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

TL;DR

This paper introduces a novel local basis reduction method combined with the Schrieffer-Wolff transformation to accurately derive effective qubit Hamiltonians in superconducting circuits, especially when standard methods fail.

Contribution

The authors develop a new local basis reduction technique that does not rely on ad hoc assumptions, improving the derivation of effective Hamiltonians for superconducting qubits.

Findings

The method outperforms existing reduction techniques in benchmark tests.

It successfully extracts non-stoquastic two-qubit Hamiltonians.

It identifies three-local interactions in multi-qubit systems.

Abstract

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.