  In this study, we explore the non-Markovian cost function for quantum error
mitigation (QEM) and the representation of two-qubit operators using Dirac
Gamma matrices, central to the structure of relativistic quantum mechanics. The
primary focus of quantum computing research, particularly with noisy
intermediate-scale quantum (NISQ) devices, is on reducing errors and
decoherence for practical application. While much of the existing research
concentrates on Markovian noise sources, the study of non-Markovian sources is
crucial given their inevitable presence in most solid-state quantum computing
devices. We introduce a non-Markovian model of quantum state evolution and a
corresponding QEM cost function for NISQ devices, considering an environment
typified by simple harmonic oscillators as a noise source. The Dirac Gamma
matrices, integral to areas of physics like quantum field theory and
supersymmetry, share a common algebraic structure with two-qubit gate
operators. By representing the latter using Gamma matrices, we are able to more
effectively analyze and manipulate these operators due to the distinct
properties of Gamma matrices. We evaluate the fluctuations of the output
quantum state for identity and SWAP gate operations in two-qubit operations
across various input states. By comparing these results with experimental data
from ion-trap and superconducting quantum computing systems, we estimate the
key parameters of the QEM cost functions. Our results reveal that as the
coupling strength between the quantum system and its environment increases, so
does the QEM cost function. This study underscores the importance of
non-Markovian models for understanding quantum state evolution and the
practical implications of the QEM cost function when assessing experimental
results from NISQ devices.
