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A Hadamard-free Clifford transformation is a circuit composed of quantumPhase (P), CZ, and CNOT gates. It is known that such a circuit can be writtenas a three-stage computation, -P-CZ-CNOT-, where each stage consists only ofgates of the specified type.  In this paper, we focus on the minimization of circuit depth by entanglinggates, corresponding to the important time-to-solution metric and the reductionof noise due to decoherence. We consider two popular connectivity maps: LinearNearest Neighbor (LNN) and all-to-all. First, we show that a Hadamard-freeClifford operation can be implemented over LNN in depth $5n$, i.e., in the samedepth as the -CNOT- stage alone. This allows us to implement arbitrary Cliffordtransformation over LNN in depth no more than $7n{-}4$, improving the bestprevious upper bound of $9n$. Second, we report heuristic evidence that onaverage a random uniformly distributed Hadamard-free Clifford transformationover $n{>}6$ qubits can be implemented with only a tiny additive overhead overall-to-all connected architecture compared to the best-known depth-optimizedimplementation of the -CNOT- stage alone. This suggests the reduction of thedepth of Clifford circuits from $2n\,{+}\,O(\log^2(n))$ to$1.5n\,{+}\,O(\log^2(n))$ over unrestricted architectures.

CNOT circuits need little help to implement arbitrary Hadamard-free Clifford transformations they generate