Notebooks

These notebooks demonstrate integration with relevant tools for design and simulation of superconducting quantum devices. Each notebook addresses a different stage of the design flow and uses a different simulation method, allowing users to choose the tools that best fit their needs.

Why multiple simulation approaches?

Designing a superconducting quantum chip involves physics at many scales. No single simulation tool covers all of them efficiently, so a practical design flow combines several complementary methods [BGGW21, KKY+19].

The notebooks in this collection are organized around four simulation categories:

1. Scattering-parameter (S-parameter) circuit models — fast, analytical or semi-analytical models for passive microwave components.

2. FEM-based electromagnetic simulations — full-wave or quasi-static solvers that capture geometry-dependent effects beyond simple analytical formulas.

3. Hamiltonian analysis — numerical or perturbative diagonalization of the quantum Hamiltonian to extract qubit parameters such as frequency, anharmonicity, and dispersive shift.

4. Pulse-level simulations — time-domain simulation of control pulses acting on the quantum system, including gate fidelity, leakage, and decoherence.

Any of these methods can be wrapped in an automated optimization loop (e.g. with Optuna) for design-space exploration.

Where each method fits in the design flow

The typical workflow when creating a chip with qpdk / gdsfactory can be summarized as follows. Each stage may loop back to earlier stages as the design is refined.

        flowchart TB
    A["Physical requirements\n(qubit frequency, coupling, T₁, …)"]
    B["Hamiltonian / perturbation analysis\n(map requirements → circuit parameters)"]
    C["Circuit / S-parameter models\n(design passive components)"]
    D["Layout with gdsfactory\n(draw the chip in qpdk)"]
    E["FEM verification\n(validate geometry with a full-wave solver)"]
    F["Pulse-level simulation\n(predict gate performance)"]
    G["Fabrication & measurement"]
    A --> B --> C --> D --> E --> F --> G
    F -.-> A
    E -.-> C
    

S-parameter circuit models

S-parameter circuit models treat microwave components as linear, frequency-dependent networks described by their scattering matrices. In qpdk these models are implemented with JAX and composed into circuits using SAX [BGGW21, GopplFB+08].

Typical use cases:

• Choosing coplanar-waveguide (CPW) resonator, capacitor, and coupling structure geometries to meet target parameters.

• Predicting resonance frequencies and quality factors of passive components.

• Simulating complete test chips with many resonators from a gdsfactory netlist.

Notebooks:

• QPDK Models — Comprehensive overview of all S-parameter models available in the qpdk model library (capacitors, inductors, waveguides, couplers, resonators).

• Circuit Simulation with QPDK — Builds and simulates composite circuits with SAX, starting from individual components and assembling a quarter-wave resonator.

• Resonator frequency estimation models — Compares analytical resonance-frequency estimates with SAX circuit simulations.

• SAX Simulation of a Resonator Test Chip — Loads a multi-resonator test chip from a YAML netlist and simulates the full S₂₁ response with SAX.

• Model Comparison to Qucs-S — Validates qpdk S-parameter models against Qucs-S reference data for various passive components.

• JAX Backend Comparison for Quantum Circuit Simulation — Benchmarks SAX circuit evaluation on CPU, GPU (CUDA), and NPU (OpenVINO) backends.

FEM-based electromagnetic simulations

Finite-element method (FEM) and full-wave solvers discretize Maxwell’s equations over the physical geometry of the device. They capture effects such as radiation, surface currents, and substrate modes that analytical models may miss [GopplFB+08].

Typical use cases:

• Extracting characteristic impedance and effective permittivity of CPW cross-sections.

• Computing eigenmode frequencies and quality factors of resonators from their physical geometry.

• Running driven-modal (port-based) S-parameter simulations of capacitors and other structures.

• Optimizing component geometry against a target specification (e.g. a desired capacitance value).

Notebooks:

• Q2D Cross-Section Impedance of a Coplanar Waveguide — Uses the Ansys Q2D quasi-static solver to extract CPW impedance from the cross-section geometry and compares the result with analytical conformal-mapping estimates.

