                 peps2.weights[1, rp1, 2] = deepcopy(peps2.weights[1, r, 1])
@@ -106,7 +109,7 @@ function su_iter(
                     direction == 1 ? gate : gate_mirrored,
                     (CartesianIndex(r, c), CartesianIndex(r, c + 1)),
                 )
-                ϵ = _su_xbond!(r, c, term, peps2, alg)
+                ϵ = _su_xbond!(r, c, term, peps2, truncation_scheme(trscheme, 1, r, c))
             end
         end
         if direction == 2
@@ -185,6 +188,7 @@ function simpleupdate(
     nnonly = is_nearest_neighbour(ham)
     use_3site = force_3site || !nnonly
     @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(peps, ham, alg; check_interval)
     else
diff --git a/src/algorithms/time_evolution/simpleupdate3site.jl b/src/algorithms/time_evolution/simpleupdate3site.jl
index 1ce887b5b..1672e2c5e 100644
--- a/src/algorithms/time_evolution/simpleupdate3site.jl
+++ b/src/algorithms/time_evolution/simpleupdate3site.jl
@@ -215,7 +215,7 @@ The arrows between `Pa`, `s`, `Pb` are
 function _proj_from_RL(
     r::AbstractTensorMap{T,S,1,1},
     l::AbstractTensorMap{T,S,1,1};
-    trunc::TensorKit.TruncationScheme=notrunc(),
+    trunc::TruncationScheme=notrunc(),
     rev::Bool=false,
 ) where {T<:Number,S<:ElementarySpace}
     rl = r * l
@@ -235,16 +235,19 @@ end
 Given a cluster `Ms` and the pre-calculated `R`, `L` bond matrices,
 find all projectors `Pa`, `Pb` and Schmidt weights `wts` on internal bonds.
 """
-function _get_allprojs(Ms, Rs, Ls, trunc::TensorKit.TruncationScheme, revs::Vector{Bool})
+function _get_allprojs(
+    Ms, Rs, Ls, trschemes::Vector{E}, revs::Vector{Bool}
+) where {E<:TruncationScheme}
     N = length(Ms)
+    @assert length(trschemes) == N - 1
     projs_errs = map(1:(N - 1)) do i
-        trunc2 = if isa(trunc, FixedSpaceTruncation)
+        trunc = if isa(trschemes[i], FixedSpaceTruncation)
             V = space(Ms[i + 1], 1)
             truncspace(isdual(V) ? V' : V)
         else
-            trunc
+            trschemes[i]
         end
-        return _proj_from_RL(Rs[i], Ls[i]; trunc=trunc2, rev=revs[i])
+        return _proj_from_RL(Rs[i], Ls[i]; trunc, rev=revs[i])
     end
     Pas = map(Base.Fix2(getindex, 1), projs_errs)
     wts = map(Base.Fix2(getindex, 2), projs_errs)
@@ -258,10 +261,10 @@ end
 Find projectors to truncate internal bonds of the cluster `Ms`
 """
 function _cluster_truncate!(
-    Ms::Vector{T}, trunc::TensorKit.TruncationScheme, revs::Vector{Bool}
-) where {T<:PEPSTensor}
+    Ms::Vector{T}, trschemes::Vector{E}, revs::Vector{Bool}
+) where {T<:PEPSTensor,E<:TruncationScheme}
     Rs, Ls = _get_allRLs(Ms)
-    Pas, Pbs, wts, ϵs = _get_allprojs(Ms, Rs, Ls, trunc, revs)
+    Pas, Pbs, wts, ϵs = _get_allprojs(Ms, Rs, Ls, trschemes, revs)
     # apply projectors
     # M1 -- (Pa1,wt1,Pb1) -- M2 -- (Pa2,wt2,Pb2) -- M3
     for (i, (Pa, Pb)) in enumerate(zip(Pas, Pbs))
@@ -322,13 +325,13 @@ In the cluster, the axes of each PEPSTensor are reordered as
 ```
 """
 function apply_gatempo!(
-    Ms::Vector{T1}, gs::Vector{T2}; trunc::TensorKit.TruncationScheme
-) where {T1<:PEPSTensor,T2<:AbstractTensorMap}
+    Ms::Vector{T1}, gs::Vector{T2}; trschemes::Vector{E}
+) where {T1<:PEPSTensor,T2<:AbstractTensorMap,E<:TruncationScheme}
     @assert length(Ms) == length(gs)
     revs = [isdual(space(M, 1)) for M in Ms[2:end]]
     @assert !all(revs)
     _apply_gatempo!(Ms, gs)
-    wts, ϵs, = _cluster_truncate!(Ms, trunc, revs)
+    wts, ϵs, = _cluster_truncate!(Ms, trschemes, revs)
     return wts, ϵs
 end
 
@@ -373,8 +376,8 @@ function get_3site_se(peps::InfiniteWeightPEPS, row::Int, col::Int)
