In order to perform simulations of fermionic systems on a quantum computer, there is an encoding needed which maps the fermionic Hamiltonian onto a qubit Hamiltonian. In this thesis, a fermionic encoding is constructed which is less constraining in terms of the required operations that realize this mapping compared to other known mappings.
In the constructed mapping, the Bravyi-Kitaev superfast encoding is used to place qubits at the edges of a graph corresponding to a physical system of n fermionic modes. From this encoding certain stabilizer constraints arise which ensure that the correct fermionic operators are being encoded throughout the entire simulation, and which are used to detect the occurrence of any errors during the simulation.
In combination with the superfast encoding, a randomized algorithm developed by Freedman-Hastings is combined with a stacking- and sewing procedure of the graph and a Vertex Coloring algorithm to obtain a cycle basis which governs the properties of these stabilizer constraints.
This constructed mapping obtains a locality and sparsity of the qubit Hamiltonian terms and stabilizer constraints of O(1), while keeping the total number of qubits O(npoly(log(n))). Here poly(log(n)) means some polynomial in log(n). It is also numerically shown that the number of qubits can be further reduced, while keeping the locality of the stabilizer constraints O(1), and the locality and sparsity of the qubit Hamiltonian terms and the sparsity of the stabilizer constraints O(poly(log(n))).
It is furthermore shown how to use this encoding to perform simulations of two fermionic systems, namely the Fermi-Hubbard model on sparse hopping graphs and the sparse SYK model.