diff --git a/dev/KrylovKit b/dev/KrylovKit
new file mode 160000
index 000000000..8bccac88a
--- /dev/null
+++ b/dev/KrylovKit
@@ -0,0 +1 @@
+Subproject commit 8bccac88a9474b47ce49bac72ae19b3806ce129f
diff --git a/ext/TensorKitChainRulesCoreExt/constructors.jl b/ext/TensorKitChainRulesCoreExt/constructors.jl
index f48ed4393..4b9dcd6b1 100644
--- a/ext/TensorKitChainRulesCoreExt/constructors.jl
+++ b/ext/TensorKitChainRulesCoreExt/constructors.jl
@@ -4,7 +4,7 @@
 @non_differentiable TensorKit.isometry(args...)
 @non_differentiable TensorKit.unitary(args...)
 
-function ChainRulesCore.rrule(::Type{<:TensorMap}, d::DenseArray, args...; kwargs...)
+function ChainRulesCore.rrule(::Type{TensorMap}, d::DenseArray, args...; kwargs...)
     function TensorMap_pullback(Δt)
         ∂d = convert(Array, unthunk(Δt))
         return NoTangent(), ∂d, ntuple(_ -> NoTangent(), length(args))...
@@ -12,6 +12,78 @@ function ChainRulesCore.rrule(::Type{<:TensorMap}, d::DenseArray, args...; kwarg
     return TensorMap(d, args...; kwargs...), TensorMap_pullback
 end
 
+# these are not the conversion to/from array, but actually take in data parameters
+# -- as a result, requires quantum dimensions to keep inner product the same:
+#   ⟨Δdata, ∂data⟩ = ⟨Δtensor, ∂tensor⟩ = ∑_c d_c ⟨Δtensor_c, ∂tensor_c⟩
+#   ⟹ Δdata = d_c Δtensor_c
+function ChainRulesCore.rrule(::Type{TensorMap{T}}, data::DenseVector,
+                              V::TensorMapSpace) where {T}
+    t = TensorMap{T}(data, V)
+    P = ProjectTo(data)
+    function TensorMap_pullback(Δt_)
+        Δt = copy(unthunk(Δt_))
+        for (c, b) in blocks(Δt)
+            scale!(b, dim(c))
+        end
+        ∂data = P(Δt.data)
+        return NoTangent(), ∂data, NoTangent()
+    end
+    return t, TensorMap_pullback
+end
+
+function ChainRulesCore.rrule(::Type{<:DiagonalTensorMap}, data::DenseVector, args...;
+                              kwargs...)
+    D = DiagonalTensorMap(data, args...; kwargs...)
+    P = ProjectTo(data)
+    function DiagonalTensorMap_pullback(Δt_)
+        # unclear if we're allowed to modify/take ownership of the input
+        Δt = copy(unthunk(Δt_))
+        for (c, b) in blocks(Δt)
+            scale!(b, dim(c))
+        end
+        ∂data = P(Δt.data)
+        return NoTangent(), ∂data, NoTangent()
+    end
+    return D, DiagonalTensorMap_pullback
+end
+
+function ChainRulesCore.rrule(::typeof(Base.getproperty), t::TensorMap, prop::Symbol)
+    if prop === :data
+        function getdata_pullback(Δdata)
+            # unclear if we're allowed to modify/take ownership of the input
+            t′ = typeof(t)(copy(unthunk(Δdata)), t.space)
+            for (c, b) in blocks(t′)
+                scale!(b, inv(dim(c)))
+            end
+            return NoTangent(), t′, NoTangent()
+        end
+        return t.data, getdata_pullback
+    elseif prop === :space
+        return t.space, Returns((NoTangent(), ZeroTangent(), NoTangent()))
+    else
+        throw(ArgumentError("unknown property $prop"))
+    end
+end
+
+function ChainRulesCore.rrule(::typeof(Base.getproperty), t::DiagonalTensorMap,
+                              prop::Symbol)
+    if prop === :data
+        function getdata_pullback(Δdata)
+            # unclear if we're allowed to modify/take ownership of the input
+            t′ = typeof(t)(copy(unthunk(Δdata)), t.domain)
+            for (c, b) in blocks(t′)
+                scale!(b, inv(dim(c)))
+            end
+            return NoTangent(), t′, NoTangent()
+        end
+        return t.data, getdata_pullback
+    elseif prop === :domain
+        return t.domain, Returns((NoTangent(), ZeroTangent(), NoTangent()))
+    else
+        throw(ArgumentError("unknown property $prop"))
+    end
+end
+
 function ChainRulesCore.rrule(::typeof(Base.copy), t::AbstractTensorMap)
     copy_pullback(Δt) = NoTangent(), Δt
     return copy(t), copy_pullback
diff --git a/ext/TensorKitChainRulesCoreExt/linalg.jl b/ext/TensorKitChainRulesCoreExt/linalg.jl
index 922900b6f..67dc5a980 100644
--- a/ext/TensorKitChainRulesCoreExt/linalg.jl
+++ b/ext/TensorKitChainRulesCoreExt/linalg.jl
@@ -113,7 +113,8 @@ function ChainRulesCore.rrule(::typeof(imag), a::AbstractTensorMap)
     return a_imag, imag_pullback
 end
 
