Transversal Clifford impossibility conjecture for logical C-QSK circuits

A stabilizer code encodes logical qubits, and a transversal Clifford gate is a physical Clifford operation that acts independently on the corresponding physical qubits. A non-trivial exponentiated Pauli operator is an exponentiated Pauli operator other than the identity case. The logical C-QSK circuit is the circuit under consideration for implementing such operators.

Transversal Clifford impossibility conjecture. There exists no stabilizer code where transversal Clifford gates realize the logical C-QSK circuit for any non-trivial exponentiated Pauli operator.

The preceding results rule out transversal Hadamard and transversal Phase implementations, and these gates generate the single-qubit Clifford group. The broader impossibility claim is presented as a conjecture and remains to be proved for arbitrary combinations of transversal Clifford gates.