  The Gell-Mann & Low theorem is a cornerstone of Quantum Field Theory (QFT)
and condensed matter physics, and many-body perturbation theory is a
foundational tool for treating interactions. However, their integration into
quantum algorithms remains a largely unexplored area of research, with current
quantum simulation algorithms predominantly operating in the Schr\"odinger
picture, leaving the potential of the interaction picture largely untapped. Our
Variational Interaction-Picture S-matrix Ansatz (VIPSA) now fills this gap,
specifically in the context of the Fermi-Hubbard model -- a canonical paradigm
in condensed matter physics which is intricately connected to phenomena such as
high-temperature superconductivity and Mott insulator transitions.
  This work offers a new conceptual perspective for variational quantum
computing based upon the Gell-Mann & Low theorem. We achieve this by employing
an innovative mathematical technique to explicitly unfold the normalized
S-matrix, thereby enabling the systematic reconstruction of the Dyson series on
a quantum computer, order by order. This method stands in contrast to the
conventional reliance on Trotter expansion for adiabatic time evolution,
marking a conceptual shift towards more sophisticated quantum algorithmic
design. We leverage the strengths of the recently developed ADAPT-VQE
algorithm, tailoring it to reconstruct perturbative terms effectively. Our
simulations indicate that this method not only successfully recovers the Dyson
series but also exhibits robust and stable convergence. We believe that our
approach shows great promise in generalizing to more complex scenarios without
increasing algorithmic complexity.
