Superconducting Qubit Parameter Calculations with scqubits#

This example demonstrates how to use scqubits [GK21] to calculate parameters for superconducting qubits and coupled resonators. The calculations help determine physical parameters like capacitances and coupling strengths needed for component design.

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import numpy as np
import scipy
import scqubits as scq
import skrf
import sympy as sp
from IPython.display import Math, display
from matplotlib import pyplot as plt

from qpdk.models.resonator import cpw_media_skrf, resonator_frequency

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SyntaxWarning: invalid escape sequence '\p'
 /home/runner/work/quantum-rf-pdk/quantum-rf-pdk/.venv/lib/python3.12/site-packages/scqubits/core/zeropi.py: 281

TunableTransmon Qubit Parameters#

A tunable transmon qubit [KYG+07] has a SQUID junction that allows tuning the effective Josephson energy through an external flux. Key parameters include:

  • \(E_{\text{J}_\text{max}}\): Maximum Josephson energy when flux = 0

  • \(E_\text{C}\): Charging energy, related to total capacitance as \(E_\text{C} = \dfrac{e^2}{2C_\Sigma}\)

  • \(d\): Asymmetry parameter between the two junctions in the SQUID

  • \(\phi\): External flux in units of flux quantum \(\phi_0\)

  • \(\alpha\): Anharmonicity, the difference between the \(E_{1→2}\) and \(E_{0→1}\) transition frequencies

See the scqubits documentation for more details.

# Create a tunable transmon with realistic parameters
transmon = scq.TunableTransmon(
    EJmax=40.0,  # Maximum Josephson energy in GHz (typical range: 20-50 GHz)
    EC=0.2,  # Charging energy in GHz (typical range: 0.1-0.5 GHz)
    d=0.1,  # Junction asymmetry (typical range: 0.05-0.2)
    flux=0.0,  # Start at flux = 0 (maximum EJ)
    ng=0.0,  # Offset charge (usually 0 or 0.5)
    ncut=30,  # Charge basis cutoff
)

# Tunable Transmon Parameters
display(
    Math(rf"""
\textbf{{Tunable Transmon Parameters:}} \\
E_{{J,\text{{max}}}} = {transmon.EJmax:.1f}\ \text{{GHz}} \\
E_C = {transmon.EC:.1f}\ \text{{GHz}} \\
d = {transmon.d:.2f} \\
\text{{Flux}} = {transmon.flux:.2f}\ \Phi_0
""")
)

# Calculate energy levels
eigenvals = transmon.eigenvals(evals_count=5)
f01 = eigenvals[1] - eigenvals[0]
f12 = eigenvals[2] - eigenvals[1]
anharmonicity = f12 - f01

display(
    Math(rf"""
\textbf{{Transmon Spectrum:}} \\
0\rightarrow 1\ \text{{frequency}}:\ \ {f01:.3f}\ \text{{GHz}} \\
1\rightarrow 2\ \text{{frequency}}:\ \ {f12:.3f}\ \text{{GHz}} \\
\text{{Anharmonicity, }} \alpha:\ \ {anharmonicity:.3f}\ \text{{GHz}}
""")
)
\[\displaystyle \textbf{Tunable Transmon Parameters:} \\ E_{J,\text{max}} = 40.0\ \text{GHz} \\ E_C = 0.2\ \text{GHz} \\ d = 0.10 \\ \text{Flux} = 0.00\ \Phi_0 \]
\[\displaystyle \textbf{Transmon Spectrum:} \\ 0\rightarrow 1\ \text{frequency}:\ \ 7.795\ \text{GHz} \\ 1\rightarrow 2\ \text{frequency}:\ \ 7.582\ \text{GHz} \\ \text{Anharmonicity, } \alpha:\ \ -0.213\ \text{GHz} \]

Parameter Sweep: Flux Dependence#

Demonstrate how the qubit frequency changes with external flux.

