Efficiency of MSC-LS at larger color code distances

Determine whether implementations of magic state cultivation with lattice surgery at color code distances greater than 3 are more efficient, in terms of spacetime overhead, than original magic state cultivation at the corresponding color code distance.

Background

The proposed protocol (MSC-LS) avoids the grafted code used in magic state cultivation by transferring the cultivated magic state into a rotated surface code via lattice surgery, and introduces code expansion and early rejection via a lookup table to reduce average spacetime overhead. The paper demonstrates significant overhead reductions at color code distance 3.

While the authors see no fundamental obstacles to extending MSC-LS to larger color code distances, it remains unresolved whether such extensions would surpass the efficiency of the original MSC protocol (which converts the color code to the grafted code) when both operate at matched color code distances.

References

However, whether such implementations are more efficient than magic state cultivation with the corresponding color code distance remains open and is an important direction for future work.

— Efficient magic state cultivation with lattice surgery  (2510.24615 - Hirano et al., 28 Oct 2025) in Section 6 (Conclusion)

Finally, it would be interesting to study whether native $$ resources provide similar savings for multi-qubit compilation, structured algorithmic subroutines, or after further logical-error reduction through cultivation or an extension of the protocols developed in this work to larger-distance 3D color codes.

— Fault-tolerant $|\sqrt{ \mathrm{T} }\rangle$ state preparation and injection for more efficient fine-grained quantum circuit synthesis  (2608.25797 - Yenilen et al., 26 Aug 2026) in Conclusion, final paragraph