  We seek to develop better upper bound guarantees on the depth of quantum CZ
gate, CNOT gate, and Clifford circuits than those reported previously. We focus
on the number of qubits $n\,{\leq}\,$1,345,000 [1], which represents the most
practical use case. Our upper bound on the depth of CZ circuits is $\lfloor n/2
+ 0.4993{\cdot}\log^2(n) + 3.0191{\cdot}\log(n) - 10.9139\rfloor$, improving
best known depth by a factor of roughly 2. We extend the constructions used to
prove this upper bound to obtain depth upper bound of $\lfloor n +
1.9496{\cdot}\log^2(n) + 3.5075{\cdot}\log(n) - 23.4269 \rfloor$ for CNOT gate
circuits, offering an improvement by a factor of roughly $4/3$ over state of
the art, and depth upper bound of $\lfloor 2n + 2.9487{\cdot}\log^2(n) +
8.4909{\cdot}\log(n) - 44.4798\rfloor$ for Clifford circuits, offering an
improvement by a factor of roughly $5/3$.