 end
 
 function _su3site_se!(
-    row::Int, col::Int, gs::Vector{T}, peps::InfiniteWeightPEPS, alg::SimpleUpdate
-) where {T<:AbstractTensorMap}
+    row::Int, col::Int, gs::Vector{T}, peps::InfiniteWeightPEPS, trschemes::Vector{E}
+) where {T<:AbstractTensorMap,E<:TruncationScheme}
     Nr, Nc = size(peps)
     @assert 1 <= row <= Nr && 1 <= col <= Nc
     rm1, cp1 = _prev(row, Nr), _next(col, Nc)
@@ -384,7 +387,7 @@ function _su3site_se!(
     coords = ((row, col), (row, cp1), (rm1, cp1))
     # weights in the cluster
     wt_idxs = ((1, row, col), (2, row, cp1))
-    wts, ϵ = apply_gatempo!(Ms, gs; trunc=alg.trscheme)
+    wts, ϵ = apply_gatempo!(Ms, gs; trschemes)
     for (wt, wt_idx) in zip(wts, wt_idxs)
         peps.weights[CartesianIndex(wt_idx)] = wt / norm(wt, Inf)
     end
@@ -414,13 +417,19 @@ function su3site_iter(
         ),
     )
     peps2 = deepcopy(peps)
+    trscheme = alg.trscheme
     for i in 1:4
         for site in CartesianIndices(peps2.vertices)
             r, c = site[1], site[2]
             gs = gatempos[i][r, c]
-            _su3site_se!(r, c, gs, peps2, alg)
+            trschemes = [
+                truncation_scheme(trscheme, 1, r, c)
+                truncation_scheme(trscheme, 2, r, _next(c, size(peps2)[2]))
+            ]
+            _su3site_se!(r, c, gs, peps2, trschemes)
         end
         peps2 = rotl90(peps2)
+        trscheme = rotl90(trscheme)
     end
     return peps2
 end
diff --git a/src/algorithms/truncation/bond_truncation.jl b/src/algorithms/truncation/bond_truncation.jl
index 40f413074..c50e1ac38 100644
--- a/src/algorithms/truncation/bond_truncation.jl
+++ b/src/algorithms/truncation/bond_truncation.jl
@@ -13,13 +13,13 @@ $(TYPEDFIELDS)
 
 The truncation algorithm can be constructed from the following keyword arguments:
 
-* `trscheme::TensorKit.TruncationScheme`: SVD truncation scheme when initilizing the truncated tensors connected by the bond.
+* `trscheme::TruncationScheme`: SVD truncation scheme when initilizing the truncated tensors connected by the bond.
 * `maxiter::Int=50` : Maximal number of ALS iterations.
 * `tol::Float64=1e-15` : ALS converges when fidelity change between two FET iterations is smaller than `tol`.
 * `check_interval::Int=0` : Set number of iterations to print information. Output is suppressed when `check_interval <= 0`. 
 """
 @kwdef struct ALSTruncation
-    trscheme::TensorKit.TruncationScheme
+    trscheme::TruncationScheme
     maxiter::Int = 50
     tol::Float64 = 1e-15
     check_interval::Int = 0
diff --git a/src/algorithms/truncation/fullenv_truncation.jl b/src/algorithms/truncation/fullenv_truncation.jl
index 7856f499b..f3fa336b3 100644
--- a/src/algorithms/truncation/fullenv_truncation.jl
+++ b/src/algorithms/truncation/fullenv_truncation.jl
@@ -13,7 +13,7 @@ $(TYPEDFIELDS)
 
 The truncation algorithm can be constructed from the following keyword arguments:
 
-* `trscheme::TensorKit.TruncationScheme` : SVD truncation scheme when optimizing the new bond matrix.
+* `trscheme::TruncationScheme` : SVD truncation scheme when optimizing the new bond matrix.
 * `maxiter::Int=50` : Maximal number of FET iterations.
 * `tol::Float64=1e-15` : FET converges when fidelity change between two FET iterations is smaller than `tol`.
 * `trunc_init::Bool=true` : Controls whether the initialization of the new bond matrix is obtained from truncated SVD of the old bond matrix. 
@@ -24,7 +24,7 @@ The truncation algorithm can be constructed from the following keyword arguments
 * [Glen Evenbly, Phys. Rev. B 98, 085155 (2018)](@cite evenbly_gauge_2018). 
