Transmon Qubits

Superconducting transmon qubits are charge-insensitive artificial atoms formed by a Josephson junction shunted by a large capacitance. The transmon spectrum is weakly anharmonic, so at least one level beyond the computational subspace must be included to capture leakage.

Single Transmon

Hamiltonian

In the rotating frame at frequency $\omega$, the transmon Hamiltonian truncated to $d$ levels is:

\[H = (\omega - \omega_{\text{frame}})\, a^\dagger a \;-\; \frac{\delta}{2}\, a^{\dagger 2} a^2 \;+\; u_x(t)\,(a + a^\dagger) \;+\; u_y(t)\, i(a^\dagger - a)\]

where $\delta$ is the anharmonicity ($\approx 200$ MHz for typical transmons) and $a$ is the annihilation operator truncated to $d$ levels. In the default rotating frame $\omega_{\text{frame}} = \omega$, the first term vanishes.

The energy spectrum is $E_n = n\omega + \tfrac{n(n-1)}{2}\,\delta$, giving transition frequencies $\omega_{01} = \omega$ and $\omega_{12} = \omega + \delta$.

Construction

using Piccolo

# 3-level transmon in the rotating frame (default)
sys = TransmonSystem(
    levels = 3,           # Truncation level (≥2)
    δ = 0.2,              # Anharmonicity (GHz)
    ω = 4.0,              # Qubit frequency (GHz) — cancels in rotating frame
    drive_bounds = [0.2, 0.2],  # Max amplitude for X, Y drives (GHz)
)
sys.levels, sys.n_drives

(3, 2)

Parameters

Lab Frame Variants

The lab_frame_type keyword selects the lab-frame Hamiltonian model:

Example: X Gate on a 3-Level Transmon

sys = TransmonSystem(levels = 3, δ = 0.2, drive_bounds = [0.2, 0.2])

# Goal: X gate in the computational subspace {|0⟩, |1⟩}
U_goal = EmbeddedOperator(:X, sys)

T, N = 20.0, 100
times = collect(range(0, T, length = N))
pulse = ZeroOrderPulse(0.05 * randn(2, N), times)
qtraj = UnitaryTrajectory(sys, pulse, U_goal)

qcp = SmoothPulseProblem(qtraj, N; Q = 100.0)
solve!(qcp; max_iter = 100, print_level = 1)
fidelity(qcp)

0.999999997239619

Multi-Transmon Systems

Hamiltonian

$N$ transmon qubits with pairwise dipole coupling in the rotating frame:

\[H = \sum_{i=1}^{N} \left[ (\omega_i - \omega_{\text{frame}})\, a_i^\dagger a_i - \frac{\delta_i}{2}\, a_i^{\dagger 2} a_i^2 \right] + \sum_{i < j} g_{ij} \left( a_i\, a_j^\dagger + a_i^\dagger\, a_j \right)\]

The coupling $g_{ij}(a_i a_j^\dagger + a_i^\dagger a_j)$ is the rotating-wave approximation of the transmon dipole interaction. In the lab frame the full form $g_{ij}(a_i + a_i^\dagger)(a_j + a_j^\dagger)$ is used instead.

Construction

sys_multi = MultiTransmonSystem(
    [4.0, 4.1],               # Qubit frequencies ωᵢ (GHz)
    [0.2, 0.22],              # Anharmonicities δᵢ (GHz)
    [0.0 0.01; 0.01 0.0];     # Coupling matrix gᵢⱼ (GHz)
    drive_bounds = 0.2,
    levels_per_transmon = 3,
)

CompositeQuantumSystem: levels = 9, n_drives = 4

Returns a CompositeQuantumSystem with tensor-product Hilbert space $\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$.

Parameters

Transmon-Cavity (Circuit QED)

Hamiltonian

A transmon dispersively coupled to a microwave cavity:

\[H = \tilde{\Delta}\, \hat{b}^\dagger \hat{b} - \chi\, \hat{a}^\dagger \hat{a}\, \hat{b}^\dagger \hat{b} - \chi'\, \hat{b}^{\dagger 2} \hat{b}^2\, \hat{a}^\dagger \hat{a} - K_q\, \hat{a}^{\dagger 2} \hat{a}^2 - K_c\, \hat{b}^{\dagger 2} \hat{b}^2\]

where $\hat{a}$ ($\hat{b}$) annihilates a transmon (cavity) excitation, $\chi$ is the dispersive shift, $K_q, K_c$ are self-Kerr nonlinearities, and $\tilde{\Delta} = \chi/2$.

The four drives are:

1. $\hat{a} + \hat{a}^\dagger$ — real transmon drive
2. $i(\hat{a}^\dagger - \hat{a})$ — imaginary transmon drive
3. $\hat{b} + \hat{b}^\dagger$ — real cavity drive
4. $i(\hat{b}^\dagger - \hat{b})$ — imaginary cavity drive

Construction

sys_cqed = TransmonCavitySystem(
    qubit_levels = 4,
    cavity_levels = 12,
    χ = 2π * 32.8e-6,       # Dispersive shift (GHz)
    K_q = 2π * 193e-3 / 2,  # Qubit self-Kerr (GHz)
    K_c = 2π * 1e-9 / 2,    # Cavity self-Kerr (GHz)
    drive_bounds = [0.5, 0.5, 1.0, 1.0],
)
sys_cqed.levels, sys_cqed.n_drives

(48, 4)

Parameters

Best Practices

1. Include Enough Levels

For single-qubit gates: $d \geq 3$ (captures leakage to $|2\rangle$). For high-fidelity ($F > 0.9999$): $d \geq 4$.

sys_3 = TransmonSystem(levels = 3, δ = 0.2)
sys_4 = TransmonSystem(levels = 4, δ = 0.2)
sys_3.levels, sys_4.levels

(3, 4)

2. Use Realistic Parameters

3. Use EmbeddedOperator for Gates

When $d > 2$, define the gate in the computational subspace:

U_goal = EmbeddedOperator(:X, sys)  # X on {|0⟩, |1⟩}, identity on |2⟩, ...

References

• Koch et al., "Charge-insensitive qubit design derived from the Cooper pair box," Phys. Rev. A76, 042319 (2007)
• Blais et al., "Circuit quantum electrodynamics," Rev. Mod. Phys.93, 025005 (2021)

See Also

• Quantum Systems Overview — General system API
• Leakage Suppression — Handling higher levels
• Operators — EmbeddedOperator details

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