Depth Optimization of CZ, CNOT, and Clifford Circuits

We seek to develop better upper bound guarantees on the depth of quantum $\text {CZ}$ gate, cnot gate, and Clifford circuits than those reported previously. We focus on the number of qubits $n\,{\leq }\,$1 345 000 (de Brugière et al., 2021), which represents the most practical use case. Our upper bound on the depth of $\text {CZ}$ circuits is $\lfloor n/2 + 0.4993{\cdot }\log ^{2}(n) + 3.0191{\cdot }\log (n) - 10.9139\rfloor$, improving the 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 the 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 $5/3$.