Trotter error with commutator scaling for the Fermi-Hubbard model, and improvements of Trotter methods by Riemannian quantum circuit optimization

Christian Mendl
Technical University of Munich

This talk is concerned with quantum simulation, i.e., approximating the quantum time evolution operator $\e^{-i t H}$. We first derive higher-order error bounds for a general Trotter product formula and apply these bounds to the time evolution operator governed by the Fermi-Hubbard Hamiltonian on one-dimensional and two-dimensional square and triangular lattices. Comparison with the actual Trotter error (evaluated on a small system) indicates that the bounds still overestimate the error. In the second part of the talk, we further improve Trotter methods via the Riemannian trust-region algorithm, used to optimize the gates in quantum circuits with Trotter splitting topologies. For the Ising and Heisenberg models on a one-dimensional lattice, we achieve orders of magnitude accuracy improvements compared to fourth-order splitting methods. The optimized circuits could also be of practical use for the time-evolving block decimation (TEBD) algorithm. (Based on arXiv:2306.10603 and arXiv:2212.07556)

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