Quantify the Non-Markovian Process with Intervening Projections in a Superconducting Processor

A Markov assumption considers a physical system memoryless to simplify its dynamics. Whereas memory effect or the non-Markovian phenomenon is more general in nature. In the quantum regime, it is challenging to define or quantify the non-Markovianity because the measurement of a quantum system often interferes with it. We simulate the open quantum dynamics in a superconducting processor, then characterize and quantify the non-Markovian process. With the complete set of intervening projections and the final measurement of the qubit, a restricted process tensor can be determined to account for the qubit-environment interaction. We apply the process tensor to predict the quantum state with memory effect, yielding an average fidelity of $99.86\%\pm 1.1\unicode{x2030}$. We further derive the Choi state of the rest process conditioned on history operations and quantify the non-Markovianity with a clear operational interpretation.