Clifford Circuit Optimization with Templates and Symbolic Pauli Gates

The Clifford group is a finite subgroup of the unitary generated by Hadamard, CNOT, and Phase gates. This plays prominent role in quantum error correction, randomized benchmarking protocols, study entanglement. Here we consider problem finding short circuit implementing given element. Our methods aim to minimize entangling gate count assuming all-to-all qubit connectivity. First, optimization based on template matching design Clifford-specific templates that leverage ability factor out Pauli SWAP Second, introduce symbolic peephole method. It works projecting full onto small subset qubits optimally recompiling projected subcircuit via dynamic programming. CNOT gates coupling chosen with remaining are expressed using Software implementation these finds circuits only 0.2% away from optimal for 6 reduces two-qubit up 64 64.7% average, compared Aaronson-Gottesman canonical form.