Theorem: Tree compilation and module-optimal streaming peak

For every $h\ge 1$:

1. batch recovery has optimal peak $3^h$, with no additional workspace, and an explicit implementation with

   $$G=9m_h,D\le 8h;$$

2. the serial streaming module class has optimal peak

   $${Q_{\mathrm{module}}}^{(h)}=h+1,$$

   with an implementation satisfying

   $$G\le 7m_h,D\le 8m_h,C_{\mathrm{cl}}\le 2h;$$

3. both implement the same channel ${\mathcal{D}}_h$.

All programmes consume $n_h-1$ syndrome qubits and return one quantum output.

Proof. For batch input, decode all child nodes at one level in parallel, measure their syndrome factors, and continue upward. There are $m_h$ decoded nodes and $h$ levels. Every node uses nine gates and at most eight rounds. Occupancy decreases after each local syndrome readout and never exceeds the initial $3^h$. The initial batch occupancy proves optimality.

For streaming, let $p_h$ be the peak in the prescribed module class. The base optimum is $p_1=2$ by Theorem thm:B5. Decode child 1, retaining its output. Decode child 2 while retaining that one qubit; this costs at most $p_{h-1}+1$. Perform the parent's early two-qubit stage and consume those two intermediate inputs, retaining only two classical bits. Decode child 3 with no other parent quantum register live. Apply the parent's late stage. Thus

For the matching module lower bound, the first child's quantum output must remain live until the second child has returned its output: neither allowed parent programme consumes a single first input alone. The serial second-child subprogramme requires at least $p_{h-1}$ live qubits on one of its histories. The retained first output is an additional qubit. Therefore $p_h\ge p_{h-1}+1$. This is a lower bound within Definition def:moduleclass, not among all possible decoders.

Every decoded node uses at most seven unitary gates and eight instruction rounds in the serial construction. At most two record bits per ancestor are retained, giving $2h$. Each local recovery is a channel identity on its logical input, including all reference systems, so composition proves recovery. Lemma lem:treenormal identifies both programmes with ${\mathcal{D}}_h$.