Constant recycled support for arbitrary numbers of outputs

Construct families of weakly r-orthogonal distillation codes with arbitrary distance d and constant-size recycled qubit support for any number of output states k.

Background

The paper develops recursive code-doubling constructions for magic-state distillation using weakly r-orthogonal binary matrices. For single-output protocols, the construction achieves arbitrary distance while using a number of recycled support qubits bounded independently of the distance. The paper also presents constructions for multiple outputs, including k-output distance-two protocols and two-output protocols of arbitrary distance.

For the two-output arbitrary-distance family, however, the authors obtain only a linear gain in qubit support under recycling rather than a distance-independent bound. They therefore leave unresolved whether the constant-support property of the single-output construction can be extended to arbitrary numbers of output states.