Employing tensor networks and Riemannian quantum circuit optimization for fermionic Hamiltonian simulation

Simulating fermionic systems is relevant for many fields like quantum chemistry or condensed matter physics. Trotterization is a simple and common tool to approximate the time evolution on a quantum computer. However, accurate simulations for long times are restricted by the resulting deep quantum circuits. The goal of this work is to increase accuracy under constant depth, utilizing only classical resources before employing any quantum device for simulation. We start from a fermionic swap network that implements a Trotter step using a brickwall circuit layout. To further increase the accuracy of the initial circuit, Riemannian optimization is employed to optimize the two-qubit gates under unitary constraints. The reference time evolution operator can efficiently be expressed as a matrix product operator for short enough times t. By interpreting the quantum circuit as a tensor network, a suitably chosen cost function and the corresponding gradient can be evaluated using tensor network methods. Executing the optimized circuit repetitively on a quantum device enables simulation times T≫t . We apply our method to molecular Hamiltonians and the 1D Fermi-Hubbard for 60 spin orbitals and t=0.3.