JCO: Optimization Framework for Nonlinear Superconducting Circuits Using a Lumped-Element Approach and Harmonic Balance

Abstract—In this contribution, we present JosephsonCircuitsOptimizer. jl (JCO), a simulation and optimization framework based on the JosephsonCircuits.jl library for Julia. It models superconducting circuits that include Josephson junctions (JJs) and other nonlinear elements within a lumped-element approach, leveraging harmonic balance—a frequency-domain technique that provides a computationally efficient alternative to traditional time-domain simulations. JCO automates the evaluation of optimal circuit parameters by implementing Bayesian optimization with Gaussian processes through a device-specific metric and identifying the optimal working point to achieve a defined performance function. This makes it well-suited for circuits with strong nonlinearity and a high-dimensional set of coupled design parameters. To demonstrate its capabilities, we focus on optimizing a Josephson Traveling-Wave Parametric Amplifier (JTWPA) based on Superconducting Nonlinear Asymmetric Inductive eLements (SNAILs), operating in the three-wave mixing (3WM) regime. The device consists of an array of unit cells, each containing a loop with multiple JJs, that amplifies weak quantum signals near the quantum noise limit. By integrating efficient simulation and optimization strategies, the framework supports the systematic development of superconducting circuits for a broad range of applications.