Non-destructive direct characterization of unitary dynamics for arbitrary qubit systems

The ability to measure unknown quantum processes is crucial in quantum information. Standard process tomography yields information on quantum processes. However, standard process tomography is complex and inefficient when only partial information regarding quantum processes or specific matrix elements is required. We propose a parameterized quantum circuit scheme for direct measurement of any specific matrix element in unitary processes. Using the target unitary process as a high-dimensional sparse matrix, the proposed method quickly determines the required matrix elements without requiring complex reconstruction algorithms. Simultaneously, our approach is nondestructive, and the target quantum state that undergoes an unknown unitary process can be retained for reuse in subsequent measurements. We extend this method to arbitrary multi-qubit systems. Using a nuclear magnetic resonance system on the SpinQ quantum cloud platform as a case study, we experimentally demonstrate the effectiveness of the method and analyze its accuracy and precision.