-function ChainRulesCore.rrule(cfg::RuleConfig, ::typeof(exp), A::AbstractTensorMap)
+function ChainRulesCore.rrule(cfg::RuleConfig{>:HasReverseMode}, ::typeof(exp),
+                              A::AbstractTensorMap)
     domain(A) == codomain(A) ||
         error("Exponential of a tensor only exist when domain == codomain.")
     P_A = ProjectTo(A)
@@ -133,3 +134,21 @@ function ChainRulesCore.rrule(cfg::RuleConfig, ::typeof(exp), A::AbstractTensorM
     end
     return C, exp_pullback
 end
+
+# define rrules for matrix functions for DiagonalTensorMap, since they access data directly.
+for f in
+    (:exp, :cos, :sin, :tan, :cot, :cosh, :sinh, :tanh, :coth, :atan, :acot, :asinh, :sqrt,
+     :log, :asin, :acos, :acosh, :atanh, :acoth)
+    f_pullback = Symbol(f, :_pullback)
+    @eval function ChainRulesCore.rrule(cfg::RuleConfig{>:HasReverseMode}, ::typeof($f),
+                                        t::DiagonalTensorMap)
+        P = ProjectTo(t) # unsure if this is necessary, should already be in pullback
+        d, pullback = rrule_via_ad(cfg, broadcast, $f, t.data)
+        function $f_pullback(Δd_)
+            Δd = P(unthunk(Δd_))
+            _, _, ∂data = pullback(Δd.data)
+            return NoTangent(), DiagonalTensorMap(∂data, t.domain)
+        end
+        return DiagonalTensorMap(d, t.domain), $f_pullback
+    end
+end
diff --git a/ext/TensorKitChainRulesCoreExt/utility.jl b/ext/TensorKitChainRulesCoreExt/utility.jl
index f270e346a..2a9f11d57 100644
--- a/ext/TensorKitChainRulesCoreExt/utility.jl
+++ b/ext/TensorKitChainRulesCoreExt/utility.jl
@@ -32,3 +32,15 @@ function (::ProjectTo{T1})(x::T2) where {S,N1,N2,T1<:AbstractTensorMap{<:Any,S,N
     end
     return y
 end
+
+function (::ProjectTo{DiagonalTensorMap{T,S,A}})(x::AbstractTensorMap) where {T,S,A}
+    x isa DiagonalTensorMap{T,S,A} && return x
+    V = space(x, 1)
+    space(x) == (V ← V) || throw(SpaceMismatch())
+    y = DiagonalTensorMap{T,S,A}(undef, V)
+    for (c, b) in blocks(y)
+        p = ProjectTo(b)
+        b .= p(block(x, c))
+    end
+    return y
+end
diff --git a/ext/TensorKitFiniteDifferencesExt.jl b/ext/TensorKitFiniteDifferencesExt.jl
index f62642560..63c9711e3 100644
--- a/ext/TensorKitFiniteDifferencesExt.jl
+++ b/ext/TensorKitFiniteDifferencesExt.jl
@@ -23,6 +23,14 @@ function FiniteDifferences.to_vec(t::AbstractTensorMap)
 end
 FiniteDifferences.to_vec(t::TensorKit.AdjointTensorMap) = to_vec(copy(t))
 
+function FiniteDifferences.to_vec(t::DiagonalTensorMap)
+    x_vec, back = to_vec(TensorMap(t))
+    function DiagonalTensorMap_from_vec(x_vec)
+        return DiagonalTensorMap(back(x_vec))
+    end
+    return x_vec, DiagonalTensorMap_from_vec
+end
+
 end
 