• HFSS Eigenmode Simulation of a CPW Resonator — Eigenmode analysis of a meandering CPW resonator in Ansys HFSS to find resonant frequencies and Q-factors.

• HFSS Driven Modal Simulation of an Interdigital Capacitor — Driven-modal S-parameter simulation of an interdigital capacitor in Ansys HFSS.

• Optuna Optimization of Interdigital Capacitor — Couples Optuna optimization with the Palace FEM solver to optimize an interdigital capacitor towards a target capacitance.

For additional electromagnetic simulation examples using Palace and Meep with gdsfactory and gdsfactory+, see the gsim documentation.

Hamiltonian analysis

Superconducting qubits are nonlinear quantum circuits whose behavior is governed by a Hamiltonian. Diagonalizing this Hamiltonian yields qubit frequencies, anharmonicities, and coupling strengths that feed back into the layout design [KYG+07, BGGW21].

Typical use cases:

• Computing transmon qubit frequency (\(\omega_{01}\)) and anharmonicity (\(\alpha\)) from Josephson energy \(E_J\) and charging energy \(E_C\).

• Calculating the dispersive shift \(\chi\) of a transmon–resonator system for readout design.

• Translating Hamiltonian-level parameters into physical layout dimensions.

Notebooks:

• Dispersive Shift of a Transmon–Resonator System with scQubits — Full numerical diagonalization of the transmon–resonator Hamiltonian with scQubits [GK21], compared against analytical perturbation theory.

• Dispersive Shift of a Transmon–Resonator System with Pymablock — Perturbative block-diagonalization with Pymablock [ADMK+25] to compute the dispersive shift symbolically and map the result to layout parameters.

• Transmon Qubit Design with NetKet — Transmon Hamiltonian analysis with NetKet (exact diagonalization and variational methods) including extraction of qubit parameters and conversion to layout dimensions.

Pulse-level simulations

Once the qubit parameters are known, pulse-level simulations model the time-domain evolution of the quantum state under microwave control pulses. These simulations predict gate fidelities, leakage to non-computational states, and the impact of decoherence [LCM22].

Typical use cases:

• Simulating single-qubit gates (e.g. X, Y) and two-qubit gates (e.g. Bell-state preparation) with realistic pulse shapes.

• Estimating leakage to higher transmon levels.

• Evaluating the effect of \(T_1\) and \(T_2\) decoherence on gate fidelity.

• Connecting physical layout parameters (frequency, anharmonicity) to gate performance.

Notebooks:

• Pulse-Level Simulation of Superconducting Qubits with QuTiP-QIP — Pulse-level simulation of transmon gates with QuTiP-QIP [LCM22], including population dynamics, leakage analysis, and decoherence effects.

Summary table

Notebook	Category	Key tools
QPDK Models	S-parameter models	qpdk, JAX
Circuit Simulation with QPDK	S-parameter models	SAX, JAX
Resonator frequency estimation models	S-parameter models	SAX
SAX Simulation of a Resonator Test Chip	S-parameter models	SAX, gdsfactory
Model Comparison to Qucs-S	S-parameter models	SAX, Qucs-S
JAX Backend Comparison for Quantum Circuit Simulation	S-parameter models	SAX, JAX, OpenVINO
Q2D Cross-Section Impedance of a Coplanar Waveguide	FEM electromagnetics	Ansys Q2D, PyAEDT
HFSS Eigenmode Simulation of a CPW Resonator	FEM electromagnetics	Ansys HFSS, PyAEDT
HFSS Driven Modal Simulation of an Interdigital Capacitor	FEM electromagnetics	Ansys HFSS, PyAEDT
Optuna Optimization of Interdigital Capacitor	FEM optimization	Optuna, Palace
Dispersive Shift of a Transmon–Resonator System with scQubits	Hamiltonian analysis	scQubits
Dispersive Shift of a Transmon–Resonator System with Pymablock	Hamiltonian analysis	Pymablock, SymPy
Transmon Qubit Design with NetKet	Hamiltonian analysis	NetKet, JAX
Pulse-Level Simulation of Superconducting Qubits with QuTiP-QIP	Pulse-level simulation	QuTiP-QIP, JAX