Spectral Hamiltonian learning in Si:P arrays

The analog quantum simulation of the Fermi-Hubbard model is a near-term application of quantum information systems in which many-body interactions are realized physically in a controlled manner. Among the many candidate methods for quantum simulation, solid-state platforms consisting of donor arrays in silicon are attractive due to the atomically precise geometric control in the impurity allocation, which allows for fine tuning of the device's quantum properties. Knowing how the device properties contribute to the objective Hamiltonian is essential to validate a quantum simulation. In this work, we develop a Hamiltonian learning protocol on systems with two to five impurities. Starting from the bound electron energies and wave function overlaps, we numerically propagate single-electron states and analyze the trajectories in the frequency domain. This allows us to calculate the tunnel couplings of the Fermi-Hubbard model and, in some cases, extract information about the shifts in on-site energies caused by the surrounding impurities. A novel method of direct Hamiltonian optimization was also developed and found to agree with accepted calculations for 2-impurity arrays. The results of this work provide an efficient way to anticipate the Hamiltonian that Si:P nanodevices can simulate and illustrate how the electron tunneling depends on the number and placement of impurities.