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In this study, we explore the non-Markovian cost function for quantum errormitigation (QEM) and the representation of two-qubit operators using DiracGamma matrices, central to the structure of relativistic quantum mechanics. Theprimary focus of quantum computing research, particularly with noisyintermediate-scale quantum (NISQ) devices, is on reducing errors anddecoherence for practical application. While much of the existing researchconcentrates on Markovian noise sources, the study of non-Markovian sources iscrucial given their inevitable presence in most solid-state quantum computingdevices. We introduce a non-Markovian model of quantum state evolution and acorresponding QEM cost function for NISQ devices, considering an environmenttypified by simple harmonic oscillators as a noise source. The Dirac Gammamatrices, integral to areas of physics like quantum field theory andsupersymmetry, share a common algebraic structure with two-qubit gateoperators. By representing the latter using Gamma matrices, we are able to moreeffectively analyze and manipulate these operators due to the distinctproperties of Gamma matrices. We evaluate the fluctuations of the outputquantum state for identity and SWAP gate operations in two-qubit operationsacross various input states. By comparing these results with experimental datafrom ion-trap and superconducting quantum computing systems, we estimate thekey parameters of the QEM cost functions. Our results reveal that as thecoupling strength between the quantum system and its environment increases, sodoes the QEM cost function. This study underscores the importance ofnon-Markovian models for understanding quantum state evolution and thepractical implications of the QEM cost function when assessing experimentalresults from NISQ devices.

Non-Markovian cost function for quantum error mitigation with Dirac Gamma matrices representation