Qubit ZooDescription
The transmon (short for “transmission-line shunted plasma oscillation qubit”) is a superconducting charge qubit derived from the Cooper pair box by dramatically increasing the ratio . Koch et al. (2007) showed that operating at (vs. for the CPB) exponentially suppresses charge dispersion — the sensitivity of transition frequencies to offset charge — while reducing anharmonicity only as a weak power law. This tradeoff is the central design insight: charge noise immunity improves exponentially while the penalty in anharmonicity is manageable.
The large ratio is achieved by shunting the Josephson junction with a large external capacitance (typically an interdigitated or pad capacitor). The qubit transition frequency is set by lithographic design, and the anharmonicity is typically to — small compared to the transition frequency () but large enough for selective microwave control.
The transmon is almost always operated inside a circuit QED architecture: coupled to a coplanar waveguide or 3D cavity resonator for dispersive readout and Purcell-filtered spontaneous emission suppression. It is the dominant qubit modality in current superconducting quantum processors (IBM, Google, Rigetti, IQM, OQC).
Figure
Hamiltonian
The full transmon Hamiltonian is identical to the Cooper pair box:
where is the charging energy, is the Josephson energy, is the Cooper pair number operator, is the offset charge, and is the superconducting phase.
In the transmon regime (), the charge dispersion of the -th level scales as:
while the anharmonicity decreases only algebraically:
The energy levels approach those of a weakly anharmonic oscillator (Duffing oscillator):
Motivation
The Cooper pair box suffered from extreme sensitivity to charge noise, limiting to nanoseconds away from the charge degeneracy point. Active feedback to maintain was impractical at scale. The transmon eliminates this problem by operating in a regime where the transition frequency is essentially flat as a function of , enabling “set and forget” operation with coherence limited by other mechanisms (dielectric loss, quasiparticles, radiation).
Key Findings
- Charge dispersion is exponentially suppressed: at , the charge dispersion of the 0→1 transition is , negligible compared to linewidths.
- Anharmonicity is sufficient for high-fidelity single-qubit gates with shaped pulses (DRAG).
- Dispersive readout via circuit QED provides QND measurement with high fidelity.
- The design is compatible with fixed-frequency operation (simpler fabrication, no flux noise) or frequency-tunable variants (split-junction transmon with flux control for two-qubit gates).
- State-of-the-art devices achieve in 3D cavities and in planar geometries.
Key Metrics
| Metric | Value | Notes | Fidelity reference |
|---|---|---|---|
| 100–500 μs | Planar: ~100–300 μs; 3D cavity: >500 μs | Koch 2007 | |
| (echo) | 100–500 μs | Often with echo | Place et al. 2021 |
| Anharmonicity | −200 to −350 MHz | Koch 2007 | |
| 20–100 | Typical operating regime | Koch 2007 | |
| 1Q gate fidelity | 99.9–99.99% | Randomized benchmarking; SOTA 99.9926% (Li et al. 2023) | Arute et al. 2019, Li et al. 2023 |
| 2Q gate fidelity (CZ/CR) | 99.0–99.9% | Cross-resonance or tunable coupler; Sycamore 99.4% CZ, Willow 99.7–99.85% | Arute et al. 2019, Google Willow 2024 |
| Readout fidelity | 97–99.9% | Dispersive, with Purcell filter + JPA | Arute et al. 2019 |
| Gate time (1Q) | 20–50 ns | DRAG pulse | Arute et al. 2019 |
| Gate time (2Q) | 100–400 ns | Depends on gate scheme | Arute et al. 2019 |
| Transition frequency | 4–6 GHz | Design-tunable | — |
| Operating temperature | 10–20 mK | Dilution refrigerator | — |
| Chip footprint per qubit | ~300 μm × 300 μm | Pad + junction + readout resonator | — |
Extracted Figures