 """
 @kwdef struct FullEnvTruncation
-    trscheme::TensorKit.TruncationScheme
+    trscheme::TruncationScheme
     maxiter::Int = 50
     tol::Float64 = 1e-15
     trunc_init::Bool = true
diff --git a/src/algorithms/truncation/truncationschemes.jl b/src/algorithms/truncation/truncationschemes.jl
index 9039d925e..3420c90b3 100644
--- a/src/algorithms/truncation/truncationschemes.jl
+++ b/src/algorithms/truncation/truncationschemes.jl
@@ -5,7 +5,11 @@ CTMRG specific truncation scheme for `tsvd` which keeps the bond space on which
 is performed fixed. Since different environment directions and unit cell entries might
 have different spaces, this truncation style is different from `TruncationSpace`.
 """
-struct FixedSpaceTruncation <: TensorKit.TruncationScheme end
+struct FixedSpaceTruncation <: TruncationScheme end
+
+struct SiteDependentTruncation{T<:TruncationScheme} <: TruncationScheme
+    trschemes::Array{T,3}
+end
 
 const TRUNCATION_SCHEME_SYMBOLS = IdDict{Symbol,Type{<:TruncationScheme}}(
     :fixedspace => FixedSpaceTruncation,
@@ -14,6 +18,7 @@ const TRUNCATION_SCHEME_SYMBOLS = IdDict{Symbol,Type{<:TruncationScheme}}(
     :truncdim => TensorKit.TruncationDimension,
     :truncspace => TensorKit.TruncationSpace,
     :truncbelow => TensorKit.TruncationCutoff,
+    :sitedependent => SiteDependentTruncation,
 )
 
 # Should be TruncationScheme but rename to avoid type piracy
@@ -25,3 +30,65 @@ function _TruncationScheme(; alg=Defaults.trscheme, η=nothing)
 
     return isnothing(η) ? alg_type() : alg_type(η)
 end
+
+function truncation_scheme(
+    trscheme::TruncationScheme, direction::Int, row::Int, col::Int; kwargs...
+)
+    return trscheme
+end
+
+function truncation_scheme(
+    trscheme::SiteDependentTruncation, direction::Int, row::Int, col::Int;
+)
+    return trscheme.trschemes[direction, row, col]
+end
+
+# Mirror a TruncationScheme by its anti-diagonal line.
+# When the number of directions is 2, it swaps the first and second direction, consistent with xbonds and ybonds, respectively.
+# When the number of directions is 4, it swaps the first and second, and third and fourth directions, consistent with the order NORTH, EAST, SOUTH, WEST.
+mirror_antidiag(trscheme::TruncationScheme) = trscheme
+function mirror_antidiag(trscheme::SiteDependentTruncation)
+    directions = size(trscheme.trschemes)[1]
+    if directions == 2
+        trschemes_mirrored = stack(
+            (
+                mirror_antidiag(trscheme.trschemes[EAST, :, :]),
+                mirror_antidiag(trscheme.trschemes[NORTH, :, :]),
+            );
+            dims=1,
+        )
+    elseif directions == 4
+        trschemes_mirrored = stack((
+            mirror_antidiag(trscheme.trschemes[EAST, :, :]),
+            mirror_antidiag(trscheme.trschemes[NORTH, :, :]),
+            mirror_antidiag(trscheme.trschemes[WEST, :, :]),
+            mirror_antidiag(trscheme.trschemes[SOUTH, :, :]),
+        ))
+    else
+        error("Unsupported number of directions for mirror_antidiag: $directions")
+    end
+    return SiteDependentTruncation(trschemes_mirrored)
+end
+
+# TODO: type piracy
+Base.rotl90(trscheme::TruncationScheme) = trscheme
+
+function Base.rotl90(trscheme::SiteDependentTruncation)
+    directions, rows, cols = size(trscheme.trschemes)
+    trschemes_rotated = similar(trscheme.trschemes, directions, cols, rows)
+
+    if directions == 2
+        trschemes_rotated[NORTH, :, :] = circshift(
+            rotl90(trscheme.trschemes[EAST, :, :]), (0, -1)
+        )
+        trschemes_rotated[EAST, :, :] = rotl90(trscheme.trschemes[NORTH, :, :])
+    elseif directions == 4
+        for dir in 1:4
+            dir′ = _prev(dir, 4)
+            trschemes_rotated[dir′, :, :] = rotl90(trscheme.trschemes[dir, :, :])
+        end
+    else
+        throw(ArgumentError("Unsupported number of directions for rotl90: $directions"))
+    end
+    return SiteDependentTruncation(trschemes_rotated)
