Design of an inductively shunted transmon qubit with tunable transverse and longitudinal coupling

Abstract

Superconducting qubits are among the most promising and versatile building blocks on the road to a functioning quantum computer. One of the main challenges in superconducting qubit architectures is to couple qubits in a well-controlled manner, especially in circuit constructions that involve many qubits. In order to avoid unwanted cross-couplings, qubits are oftentimes coupled via harmonic resonators, which act as buses that mediate the interaction. This thesis is set in the framework of superconducting transmon-type qubit architectures with special focus on two important types of coupling between qubits and harmonic resonators: transverse and longitudinal coupling. While transverse coupling naturally appears in transmon-like circuit constructions, longitudinal coupling is much harder to implement and hardly ever the only coupling term present. Nevertheless, we will see that longitudinal coupling offers some remarkable advantages with respect to scalability and readout. This thesis will focus on a design, which combines both these coupling types in a single circuit and provides the possibility to choose between pure transverse and pure longitudinal or have both at the same time. The ability to choose between transverse and longitudinal coupling in the same circuit provides the flexibility to use one for coupling to the next qubit and one for readout, or vice versa. We will start with an introduction to circuit quantization, where we will explain how to describe and analyze superconducting electrical circuits in a systematic way and discuss which characteristic circuit elements make up qubits and resonators. We will then introduce the two types of coupling between qubit and resonator which are provided in our design: transverse and longitudinal coupling. In order to show that longitudinal coupling has some remarkable advantages with respect to the scalability of a circuit, we will discuss a scalable qubit architecture, which can be implemented with our design. Translating this discussion from the Hamiltonian level to the language of circuit quantization, we will show how to design circuits with specifically tailored couplings. Having introduced these basic concepts, we will focus on our circuit design that consists of an inductively shunted transmon qubit with tunable coupling to an embedded harmonic mode. Using a symmetric design, static transverse coupling terms are canceled out, while the parity of the only remaining coupling term can be tuned via an external flux. The distinctive feature of the tunable design is that the transverse coupling disappears when the longitudinal is maximal and vice versa. Subsequently, we will turn to the implementation of our circuit design, discuss how to choose the parameters, and present an adapted alternative circuit, where coupling strength and anharmonicity scale better than in the original circuit. Furthermore, we show how the anharmonicity and the coupling can be boosted by additional flux-biasing. We will see that for conveniently chosen parameters longitudinal and transverse coupling have comparable values, while all other coupling terms can be suppressed. In addition, we present a proposal for an experimental device that will serve as a prototype for a first experiment. Coming back to the scalable architecture mentioned above, we will show how our design can be scaled up to a grid, which can be done in modular fashion with strictly local couplings. In such a grid of fixed-frequency qubits and resonators with a particular pattern of always-on interactions, coupling is strictly confined to nearest and next-nearest neighbor resonators; there is never any direct qubit-qubit coupling. We will conclude the thesis discussing different possibilities to do readout with our circuit design, including a short discussion of the coupling between the circuit and the environment, and the influence of dissipation.