A Simple Method for Compiling Quantum Stabilizer Circuits

Stabilizer circuits play an important role in quantum error correction protocols, and will be vital for ensuring fault tolerance in future quantum hardware. While stabilizer circuits are defined on the Clifford generating set, $\{H, S, CX\}$, not all of these gates are native to quantum hardware. As such they must be compiled into the native gateset, with the key difference across hardware archetypes being the native two-qubit gate. Here we introduce an intuitive and accessible method for Clifford gate compilation. While multiple open source solutions exist for quantum circuit compilation, these operate on arbitrary quantum gates. By restricting ourselves to Clifford gates, the compilation process becomes almost trivial and even large circuits can be compiled manually. The core idea is well known: if two Clifford circuits conjugate Paulis identically, they are equivalent. Compilation is then reduced to ensuring that the instantaneous Pauli conjugation is correct for each qubit at every timestep. This is Tableaux Manipulation, so called as we directly interrogate stabilizer tableaux to ensure correct Pauli conjugation. We provide a brief explanation of the process along with a worked example to build intuition; we finally show some comparisons for compiling large circuits to open source software, and highlight that this method ensures a minimal number of quantum gates are employed.