Is Circuit Depth Accurate for Comparing Quantum Circuit Runtimes?

Although quantum circuit depth is commonly used to approximate circuit runtimes, it overlooks a prevailing trait of current hardware implementation: different gates have different execution times. Recognizing the potential for discrepancies, we investigate depth's accuracy for comparing runtimes between compiled versions of the same circuit. In particular, we assess the accuracy of traditional and multi-qubit depth for (1) predicting relative differences in runtime and (2) identifying compiled circuit version(s) with the shortest runtime. Finding that circuit depth is not accurate for either task, we introduce a new metric, gate-aware depth, that weights gates' contributions to runtime using an architecture's average gate execution times. Using average gate times allows gate-aware depth to capture variations by gate type without requiring exact knowledge of all gate times, increasing accuracy while maintaining portability across devices of the same architecture. Compared to traditional and multi-qubit depth, gate-aware depth reduces the average relative error of predictions in task (1) by 68 and 18 times and increases the average number of correct identifications in task (2) by 20 and 43 percentage points, respectively. Finally, we provide gate-aware depth weight configurations for current IBM Eagle and Heron architectures.