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Clifford circuits play an important role in quantum computation. Gottesmanand Chuang proposed a gate teleportation protocol so that a quantum circuit canbe implemented by the teleportation circuit with specific ancillary qubits. Inparticular, an $n$-qubit Clifford circuit $U$ can be implemented by preparingan ancillary stabilizer state $(I\otimes U)|\Phi^+\rangle^{\otimes n}$ forteleportation and doing a Pauli correction conditioned on the measurement. Inthis paper, we provide an alternative procedure to implement a Clifford circuitthrough Pauli measurements, by preparing $O(1)$ ancillas that areCalderbank-Shor-Steane (CSS) stabilizer states. That is to say, $O(1)$ CSSstates are sufficient to implement any Clifford circuit. As an application tofault-tolerant quantum computation, any Clifford circuit can be implemented by$O(1)$ steps of Steane syndrome extraction if clean CSS stabilizer states areavailable.

Depth reduction for quantum Clifford circuits through Pauli measurements