  Magic states are eigenstates of non-Pauli operators. One way of suppressing
errors present in magic states is to perform parity measurements in their
non-Pauli eigenbasis and postselect on even parity. Here we develop new
protocols based on non-Pauli parity checking, where the measurements are
implemented with the aid of pre-distilled multiqubit resource states. This
leads to a two step process: pre-distillation of multiqubit resource states,
followed by implementation of the parity check. These protocols can prepare
single-qubit magic states that enable direct injection of single-qubit axial
rotations without subsequent gate-synthesis and its associated overhead. We
show our protocols are more efficient than all previous comparable protocols
with quadratic error reduction, including the protocols of Bravyi and Haah.