 # TODO: Investigate why the approach below doesn't work
diff --git a/test/ad.jl b/test/ad.jl
index 93251eb03..044e89534 100644
--- a/test/ad.jl
+++ b/test/ad.jl
@@ -3,6 +3,7 @@ using ChainRulesTestUtils
 using FiniteDifferences: FiniteDifferences
 using Random
 using LinearAlgebra
+using Zygote
 
 const _repartition = @static if isdefined(Base, :get_extension)
     Base.get_extension(TensorKit, :TensorKitChainRulesCoreExt)._repartition
@@ -15,6 +16,10 @@ end
 function ChainRulesTestUtils.rand_tangent(rng::AbstractRNG, x::AbstractTensorMap)
     return randn!(similar(x))
 end
+function ChainRulesTestUtils.rand_tangent(rng::AbstractRNG, x::DiagonalTensorMap)
+    V = x.domain
+    return DiagonalTensorMap(randn(eltype(x), reduceddim(V)), V)
+end
 ChainRulesTestUtils.rand_tangent(::AbstractRNG, ::VectorSpace) = NoTangent()
 function ChainRulesTestUtils.test_approx(actual::AbstractTensorMap,
                                          expected::AbstractTensorMap, msg=""; kwargs...)
@@ -152,6 +157,46 @@ Vlist = ((ℂ^2, (ℂ^3)', ℂ^3, ℂ^2, (ℂ^2)'),
             test_rrule(TensorMap, convert(Array, T1), codomain(T1), domain(T1);
                        fkwargs=(; tol=Inf))
         end
+
+        test_rrule(Base.getproperty, T1, :data)
+        test_rrule(TensorMap{scalartype(T1)}, T1.data, T1.space)
+        test_rrule(Base.getproperty, T2, :data)
+        test_rrule(TensorMap{scalartype(T2)}, T2.data, T2.space)
+    end
+
+    @timedtestset "Basic utility (DiagonalTensor)" begin
+        for v in V
+            rdim = reduceddim(v)
+            D1 = DiagonalTensorMap(randn(rdim), v)
+            D2 = DiagonalTensorMap(randn(rdim), v)
+            D = D1 + im * D2
+            T1 = TensorMap(D1)
+            T2 = TensorMap(D2)
+            T = T1 + im * T2
+
+            # real -> real
+            P1 = ProjectTo(D1)
+            @test P1(D1) == D1
+            @test P1(T1) == D1
+
+            # complex -> complex
+            P2 = ProjectTo(D)
+            @test P2(D) == D
+            @test P2(T) == D
+
+            # real -> complex 
+            @test P2(D1) == D1 + 0 * im * D1
+            @test P2(T1) == D1 + 0 * im * D1
+
+            # complex -> real
+            @test P1(D) == D1
+            @test P1(T) == D1
+
+            test_rrule(DiagonalTensorMap, D1.data, D1.domain)
+            test_rrule(DiagonalTensorMap, D.data, D.domain)
+            test_rrule(Base.getproperty, D, :data)
+            test_rrule(Base.getproperty, D1, :data)
+        end
     end
 
     @timedtestset "Basic Linear Algebra with scalartype $T" for T in eltypes
@@ -196,6 +241,21 @@ Vlist = ((ℂ^2, (ℂ^3)', ℂ^3, ℂ^2, (ℂ^2)'),
         test_rrule(LinearAlgebra.dot, A, B)
     end
 
+    @timedtestset "Matrix functions ($T)" for T in eltypes
+        for f in (sqrt, exp)
+            check_inferred = false # !(T <: Real) # not type-stable for real functions
+            t1 = randn(T, V[1] ← V[1])
+            t2 = randn(T, V[2] ← V[2])
+            d = DiagonalTensorMap{T}(undef, V[1])
+            (T <: Real && f === sqrt) ? randexp!(d.data) : randn!(d.data)
+            d2 = DiagonalTensorMap{T}(undef, V[1])
+            (T <: Real && f === sqrt) ? randexp!(d2.data) : randn!(d2.data)
+            test_rrule(f, t1; rrule_f=Zygote.rrule_via_ad, check_inferred)
+            test_rrule(f, t2; rrule_f=Zygote.rrule_via_ad, check_inferred)
+            test_rrule(f, d; check_inferred, output_tangent=d2)
+        end
+    end
+
     symmetricbraiding &&
         @timedtestset "TensorOperations with scalartype $T" for T in eltypes
             atol = precision(T)
diff --git a/test/runtests.jl b/test/runtests.jl
index 1f06191cc..d0cd9945b 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -60,7 +60,8 @@ include("spaces.jl")
 include("tensors.jl")
 include("diagonal.jl")
 include("planar.jl")
-if !(Sys.isapple()) # TODO: remove once we know why this is so slow on macOS
+# TODO: remove once we know AD is slow on macOS CI
+if !(Sys.isapple() && get(ENV, "CI", "false") == "true")
     include("ad.jl")
 end
 include("bugfixes.jl")