Depth reduction for quantum Clifford circuits through Pauli measurements

Clifford circuits play an important role in quantum computation. Gottesman and Chuang proposed a gate teleportation protocol so that a quantum circuit can be implemented by the teleportation circuit with specific ancillary qubits. In particular, an n-qubit Clifford circuit U can be implemented by preparing an ancillary stabilizer state $I \otimes U |\Psi\rightangle^{\otimes n}$ for teleportation and doing a Pauli correction conditioned on the measurement. In this paper, we provide an alternative procedure to implement a Clifford circuit through Pauli measurements, by preparing O(1) ancillas that are Calderbank-Shor-Steane (CSS) stabilizer states. That is to say, O(1) CSS states are sufficient to implement any Clifford circuit. As an application to fault-tolerant quantum computation, any Clifford circuit can be implemented by O(1) steps of Steane syndrome extraction if clean CSS stabilizer states are available.