We analyze the coupling of two flux qubits with a general many-body projector into the low-energy subspace. Specifically, we extract the effective Hamiltonians that controls the dynamicsof two qubits when they are coupled via a capacitor and/or via a Josephson junction. While the capacitor induces a static charge coupling tunable by design, the Josephson junction produces a magnetic-like interaction easily tunable by replacing the junction with a SQUID. Those two elements allow to engineer qubits Hamiltonians with XX, YY and ZZ interactions, including ultra-strongly coupled ones. We present an exhaustive numerical study for two three-Josephson junctions flux qubit that can be directly used in experimental work. The method developed here, namely the numerical tool to extract qubit effective Hamiltonians at strong coupling, can be applied to replicate our analysis for general systems of many qubits and any type of coupling.
We have analyzed and proposed coupling mechanisms between Three Josephson Junction Flux Qubits (3JJQ). For this, we have developed a numerical method to extract the effective Hamiltonianof a system of coupled qubits via the Schrieffer-Wolff transformation (SWT). We then give a comprehensive introduction to the 3JJQ, and study it analytically by approximating its potential with a Harmonic well. With a clear understanding of the 3JJQs, we use the SWT to gain intuition about their effective dipolar interaction with the electromagnetic field, and use that intuition to propose and study analytically and numerically the capacitive coupling of two 3JJQs via a non-tunable capacitor, and the inductive coupling of two 3JJQs via a tunable Josephson Junction (dc-SQUID), showing that we are able to reproduce non-stoquastic Hamiltonians in the strong-coupling regime.
A flux qubit can interact strongly when it is capacitively coupled to other circuit elements. This interaction can be separated in two parts, one acting on the qubit subspaces and onein which excited states mediate the interaction. The first term dominates the interaction between the flux qubit and an LC-resonator, leading to ultrastrong couplings of the form σy(a+a†), which complement the inductive σxi(a†−a) coupling. However, when coupling two flux qubits capacitively, all terms need to be taken into account, leading to complex non-stoquastic ultrastrong interaction of the σyσy, σzσz and σxσx type. Our theory explains all these interactions, describing them in terms of general circuit properties—coupling capacitances, qubit gaps, inductive, Josephson and capactive energies—, that apply to a wide variety of circuits and flux qubit designs.