For more details, see the scqubits-examples repository

# Sweep flux and calculate qubit frequency
transmon.plot_evals_vs_paramvals(
    "flux", np.linspace(-1.1, 1.1, 201), subtract_ground=True
)
plt.show()
transmon.plot_wavefunction(esys=None, which=(0, 1, 2, 3, 8), mode="real")
plt.show()
../_images/dab724a14a13cd9f9f6e55d962599beb33974f63fadbe573fe6fd535b5143153.svg ../_images/fc5ad5c86f5dd82cd4c6225bcef8907f1c7703b3d9dd44f47599c91f64029ddd.svg

Coupled Qubit-Resonator System#

When a transmon is coupled to a resonator, the interaction can be described by the Jaynes-Cummings model [SK93] :

(1)#\[\mathcal{H} = \omega_r a^{\dagger} a + \frac{\omega_q}{2} \sigma_z + g (a^{\dagger} \sigma^- + a \sigma^+),\]

where \(\omega_r\) is the resonator frequency, \(\omega_q\) is the qubit frequency, and \(g\) is the coupling strength. The operators \(a^{\dagger}\) and \(a\) are the creation and annihilation operators for the resonator, while \(\sigma_z\), \(\sigma^-\), and \(\sigma^+\) are the Pauli and ladder operators for the qubit.

The coupling strength \(g\) depends on the overlap between the qubit and resonator modes and affects the system dynamics.

# Resonator (Oscillator) Parameters
resonator = scq.Oscillator(
    E_osc=6.5,  # Resonator frequency in GHz (typical range: 4–8 GHz)
)

display(
    Math(rf"""
\textbf{{Resonator Parameters:}} \\
\text{{Frequency}}:\ \ {resonator.E_osc:.1f}\ \text{{GHz}} \\
""")
)

# Create a coupled system using HilbertSpace
hilbert_space = scq.HilbertSpace([transmon, resonator])

# Add interaction between qubit and resonator
# The interaction is typically g(n_qubit * (a + a†)) where n is the charge operator and a is the resonator annihilation operator
g_coupling = 0.15  # Coupling strength in GHz (typical range: 0.05–0.3 GHz)
interaction = scq.InteractionTerm(
    g_strength=g_coupling,
    operator_list=[
        (0, transmon.n_operator),  # (subsystem_index, operator)
        (1, resonator.annihilation_operator() + resonator.creation_operator()),
    ],
)
hilbert_space.interaction_list = [interaction]


display(Math(f"g_{{\\text{{Qubit–Resonator}}}} = {g_coupling:.3f}\\,\\text{{GHz}}"))

# Calculate the coupled system eigenvalues
coupled_eigenvals = hilbert_space.eigenvals(evals_count=10)
print(f"  Coupled system ground state: {coupled_eigenvals[0]:.3f} GHz")
\[\displaystyle \textbf{Resonator Parameters:} \\ \text{Frequency}:\ \ 6.5\ \text{GHz} \\ \]
\[\displaystyle g_{\text{Qubit–Resonator}} = 0.150\,\text{GHz}\]
  Coupled system ground state: -36.054 GHz

Physical (Lumped-element) Parameter Extraction#

From the quantum model, we can extract physical parameters relevant for PDK design.

The coupling strength \(g\) (in the dispersive limit \(g\ll \omega_q, \omega_r\)) can be related to a coupling capacitance \(C_c\) via [Sav23]:

where \(C_\Sigma\) is the total qubit capacitance, \(C_{\text{q}}\) is the capacitance between the qubit pads, and \(C_r\) is the total capacitance of the resonator.

# Calculate physical capacitance
# The charging energy EC = e²/(2C_total), so C_Σ = e²/(2*EC)
EC_joules = transmon.EC * 1e9 * scipy.constants.h  # Convert GHz to Joules

# Total capacitance of the transmon
C_Σ = scipy.constants.e**2 / (2 * EC_joules)

# Josephson inductance - use typical realistic value
# For EJ ~ 40 GHz, typical LJ ~ 0.8-1.0 nH
# LJ scales inversely with EJ: LJ ~ 1.0 nH * (25 GHz / EJ)
LJ_typical = 1.0e-9 * (25.0 / transmon.EJmax)  # Typical scaling

# Compute coupling capacitance from coupling strength
C_c_sym, C_Σ_sym, C_r_sym, omega_q_sym, omega_r_sym, g_sym = sp.symbols(
    "C_c C_Σ C_r omega_q omega_r g", real=True, positive=True
)
equation = sp.Eq(
    g_sym,
    0.5 * (C_c_sym / sp.sqrt(C_Σ_sym * C_r_sym)) * sp.sqrt(omega_q_sym * omega_r_sym),
)
solution = sp.solve(equation, C_c_sym)
C_c_sol = next(sol for sol in solution if sol.is_real and sol > 0)
display(Math(f"C_\\text{{c}} = {sp.latex(C_c_sol)}"))


