  Characterizing quantum dynamics is a cornerstone pursuit across quantum
physics, quantum information science, and quantum computation. The precision of
quantum gates in manipulating input basis states and their intricate
superpositions is paramount. In this study, we conduct a thorough analysis of
the SQSCZ gate, a universal two-qubit entangling gate, using real quantum
hardware. This gate is a fusion of the square root of SWAP ($\sqrt{SWAP}$) and
the square root of controlled-Z ($\sqrt{CZ}$) gates, serves as a foundational
element for constructing universal gates, including the controlled-NOT gate. we
begin by explaining the theory behind quantum process tomography (QPT),
exploring the \textit{Choi-Jamiolkowski} isomorphism or the Choi matrix
representation of the quantum process, along with a QPT algorithm utilizing
Choi representation. Subsequently, we provide detailed insights into the
experimental realization of the SQSCZ gate using a transmon-based
superconducting qubit quantum computer. To comprehensively assess the gate's
performance on a noisy intermediate-scale quantum (NISQ) computer, we conduct
QPT experiments across diverse environments, employing both IBM Quantum's
simulators and IBM Quantum's real quantum computer. Leveraging the Choi matrix
in our QPT experiments allows for a comprehensive characterization of our
quantum operations. Our analysis unveils commendable fidelities and noise
properties of the SQSCZ gate, with process fidelities reaching $97.27098\%$ and
$88.99383\%$, respectively. These findings hold promising implications for
advancing both theoretical understanding and practical applications in the
realm of quantum computation.
