  Reducing the impact of errors and decoherence in near-term quantum computers,
such as noisy intermediate-scale quantum (NISQ) devices, is critical for their
practical implementation. These factors significantly limit the applicability
of quantum algorithms, necessitating a comprehensive understanding of their
physical origins to establish effective error mitigation strategies. In this
study, we present a non-Markovian model of quantum state evolution and a
quantum error mitigation cost function tailored for NISQ devices interacting
with an environment represented by a set of simple harmonic oscillators as a
noise source. Employing the projection operator formalism and both advanced and
retarded propagators in time, we derive the reduced-density operator for the
output quantum states in a time-convolutionless form by solving the quantum
Liouville equation. We examine the output quantum state fluctuations for both
identity and controlled-NOT (CNOT) gate operations in two-qubit operations
using a range of input states. Subsequently, these results are compared with
experimental data from ion-trap and superconducting quantum computing systems
to estimate the crucial parameters of the cost functions for quantum error
mitigation. Our findings reveal that the cost function for quantum error
mitigation increases as the coupling strength between the quantum system and
its environment intensifies. This study underscores the significance of
non-Markovian models in understanding quantum state evolution and highlights
the practical implications of the quantum error mitigation cost function when
assessing experimental results from NISQ devices.