# Use a typical value for resonator capacitance to ground
resonator_media = cpw_media_skrf(width=10, gap=6)(
    frequency=skrf.Frequency.from_f([5], unit="GHz")
)


def _objective(length: float) -> float:
    """Find resonator length for target frequency using SciPy."""
    freq = resonator_frequency(
        length=length, media=resonator_media, is_quarter_wave=True
    )
    return (freq - resonator.E_osc * 1e9) ** 2  # MSE


length_initial = 4000.0  # Initial guess in µm
result = scipy.optimize.minimize(_objective, length_initial, bounds=[(1000, 20000)])
length = result.x[0]
print(
    f"Optimization success: {result.success}, message: {result.message}, nfev: {result.nfev}"
)
display(
    Math(
        f"\\textrm{{Resonator length at width 10 µm and gap 6 µm}}\\,{resonator.E_osc:.1f}\\,\\mathrm{{GHz}}: {length:.1f}\\,\\mathrm{{µm}}"
    )
)
\[\displaystyle C_\text{c} = \frac{2.0 \sqrt{C_{r}} \sqrt{C_{Σ}} g}{\sqrt{\omega_{q}} \sqrt{\omega_{r}}}\]
Optimization success: True, message: CONVERGENCE: RELATIVE REDUCTION OF F <= FACTR*EPSMCH, nfev: 24
\[\displaystyle \textrm{Resonator length at width 10 µm and gap 6 µm}\,6.5\,\mathrm{GHz}: 4681.9\,\mathrm{µm}\]

The total capacitance of the resonator can be estimated from its characteristic impedance \(Z_0\) and phase velocity \(v_p\) as [GopplFB+08]:

(2)#\[C_r = \frac{l}{\mathrm{Re}(Z_0 v_p)}\]

where \(l\) is the resonator length.

# Get total capacitance to ground of the resonator (in isolation, we disregard effect of coupling to qubits)
C_r = 1 / np.real(resonator_media.z0 * resonator_media.v_p).mean() * length * 1e-6  # F

# Substitute and evaluate numerically
C_c_num = C_c_sol.subs(
    {
        g_sym: g_coupling,
        C_Σ_sym: C_Σ,
        C_r_sym: C_r,
        omega_q_sym: f01,
        omega_r_sym: resonator.E_osc,
    }
).evalf()

display(
    Math(rf"""
\textbf{{Physical Parameters for PDK Design:}}\\
\text{{Total qubit capacitance:}}~{C_Σ * 1e15:.1f}~\mathrm{{fF}}\\
\text{{Josephson inductance:}}~{LJ_typical * 1e9:.2f}~\mathrm{{nH}}\\
\text{{Estimated target qubit–resonator coupling capacitance:}}~{C_c_num * 1e15:.1f}~\mathrm{{fF}}
""")
)
\[\displaystyle \textbf{Physical Parameters for PDK Design:}\\ \text{Total qubit capacitance:}~96.9~\mathrm{fF}\\ \text{Josephson inductance:}~0.62~\mathrm{nH}\\ \text{Estimated target qubit–resonator coupling capacitance:}~11.6~\mathrm{fF} \]

References#

[GK21]

Peter Groszkowski and Jens Koch. Scqubits: a Python package for superconducting qubits. Quantum, 5:583, November 2021. URL: https://quantum-journal.org/papers/q-2021-11-17-583/ (visited on 2025-09-18), doi:10.22331/q-2021-11-17-583.

[GopplFB+08]

M. Göppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M. Fink, P. J. Leek, G. Puebla, L. Steffen, and A. Wallraff. Coplanar waveguide resonators for circuit quantum electrodynamics. Journal of Applied Physics, 104(11):113904, December 2008. URL: https://pubs.aip.org/jap/article/104/11/113904/145728/Coplanar-waveguide-resonators-for-circuit-quantum (visited on 2025-09-18), doi:10.1063/1.3010859.

[KYG+07]

Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, Alexandre Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. Charge-insensitive qubit design derived from the Cooper pair box. Physical Review A, 76(4):042319, October 2007. URL: https://link.aps.org/doi/10.1103/PhysRevA.76.042319 (visited on 2025-09-04), doi:10.1103/PhysRevA.76.042319.

[Sav23]

Niko Savola. Design and modelling of long-coherence qubits using energy participation ratios. Master's thesis, Aalto University, February 2023. URL: http://urn.fi/URN:NBN:fi:aalto-202305213270.

[SK93]

Bruce W. Shore and Peter L. Knight. The Jaynes–Cummings Model. Journal of Modern Optics, 40(7):1195–1238, July 1993. URL: http://www.tandfonline.com/doi/abs/10.1080/09500349314551321 (visited on 2023-02-01), doi:10.1080/09500349314551321.