Fermions and Gaussianity ; Resources and Simulability

          @misc{ pirsa_PIRSA:25030179,
            doi = {10.48660/25030179},
            url = {https://pirsa.org/25030179},
            author = {},
            keywords = {Quantum Information},
            language = {en},
            title = {Fermions and Gaussianity ; Resources and Simulability},
            publisher = {Perimeter Institute},
            year = {2025},
            month = {mar},
            note = {PIRSA:25030179 see, \url{https://pirsa.org}}
          }
          

Abstract

Matchgates are a well studied class of quantum circuits tied to the time dynamics of Free Fermion Hamiltonians. It is important to note however that Matchgates specifically come from representing Free Fermions with the Jordan-Wigner encoding. When we represent our fermionic systems with other encodings besides Jordan-Wigner, we still are considering the time dynamics of Free Fermion solvable Hamiltonians, but we can introduce complexity in how we encode our fermionic information. This gives us a test ground for clarifying what physical properties make time dynamics hard to simulate, even when Hamiltonians can be exactly diagonalized. In this talk I will discuss the theory behind matchgates, fermionic encodings, and recent results in the simulability of Clifford/matchgate hybrid circuits (arxiv:2312.08447, arxiv:2410.10068). These results clarify resources for Free Fermions represented beyond the Jordan-Wigner encoding, as well as an overall perspective of what it means for a state to be Gaussian.