[jack-dunham]

Stacked on blocked-message-update (#1), which is stacked on bp-gate-stages and jd/quadratic-form.

Summary

• BufferedBPGateUpdate is an opt-in subalgorithm of BPApplyGate. Its bp_gate_factorize! and bp_gate_restore! absorb the message roots in place, so each vertex holds two tensors of its own size instead of one per message.
• absorb_matrices!(alg, ψ, matrices...) overwrites ψ with ψ times each operator in turn, alternating between ψ and one scratch tensor of its size. With a TensorOperationsContract, every contraction goes through TensorOperations.
• The stages use upstream's message_gauge roots (Use the asymmetric Gram root for the BP simple-update gauge ITensor/ITensorNetworksNext.jl#204). Each (x, y) pair is wrapped as operators on the bond name, so absorb_matrices! keeps ψ's index names while the leg holds the rank space.
• Dense storage only: graded storage, or messages of a wider element type than the tensor, throw an ArgumentError.

The first commit consolidates the buffered parts of several earlier bp-gate-stages commits, rewritten for the current gate API.

Tests

• BufferedBPGateUpdate matches SimpleBPGateUpdate on a 2×3 grid in Float32, Float64 and ComplexF64, with the default allocator and with a BufferAllocator.
• apply_operators leaves its input unchanged, and apply_operator! names the new bond as requested.
• absorb_matrices! matches apply for 0 to 3 matrices, with MatricizeContract and TensorOperationsContract.
• Graded storage and wider message element types throw.
• test_aqua.jl, test_apply_operator.jl, test_prod_tensors.jl, test_blockedmessageupdate.jl, test_beliefpropagation.jl, test_normnetwork.jl and test_quadraticformnetwork.jl pass (1266 tests).

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diff --git a/src/apply/apply_operators.jl b/src/apply/apply_operators.jl
index dc9ae4b..b3d9697 100644
--- a/src/apply/apply_operators.jl
+++ b/src/apply/apply_operators.jl
@@ -529,7 +529,8 @@ function bp_gate_factorize!(
         subalgorithm::SimpleBPGateUpdate, op::AbstractITensor, state, env, v, w
     )
     roots = [
-        message_gauge(m; alg = subalgorithm.eigh_alg) for m in incoming_messages(env, v => w)
+        message_gauge(m; alg = subalgorithm.eigh_alg) for
+            m in incoming_messages(env, v => w)
     ]
     ψ = absorb_matrices!(subalgorithm.contract_alg, state[v], first.(roots)...)
     bondname = only(intersect(names(state[v]), names(env[w => v])))
diff --git a/test/test_apply_operator.jl b/test/test_apply_operator.jl
index 5f78ff1..6148c68 100644
--- a/test/test_apply_operator.jl
+++ b/test/test_apply_operator.jl
@@ -226,7 +226,8 @@ end
         end
         buffered = BPApplyGate(; subalgorithm = buffered_update)
         simple_state, _ = apply_operator(gates[1], network, complex_env)
-        buffered_state, _ = apply_operator(gates[1], network, complex_env; alg = buffered)
+        buffered_state, _ =
+            apply_operator(gates[1], network, complex_env; alg = buffered)
         @test prod(buffered_state) ≈ prod(simple_state) rtol = rtol
     end