General low‑overhead high‑rate code‑surgery construction for arbitrary logical operator sets

Construct a high‑rate code‑surgery scheme that, for any quantum low‑density parity‑check (qLDPC) code, can measure an arbitrary set of pairwise logically disjoint logical Pauli operators, including Y‑type and mixed XZ‑type operators, in parallel with guaranteed low overhead; specifically, design a construction whose ancilla size is asymptotically bounded by the size of the data code block (i.e., |A| = Õ(n)).

Background

Code surgery generalizes lattice surgery to arbitrary qLDPC codes by coupling a data code block to an ancilla system to implement logical Pauli product measurements. Recent work provides low‑rate gadgets for single logical operators and high‑rate gadgets that can measure multiple operators in parallel with overhead scaling near the code size for certain cases.

However, a universally applicable, provably low‑overhead high‑rate construction that can measure any arbitrary set of logically disjoint logical Pauli operators—including Y and mixed XZ types—on any qLDPC code remains unavailable. Establishing such a construction would enable broad, scalable parallel logical operations with bounded ancilla overhead across code families.

References

Note that, although we do not know such a construction yet that can measure an arbitrary set of logically non-overlapping logicals (including Y- and XZ-type) for any code with a guaranteed low overhead, we expect that more advanced schemes based on Refs. can approach this goal, at least for certain structured codes.

— Shor's algorithm is possible with as few as 10,000 reconfigurable atomic qubits  (2603.28627 - Cain et al., 30 Mar 2026) in Appendix, Section “Surgery,” Subsection “Description” (High‑rate surgery paragraph)