  An open question in designing superconducting quantum circuits is how best to
reduce the full circuit Hamiltonian which describes their dynamics to an
effective two-level qubit Hamiltonian which is appropriate for manipulation of
quantum information. Despite advances in numerical methods to simulate the
spectral properties of multi-element superconducting circuits, the literature
lacks a consistent and effective method of determining the effective qubit
Hamiltonian. Here we address this problem by introducing a novel local basis
reduction method. This method does not require any ad hoc assumption on the
structure of the Hamiltonian such as its linear response to applied fields. We
numerically benchmark the local basis reduction method against other
Hamiltonian reduction methods in the literature and report specific examples of
superconducting qubits, including the capacitively-shunted flux qubit, where
the standard reduction approaches fail. By combining the local basis reduction
method with the Schrieffer-Wolff transformation we further extend its
applicability to systems of interacting qubits and use it to extract both
non-stoquastic two-qubit Hamiltonians and three-local interaction terms in
three-qubit Hamiltonians.