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index 7d93eda49..787964a9b 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -61,6 +61,9 @@ end
         @time @safetestset "Cluster truncation with projectors" begin
             include("timeevol/cluster_projectors.jl")
         end
+        @time @safetestset "Time evolution with site-dependent truncation" begin
+            include("timeevol/sitedep_truncation.jl")
+        end
     end
     if GROUP == "ALL" || GROUP == "UTILITY"
         @time @safetestset "LocalOperator" begin
diff --git a/test/timeevol/cluster_projectors.jl b/test/timeevol/cluster_projectors.jl
index 4ad3f60f6..91b2994e8 100644
--- a/test/timeevol/cluster_projectors.jl
+++ b/test/timeevol/cluster_projectors.jl
@@ -30,7 +30,7 @@ Vspaces = [
         revs = [isdual(space(M, 1)) for M in Ms1[2:end]]
         # no truncation
         Ms2 = deepcopy(Ms1)
-        wts2, ϵs, = _cluster_truncate!(Ms2, FixedSpaceTruncation(), revs)
+        wts2, ϵs, = _cluster_truncate!(Ms2, fill(FixedSpaceTruncation(), N-1), revs)
         @test all((ϵ == 0) for ϵ in ϵs)
         absorb_wts_cluster!(Ms2, wts2)
         for (i, M) in enumerate(Ms2)
@@ -41,7 +41,7 @@ Vspaces = [
         @test all(lorths) && all(rorths)
         # truncation on one bond
         Ms3 = deepcopy(Ms1)
-        wts3, ϵs, = _cluster_truncate!(Ms3, truncspace(Vns), revs)
+        wts3, ϵs, = _cluster_truncate!(Ms3, fill(truncspace(Vns), N-1), revs)
         @test all((i == n) || (ϵ == 0) for (i, ϵ) in enumerate(ϵs))
         absorb_wts_cluster!(Ms3, wts3)
         for (i, M) in enumerate(Ms3)
diff --git a/test/timeevol/sitedep_truncation.jl b/test/timeevol/sitedep_truncation.jl
new file mode 100644
index 000000000..ca62daca9
--- /dev/null
+++ b/test/timeevol/sitedep_truncation.jl
@@ -0,0 +1,54 @@
+using Test
+using LinearAlgebra
+using Random
+using TensorKit
+using PEPSKit
+using PEPSKit: NORTH, EAST
+
+function get_bonddims(wpeps::InfiniteWeightPEPS)
+    xdims = collect(dim(domain(t, EAST)) for t in wpeps.vertices)
+    ydims = collect(dim(domain(t, NORTH)) for t in wpeps.vertices)
+    return stack([xdims, ydims]; dims=1)
+end
+
+@testset "Simple update: bipartite 2-site" begin
+    Nr, Nc = 2, 2
+    ham = real(heisenberg_XYZ(InfiniteSquare(Nr, Nc); Jx=1.0, Jy=1.0, Jz=1.0))
+    Random.seed!(100)
+    wpeps0 = InfiniteWeightPEPS(rand, Float64, ℂ^2, ℂ^10; unitcell=(Nr, Nc))
+    normalize!.(wpeps0.vertices, Inf)
+    # set trscheme to be compatible with bipartite structure
+    bonddims = stack([[6 4; 4 6], [5 7; 7 5]]; dims=1)
+    trscheme = SiteDependentTruncation(collect(truncdim(d) for d in bonddims))
+    alg = SimpleUpdate(1e-2, 1e-14, 4, trscheme)
+    wpeps, = simpleupdate(wpeps0, ham, alg; bipartite=true)
+    @test get_bonddims(wpeps) == bonddims
+    # check bipartite structure is preserved
+    for col in 1:2
+        cp1 = PEPSKit._next(col, 2)
+        @test (
+            wpeps.vertices[1, col] == wpeps.vertices[2, cp1] &&
+            wpeps.weights[1, 1, col] == wpeps.weights[1, 2, cp1] &&
+            wpeps.weights[2, 1, col] == wpeps.weights[2, 2, cp1]
+        )
+    end
+end
+
+@testset "Simple update: generic 2-site and 3-site" begin
+    Nr, Nc = 3, 4
+    ham = real(heisenberg_XYZ(InfiniteSquare(Nr, Nc); Jx=1.0, Jy=1.0, Jz=1.0))
+    Random.seed!(100)
+    wpeps0 = InfiniteWeightPEPS(rand, Float64, ℂ^2, ℂ^10; unitcell=(Nr, Nc))
+    normalize!.(wpeps0.vertices, Inf)
+    # Site dependent truncation
+    bonddims = rand(2:8, 2, Nr, Nc)
+    @show bonddims
+    trscheme = SiteDependentTruncation(collect(truncdim(d) for d in bonddims))
+    alg = SimpleUpdate(1e-2, 1e-14, 2, trscheme)
+    # 2-site SU
+    wpeps, = simpleupdate(wpeps0, ham, alg; bipartite=false)
+    @test get_bonddims(wpeps) == bonddims
+    # 3-site SU
+    wpeps, = simpleupdate(wpeps0, ham, alg; bipartite=false, force_3site=true)
+    @test get_bonddims(wpeps) == bonddims
+end