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An open question in designing superconducting quantum circuits is how best toreduce the full circuit Hamiltonian which describes their dynamics to aneffective two-level qubit Hamiltonian which is appropriate for manipulation ofquantum information. Despite advances in numerical methods to simulate thespectral properties of multi-element superconducting circuits, the literaturelacks a consistent and effective method of determining the effective qubitHamiltonian. Here we address this problem by introducing a novel local basisreduction method. This method does not require any ad hoc assumption on thestructure of the Hamiltonian such as its linear response to applied fields. Wenumerically benchmark the local basis reduction method against otherHamiltonian reduction methods in the literature and report specific examples ofsuperconducting qubits, including the capacitively-shunted flux qubit, wherethe standard reduction approaches fail. By combining the local basis reductionmethod with the Schrieffer-Wolff transformation we further extend itsapplicability to systems of interacting qubits and use it to extract bothnon-stoquastic two-qubit Hamiltonians and three-local interaction terms inthree-qubit Hamiltonians.

Effective Hamiltonians for interacting superconducting qubits -- local basis reduction and the Schrieffer-Wolff transformation