Simulation Framework for the Automated Search of Optimal Parameters Using Physically Relevant Metrics in Nonlinear Superconducting Quantum Circuits

TL;DR

The paper addresses the computational challenge of optimizing highly nonlinear superconducting circuits with large design spaces. It introduces JosephsonCircuitOptimizer.jl (JCO), a two-stage framework built on JosephsonCircuits.jl that uses harmonic balance for steady-state, plus Bayesian optimization with Gaussian processes to minimize a device metric $\mathcal{M}(S(\mathbf{p}))$ over the design space $\mathcal{P}$ and then maximize the performance $\mathcal{F}(X(\mathbf{q}; \mathbf{p}^*))$ over the working space $\mathcal{Q}$ using driven-X-parameters $X(\mathbf{q}; \mathbf{p}^*)$. The method is demonstrated on a SNAIL-based JTWPA in the three-wave mixing regime, achieving a phase-matched design with approximately 20 dB gain at an optimized working point $\mathbf{q}^*$, after substantial linear and nonlinear simulations. The framework provides a scalable, reproducible workflow for designing quantum-limited amplifiers and related superconducting devices, with planned extensions to Kerr corrections and dimensionality reduction to further enhance performance and generality.

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 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.

Figures (5)

• Figure 1: Workflow of the JCO framework. The framework works as follows: (1) Linear simulations are performed across the device parameter space $\mathcal{P}$, uniformly sampled. Each configuration is assigned a metric value $\mathcal{M}$. (2) Bayesian optimization refines the sampling in regions surrounding local minima of $\mathcal{M}$ identified in the previous stage, in order to find the optimal parameters $\mathbf{p}^*$ minimizing $\mathcal{M}$. (3) Nonlinear simulations are performed by sweeping over the space of physical quantities $\mathcal{Q}$ (input frequencies and tone amplitudes) with $\mathbf{p}^*$ fixed. The process can be repeated to apply nonlinear corrections. The final outcome is the optimal device parameters $\mathbf{p}^*$ and working point $\mathbf{q}^*$ that maximizes the performance function $\mathcal{F}$. In this example, the first two stages explore two parameters taking 14 and 10 possible values, yielding $14 \times 10$ uniformly distributed configurations within $\mathcal{P}$. Each point $\mathbf{p}$ on the graph corresponds to a distinct circuit configuration, color-coded according to its metric value $\mathcal{M}$. In the third stage, two physical quantities are varied with $7 \times 8$ configurations in the $\mathcal{Q}$ space. Each point represents a different working point $\mathbf{q}$, color-coded according to the performance function $\mathcal{F}$.
• Figure 2: (a) Correlation matrix of the device parameters and (b) 1D histogram of the parameter count weighted by the device-specific metric, for the use case presented in Section \ref{['ch:3']} in a larger device parameter space. The description of each parameter is provided in Fig. \ref{['fig:snail-scheme']}. The correlation matrix is computed over a subset of configurations filtered by a metric below a specified cutoff. These visualizations highlight parameter inter-dependencies and indicate which values contribute most to optimal circuit performance. After the optimization stage, the red lines define the best parameter configuration.
• Figure 3: Schematic of a SNAIL-based JTWPA. The device is composed of $N = N_{\text{macro}} \times P$ cells arranged in a 1D chain. Each macrocell $N_{\text{macro}}$ consists of multiple unit cells: $P - 1$ unloaded cells and one loaded cell, where $P$ denotes the loading pitch. Each unit cell includes a SNAIL loop formed by two parallel branches. The first branch hosts a single small JJ with area $A_J$ and critical current density $\rho_{Ic}$, yielding a critical current $I_c = \rho_{Ic} A_J$. The second branch contains three larger JJs, each with area $A_J/\alpha$, where $\alpha \in (0,1)$ is the asymmetry parameter. The loop is threaded by an external DC magnetic flux $\Phi_{\mathrm{ext}}$, delivered via a dedicated flux line, which tunes the effective nonlinearity of the structure. Each cell is shunted to ground via a gate capacitance $C_g$, determined by the dielectric thickness $t$. Loaded and unloaded cells differ in their inductance and capacitance to modify the impedance and dispersion properties of the device. Particularly, $L_\ell$ is the ratio between the inductance of the small junction inductance of the loaded and unloaded cell, which is proportional to the ratio between the small junction areas. $C_\ell$ is the ratio between the ground capacitances of the loaded and unloaded cell.
• Figure 4: Dispersion relation of the optimized SNAIL-based JTWPA, obtained by minimizing the device-specific metric to an optimal circuit configuration $\mathbf{p}^*$. The selected optimal design parameters are $\mathbf{p}^* = \{ A_J = 0.49\,µ m^2, \rho_{Ic} = 0.9\,µ A/µ m^2, \alpha = 0.23, t = 9\,\mathrm{nm}, L_\ell = 1.5, C_\ell = 1, P = 3 \}$. The orange line shows the linear relation $k_\mathrm{p} = 2k_\mathrm{p/2}$. Vertical black and dashed lines indicate $f_\mathrm{p}$ and $f_\mathrm{p}/2$, while blue lines define the signal bandwidth $f_\mathrm{BW}$.
• Figure 5: Gain profile of the SNAIL-based JTWPA at the optimal working point $\mathbf{q}^*$, obtained by maximizing the performance function $\mathcal{F}(\mathbf{q}, \mathbf{p}^*)$. The selected working point is at $A_{\mathrm{DC}}^{\mathrm{NL}} = 212\,µ A$ and $A_{\mathrm{p}}^{\mathrm{NL}} = 0.1\,µ A$. At this operating condition, the amplifier achieves a gain of approximately 20 dB. The vertical lines correspond to those in Fig. \ref{['fig:disp']}.