Nonreciprocal circuit quantization in the singular limit : design of a passively protected superconducting qubit encoding a quantum error-correcting code

Rymarz, Martin Kevin; DiVincenzo, David P. (Thesis advisor); Hassler, Fabian (Thesis advisor)

Aachen : RWTH Aachen University (2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2023


The practical realization of a quantum computer is a truly fascinating and challenging task, which, over the past decades, constantly attracted increasing attention in both the scientific community but also the overall society. Currently, there exist many different architectures that promise a possible realization of a universal quantum computer, capable of exponentially accelerating certain computational tasks and simulating complex quantum systems, which, on a classical computer, is practically impossible. A particular appealing state-of-the-art platform for quantum computation are superconducting electrical networks, which mainly build on the nonlinearity of the Josephson effect in order to reliably encode and process quantum information. The theoretical description and exploration of such systems is the content of this thesis. More specifically, we propose a novel circuit design for a nonreciprocal and passively protected superconducting qubit that, for the first time, effectively hardware-encodes the bosonic Gottesman-Kitaev-Preskill quantum error-correcting code, which we assign to a specific point in the Hofstadter butterfly. The proposed qubit design has a twofold (quasi-)degenerate ground space that is spanned by eigenfunctions with disjoint support. We find that the qubit is inherently protected against common types of noise, and we demonstrate that it is suitable for universal quantum computation. In the process of elaborating the proposed superconducting qubit design, we extend the scope of circuit quantization theory in two key aspects: To begin with, we generalize the conventional literature on circuit quantization to incorporate general nonlinear inductors and capacitors, as well as the gyrator. The latter is a prototypical nonreciprocal network element, whose integration into the toolbox for quantum circuit design opens intriguing new possibilities as it captures, up until now, unexplored effects. Moreover, we develop the analysis and a classification of (nearly) singular electrical networks, both reciprocal and nonreciprocal. In this context, we demonstrate the failure of the Dirac-Bergmann theory for the quantization of inherently constrained systems when applied to physically realistic electrical networks, and we show the prevalent procedure, eliminating variables by manipulation of the classical Kirchhoff's conservation laws, to be invalid. Instead, we propose a correct treatment of such systems by means of a regularized Born-Oppenheimer approach. Based on the results of Kato-Rellich perturbation theory, we mathematically rigorously prove the validity of the underlying nonanalytic perturbation theory, allowing an identification of universal fixed points that we use to classify generic nonlinear inductors by their quantum dynamics as the singular limit of a vanishingly small intrinsic capacitance is approached.


  • JARA-Institute for Quantum Information [080043]
  • Department of Physics [130000]
  • Chair of Theoretical Physics [137310]