Non-Markovian Cost Function for Quantum Error Mitigation

In near-term quantum computers like noisy intermediate-scale quantum (NISQ) devices, reducing the impact of errors and decoherence is critical for practical implementation. Existing studies have primarily focused on Markovian noise sources; however, understanding the relationship between quantum error mitigation (QEM) and non-Markovian noise sources is essential, as these effects are practically unavoidable in most solid-state devices used for quantum computing. Here, a non-Markovian model of quantum state evolution and a QEM cost function of controlled-NOT (CNOT) gate operation are presented for NISQ devices interacting with an environment characterized by simple harmonic oscillators as a noise source. Using the projection operator formalism and both advanced and retarded propagators in time, the reduced-density operator is derived for output quantum states in a time-convolutionless form by solving the quantum Liouville equation. Output quantum state fluctuations are analyzed for identity and CNOT gate operations in two-qubit operations across various input states and compare these results with experimental data from ion-trap and superconducting quantum computing systems to estimate the key parameters of the QEM cost functions. These findings demonstrate that the QEM cost function increases as the coupling strength between the quantum system and its environment intensifies. This study highlights the significance of non-Markovian models for understanding quantum state evolution and the practical implications of the QEM cost function in assessing experimental results from NISQ devices.