diff --git a/src/formnetworks/bilinearformnetwork.jl b/src/formnetworks/bilinearformnetwork.jl
index 19ca7568..11d298c5 100644
--- a/src/formnetworks/bilinearformnetwork.jl
+++ b/src/formnetworks/bilinearformnetwork.jl
@@ -63,9 +63,10 @@ function BilinearFormNetwork(
   kwargs...,
 )
   @assert issetequal(flatten_siteinds(bra), flatten_siteinds(ket))
+  link_space = isempty(flatten_siteinds(bra)) ? 1 : nothing
   operator_inds = union_all_inds(siteinds(ket), dual_site_index_map(siteinds(ket)))
   # TODO: Define and use `identity_network` here.
-  O = ITensorNetwork(Op("I"), operator_inds; link_space=nothing)
+  O = ITensorNetwork(Op("I"), operator_inds; link_space)
   return BilinearFormNetwork(O, bra, ket; dual_site_index_map, kwargs...)
 end
 
diff --git a/src/gauging.jl b/src/gauging.jl
index 9f394243..2ad9d0f4 100644
--- a/src/gauging.jl
+++ b/src/gauging.jl
@@ -25,10 +25,7 @@ function Base.copy(ψ::VidalITensorNetwork)
   return VidalITensorNetwork(copy(site_tensors(ψ)), copy(bond_tensors(ψ)))
 end
 
-function default_norm_cache(ψ::ITensorNetwork)
-  ψψ = norm_sqr_network(ψ)
-  return BeliefPropagationCache(ψψ, group(v -> first(v), vertices(ψψ)))
-end
+default_norm_cache(ψ::ITensorNetwork) = BeliefPropagationCache(QuadraticFormNetwork(ψ))
 
 function ITensorNetwork(
   ψ_vidal::VidalITensorNetwork; (cache!)=nothing, update_gauge=false, update_kwargs...