2 Fast 2 Surgery: Fast surgery on QLDPC codes with $\tilde{O}(n(k+d))$ space overhead

We show that the fast surgery framework of Baspin-Berent-Cohen can be instantiated using at most $\tilde{O}(n(k+d))$ ancilla qubits, improving on the previously achievable $O(n\cdot k\cdot d)$ overhead. This result also establishes that fast surgery can be performed at a cost comparable to the $\tilde{O}(nk)$ overhead required for regular surgery. We build the surgery ancilla by first starting with a dense, but expanding system; and then applying expansion-preserving sparsification methods from gauging and code embeddings.