Geometric Algebra Jordan–Wigner Transformation for Quantum Simulation

Quantum simulation qubit models of electronic Hamiltonians rely on specific transformations in order to take into account the fermionic permutation properties of electrons. These transformations (principally the Jordan–Wigner transformation (JWT) and the Bravyi–Kitaev transformation) correspond in a quantum circuit to the introduction of a supplementary circuit level. In order to include the fermionic properties in a more straightforward way in quantum computations, we propose to use methods issued from Geometric Algebra (GA), which, due to its commutation properties, are well adapted for fermionic systems. First, we apply the Witt basis method in GA to reformulate the JWT in this framework and use this formulation to express various quantum gates. We then rewrite the general one and two-electron Hamiltonian and use it for building a quantum simulation circuit for the Hydrogen molecule. Finally, the quantum Ising Hamiltonian, widely used in quantum simulation, is reformulated in this framework.