Microwave Simulations of Superconducting Circuits using Coupled Scattering Parameter Calculation and Contour Integration

Abstract

In this project we investigate and build upon a method for microwave simulations of superconducting circuits containing both linear components and nonlinear Josephson junctions. The main idea behind this method is to connect frequency responses of arbitrary linear microwave components in a closed network, and find solutions to the corresponding nonlinear eigenvalueproblem using an efficient method based on complex analysis and numerical linear algebra. The frequency responses of individual microwave components can be extracted in isolation as scattering parameters from simulations using the conventional finite element method (FEM), or from well-known lumped-element models. From the solutions to the linear network simulation, analysis for weakly anharmonic quantum circuits based on the energy participation ratio in the Josephson junctions is applied to correct for nonlinear effects. The method as a whole enables simulations of large-scale systems to a significantly lesser cost but with similar accuracy in predicted experimental parameters compared to full-wave FEM, and provides with a solid foundation for the development of advanced chips for quantum computers.

Popular Abstract

In the ongoing race towards the first quantum computer with a logical quantum bit (or qubit - a superposition of 0 and 1), large efforts are being put into iteratively building and improving the prototype machines hosing this complex technology. A major part of this endeavor is developing the right tools used to design and predict the behavior of the machines. We have developed a new and efficient simulation method for predicting key metrics of superconducting qubit designs, and these predictions agree well with experimentally measured values.

At the heart of a superconducting quantum computer is a small chip, consisting of a planar superconducting circuit layer deposited on a low-loss dielectric substrate. This chip is hooked up to microwave equipment and cooled down in a dilution refrigerator to ultra-cold temperatures of around 10 mK. In this isolated environment quantum effects can occur with little unintentional perturbation, which allows quantum systems to be controlled and evolved according to our plans through the exchange of microwave pulses. Having efficient tools to accurately design and model these delicate systems is critical in the challenge of making the systems better and bigger.

The prototyping rate of superconducting circuit designs for quantum computers is today limited by several factors. One of these is the time it takes to predict the electromagnetic characteristics of designs through simulation. The solutions that exist today can often be categorized into two distinct types: 1) full-wave simulations, that take a geometrical model of a circuit as an input and solves Maxwell’s equations in its physical volume, and 2) lumped element simulations, that does not take into account the geometry and physical distances, but solves equations given a simplified topological model of a circuit consisting of a network of standard electric circuit elements.

Modeling a full circuit in a single geometrical model can be very costly, as the details required to accurately represent a design become many. Similarly, modeling the full system only as a topology using standard electromagnetic circuit elements would have to be very densely packed, difficult to construct and maintain by hand, and often not precise enough.

Our approach is based on a hybrid of these two types, harnessing their strengths while ridding some of their weaknesses. Instead of a single geometry, we dissect a circuit into pieces that can be accurately modeled one by one. These geometrical pieces are then assembled like a puzzle into a topological model that reconstructs the full circuit. This allows us to efficiently construct models of large chips while maintaining high accuracy in the simulation. By applying powerful mathematical methods from complex analysis and numerical linear algebra, we have implemented a tool chain that computes and predicts behaviors for large systems with little computational effort.

The results are promising. Predicted values such as resonance frequencies, losses and coupling rates are well aligned with what we measure experimentally in the lab at Alice & Bob and come to a fraction of the overall computational cost of conventional methods. However, more exact knowledge about the material properties of the circuits could potentially improve the simulation accuracy. Moreover, extended modeling of nontrivial quantum effects could further improve the understanding of some experimental results. Finally, handling crosstalk between components that are closely positioned in the circuit geometry but not directly connected in the topology is something that could be investigated.

In conclusion, we have shown that our method provides a path towards accurate, large-scale superconducting circuit simulations, using a tool chain that provides a solid foundation for the development of advanced chips for quantum computers.

• author: Hallén, Mattis
• supervisor: Peter Samuelsson; Sebastién Jezouin
• organization: Mathematical Physics; Department of Physics
• alternative title: Mikrovågssimulationer av supraledande kretsar med kopplade S-parametrar och konturintegrering
• course: PHYM01 20241
• year: 2024
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• keywords: Superconducting Circuits, Microwave Simulations, Finite Element Method, Numerical Linear Algebra, Complex Analysis, Quantum Computers
• language: English
• id: 9171442
• date added to LUP: 2024-08-13 08:08:37
• date last changed: 2024-08-13 08:08:37