add rrule for flip

ext/TensorKitChainRulesCoreExt/linalg.jl

@@ -83,6 +83,12 @@
     return tA, twist_pullback
 end
 
+function ChainRulesCore.rrule(::typeof(flip), A::AbstractTensorMap, is; inv::Bool=false)
+    tA = flip(A, is; inv)
+    flip_pullback(ΔA) = NoTangent(), flip(unthunk(ΔA), is; inv=!inv), NoTangent()
+    return tA, flip_pullback
+end
+
 function ChainRulesCore.rrule(::typeof(dot), a::AbstractTensorMap, b::AbstractTensorMap)
     dot_pullback(Δd) = NoTangent(), @thunk(b * Δd'), @thunk(a * Δd)
     return dot(a, b), dot_pullback

src/fusiontrees/manipulations.jl

@@ -243,24 +243,46 @@ end
 # -> A-move (foldleft, foldright) is complicated, needs to be reexpressed in standard form
 
 # flip a duality flag of a fusion tree
-function flip(f₁::FusionTree{I,N₁}, f₂::FusionTree{I,N₂}, i::Int) where {I<:Sector,N₁,N₂}
+function flip(f₁::FusionTree{I,N₁}, f₂::FusionTree{I,N₂}, i::Int;
+              inv::Bool=false) where {I<:Sector,N₁,N₂}
     @assert 0 < i ≤ N₁ + N₂
     if i ≤ N₁
         a = f₁.uncoupled[i]
-        fs = frobeniusschur(a) * twist(a)
-        factor = f₁.isdual[i] ? fs : one(fs)
+        χₐ = frobeniusschur(a)
+        θₐ = twist(a)
+        if !inv
+            factor = f₁.isdual[i] ? χₐ * θₐ : one(θₐ)
+        else
+            factor = f₁.isdual[i] ? one(θₐ) : χₐ * conj(θₐ)
+        end
         isdual′ = TupleTools.setindex(f₁.isdual, !f₁.isdual[i], i)
         f₁′ = FusionTree{I}(f₁.uncoupled, f₁.coupled, isdual′, f₁.innerlines, f₁.vertices)
         return SingletonDict((f₁′, f₂) => factor)
     else
         i -= N₁
         a = f₂.uncoupled[i]
-        factor = f₂.isdual[i] ? frobeniusschur(a) : twist(a)
+        χₐ = frobeniusschur(a)
+        θₐ = twist(a)
+        if !inv
+            factor = f₂.isdual[i] ? χₐ * one(θₐ) : θₐ
+        else
+            factor = f₂.isdual[i] ? conj(θₐ) : χₐ * one(θₐ)
+        end
         isdual′ = TupleTools.setindex(f₂.isdual, !f₂.isdual[i], i)
         f₂′ = FusionTree{I}(f₂.uncoupled, f₂.coupled, isdual′, f₂.innerlines, f₂.vertices)
         return SingletonDict((f₁, f₂′) => factor)
     end
 end
+function flip(f₁::FusionTree{I,N₁}, f₂::FusionTree{I,N₂}, ind;
+              inv::Bool=false) where {I<:Sector,N₁,N₂}
+    f₁′, f₂′ = f₁, f₂
+    factor = one(sectorscalartype(I))
+    for i in ind
+        (f₁′, f₂′), s = only(flip(f₁′, f₂′, i; inv))
+        factor *= s
+    end
+    return SingletonDict((f₁′, f₂′) => factor)
+end
 
 # change to N₁ - 1, N₂ + 1
 function bendright(f₁::FusionTree{I,N₁}, f₂::FusionTree{I,N₂}) where {I<:Sector,N₁,N₂}

src/tensors/indexmanipulations.jl

@@ -5,17 +5,19 @@
 
 Return a new tensor that is isomorphic to `t` but where the arrows on the indices `i` that satisfy
 `i ∈ I` are flipped, i.e. `space(t′, i) = flip(space(t, i))`.
+
+!!! note
+    The isomorphism that `flip` applies to each of the indices `i ∈ I` is such that flipping two indices
+    that are afterwards contracted within an `@tensor` contraction will yield the same result as without
+    flipping those indices first. However, `flip` is not involutary, i.e. `flip(flip(t, I), I) != t` in
+    general. To obtain the original tensor, one can use the `inv` keyword, i.e. it holds that
+    `flip(flip(t, I), I; inv=true) == t`.
 """
-function flip(t::AbstractTensorMap, I)
+function flip(t::AbstractTensorMap, I; inv::Bool=false)
     P = flip(space(t), I)
     t′ = similar(t, P)
     for (f₁, f₂) in fusiontrees(t)
-        f₁′, f₂′ = f₁, f₂
-        factor = one(scalartype(t))
-        for i in I
-            (f₁′, f₂′), s = only(flip(f₁′, f₂′, i))
-            factor *= s
-        end
+        (f₁′, f₂′), factor = only(flip(f₁, f₂, I; inv))
         scale!(t′[f₁′, f₂′], t[f₁, f₂], factor)
     end
     return t′

test/ad.jl

@@ -224,6 +224,9 @@ Vlist = ((ℂ^2, (ℂ^3)', ℂ^3, ℂ^2, (ℂ^2)'),
         test_rrule(twist, A, 1)
         test_rrule(twist, A, [1, 3])
 
+        test_rrule(flip, A, 1)
+        test_rrule(flip, A, [1, 3, 4])
+
         D = randn(T, V[1] ⊗ V[2] ← V[3])
         E = randn(T, V[4] ← V[5])
         symmetricbraiding && test_rrule(⊗, D, E)

test/tensors.jl

@@ -324,6 +324,13 @@ for V in spacelist
                 @test HrA12array ≈ convert(Array, HrA12)
             end
         end
+        @timedtestset "Index flipping: test flipping inverse" begin
+            t = rand(ComplexF64, V1 ⊗ V1' ← V1' ⊗ V1)
+            for i in 1:4
+                @test t ≈ flip(flip(t, i), i; inv=true)
+                @test t ≈ flip(flip(t, i; inv=true), i)
+            end
+        end
         @timedtestset "Index flipping: test via explicit flip" begin
             t = rand(ComplexF64, V1 ⊗ V1' ← V1' ⊗ V1)
             F1 = unitary(flip(V1), V1)