Figure 5

Schematic diagram of achieving the operation \(HR_{Z}(\theta )\) in Protocol 1, where angle θ is restricted to \(B \equiv \{0, \frac{2\pi}{4}, \frac{4\pi}{4}, \frac{6\pi}{4}\}\). The implementation of each \(HR_{Z}(\theta )\) operation requires the participation of two ancillary qubits. Each ancilla is produced by Alice’s X or Z-basis measurement on half of the Bell state \(\frac{1}{\sqrt{2}}(|00\rangle +|11\rangle )\). In addition, Alice needs to perform X or Y-basis measurements on the ancillary qubits that have been manipulated by Bob to drive the computation on register qubits. Real number \(S_{i}\) denotes the ith measurement result