- Compilation/
Pauli Product Rotations
Pauli Product Rotations
Static gates
We list typical static gates with their PPR decomposition and cost. For the cost, we count Clifford angles (\frac{\pi}{4} (2k+1) for k \in \mathbb{Z}) and non-Clifford angles with T corresponding to odd multiples of \frac{\pi}{8} and \sqrt{T} to odd multiples of \frac{\pi}{16} and so on.
| Operator | PPR | cost |
|---|---|---|
| S | e^{i \frac{\pi}{4}} e^{-i \frac{\pi}{4} Z} | 1 |
| T | e^{i \frac{\pi}{8}} e^{-i \frac{\pi}{8} Z} | 1T |
| Hadamard | e^{i \frac{\pi}{2}} e^{-i \frac{\pi}{4} Z} e^{-i \frac{\pi}{4} X} e^{-i \frac{\pi}{4} Z} | 3 |
| SX | e^{i \frac{\pi}{4}} e^{-i \frac{\pi}{4} X} | 1 |
| CNOT | e^{-i \frac{\pi}{4}} e^{-i \frac{\pi}{4} ZX} e^{i \frac{\pi}{4} ZI} e^{i \frac{\pi}{4} IX} | 3 |
| CZ | e^{-i \frac{\pi}{4}} e^{-i \frac{\pi}{4} ZZ} e^{i \frac{\pi}{4} ZI} e^{i \frac{\pi}{4} IZ} | 3 |
| CY | e^{-i \frac{\pi}{4}} e^{-i \frac{\pi}{4} ZY} e^{i \frac{\pi}{4} ZI} e^{i \frac{\pi}{4} IY} | 3 |
| CH | e^{-i \frac{\pi}{8} IY} \ \text{CZ} \ e^{i \frac{\pi}{8} IY} | 2T + 3 |
| SWAP | e^{i \frac{\pi}{4}} e^{-i \frac{\pi}{4} ZZ} e^{-i \frac{\pi}{4} XX} e^{-i \frac{\pi}{4} YY} | 3 |
| ISWAP | e^{i \frac{\pi}{4} XX} e^{i \frac{\pi}{4} YY} | 2 |
| SISWAP | e^{i \frac{\pi}{8}XX} e^{i \frac{\pi}{8}YY} | 2T |
| CSWAP | e^{i \frac{\pi}{8}} e^{-i \frac{\pi}{8} ZII} e^{-i \frac{\pi}{8} IXX} e^{-i \frac{\pi}{8} IYY} e^{-i \frac{\pi}{8} IZZ} e^{i \frac{\pi}{8} ZXX} e^{i \frac{\pi}{8} ZYY} e^{i \frac{\pi}{8} ZZZ} | 7T |
| Toffoli | e^{i \frac{\pi}{8}} e^{-i \frac{\pi}{8} Z_0}e^{-i \frac{\pi}{8} Z_1}e^{-i \frac{\pi}{8} X_2} e^{-i \frac{\pi}{8} ZZX} e^{i \frac{\pi}{8} IZX}e^{i \frac{\pi}{8} ZIX} e^{i \frac{\pi}{8} ZZI} | 7T |
MultiControlledX
The MultiControlledX gate is decomposed into PPRs using all operators from \{Z_0, I_0\}\otimes\{Z_1, I\} \otimes \cdots \{Z_{n-2}, I\}\otimes \{X_{n-1}, I\} for n qubits and n-1 controls. Terms with odd number of Paulis have a minus in front of the phase \frac{\pi}{2^{n}}. For example, for n=4 we have the following:
Note how we sorted the generators by their Pauli weight (i.e., the number of non-trivial Pauli operators in the Pauli word). This is just for better readability. Because all generators commute, we can trivially decouple all generators into individual PPRs.