Accuracy of the effective Hamiltonian in the quantum simulation experiments

Superconducting qubits and cold atom systems can simulate a variety of quantum Hamiltonians by implementing the couplings directly, as opposed to Trotterization of the quantum evolution and gate-based simulation. In this approach, the effective Hamiltonian is often calibrated experimentally to match the desired properties. We develop a rigorous mathematical foundation for the calibration of the building blocks to obtain a target effective Hamiltonian. The building blocks we consider are both superconducting circuits and cold atoms. For both approaches we provide an error estimate: the difference in the norm between the obtained effective Hamiltonian and the target. Our approach extends to time-dependent Hamiltonians via a version of adiabatic theorem.