  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 $(I\otimes U)|\Phi^+\rangle^{\otimes n}$ 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.
