In fault-tolerant quantum computing (FTQC) based on surface code architectures, in particular those that are described by the Game of Surface Codes, quantum programs get compiled to Pauli product rotations (PPRs) and Pauli product measurements (PPMs).
A PPR can be written like [1]
P_\phi = e^{-i \phi P},
in terms of its Pauli word, or Pauli product, generator P. For example (XIY)_{\frac{\pi}{4}} = \exp\left(-i \frac{\pi}{4} X \otimes \mathbb{I} \otimes Y\right), though we typically omit the explicit tensor products.
To save on classical compilation times, it is useful to write common gates and subroutines directly in this representation. Here, we bookkeep the PPR decompositions of common gates and circuits.
Inputs
- Arbitrary gate or sub-circuit
Outputs
Example
The (Clifford + T) gate set is an important example, as all circuits can be approximated by this gate set. Its gates have the following PPR decompositions:
\begin{align*}
S &= e^{i \frac{\pi}{4}} e^{-i \frac{\pi}{4} Z} \\
T &= e^{i \frac{\pi}{8}} e^{-i \frac{\pi}{8} Z} \\
H &= e^{i \frac{\pi}{2}} e^{-i \frac{\pi}{4} Z} e^{-i \frac{\pi}{4} X} e^{-i \frac{\pi}{4} Z} \\
\text{CNOT} &= e^{i \frac{\pi}{4}} e^{-i \frac{\pi}{4} ZX} e^{i \frac{\pi}{4} ZI} e^{i \frac{\pi}{4} IX}. \\
\end{align*}
Note that we have explicitly stated the global phases that are omitted in Fig. 5 of [1].
Typical usage
FTQC scenarios using surface codes like those described by the Game of Surface Codes [1].
References
[1] "A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery", Daniel Litinski, arXiv:1808.02892, Quantum 3, 128 (2019).