Transmon

Figure

Description

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 $E_{J} / E_{C}$ . Koch et al. (2007) showed that operating at $E_{J} / E_{C} \sim 50 - 100$ (vs. $\sim 1$ for the CPB) exponentially suppresses charge dispersion — the sensitivity of transition frequencies to offset charge $n_{g}$ — 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 $E_{J} / E_{C}$ ratio is achieved by shunting the Josephson junction with a large external capacitance (typically an interdigitated or pad capacitor). To leading order, the qubit transition obeys $ℏ ω_{01} \approx 8 E_{J} E_{C} - E_{C}$ , and the angular-frequency anharmonicity $α = ω_{12} - ω_{01}$ obeys $ℏ α \approx - E_{C}$ . Thus $α /2 π$ is typically $- 200$ to $- 350 MHz$ — small compared with $ω_{01} /2 π \sim 4 - 6 GHz$ 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).

Hamiltonian

The full transmon Hamiltonian is identical to the Cooper pair box:

$H = 4 E_{C} (\overset{n}{^} - n_{g})^{2} - E_{J} cos \overset{φ}{^}$

where $E_{C} = e^{2} /2 C_{Σ}$ is the charging energy, $E_{J}$ is the Josephson energy, $\overset{n}{^}$ is the Cooper pair number operator, $n_{g}$ is the offset charge, and $\overset{φ}{^}$ is the superconducting phase.

In the transmon regime ( $E_{J} / E_{C} ≫ 1$ ), the charge dispersion of the $m$ -th level scales as:

$ϵ_{m} \propto e^{- 8 E_{J} / E_{C}}$

while the anharmonicity decreases only algebraically:

$ℏ α \approx - E_{C}, α \equiv ω_{12} - ω_{01}$

The energy levels approach those of a weakly anharmonic oscillator (Duffing oscillator):

$E_{m} \approx - E_{J} + 8 E_{J} E_{C} (m + \frac{1}{2}) - \frac{E _{C}}{12} (6 m^{2} + 6 m + 3)$

Motivation

The Cooper pair box suffered from extreme sensitivity to $1/ f$ charge noise, limiting $T_{2}$ to nanoseconds away from the charge degeneracy point. Active feedback to maintain $n_{g} = 1/2$ 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 $n_{g}$ , enabling “set and forget” operation with coherence limited by other mechanisms (dielectric loss, quasiparticles, radiation).

Experimental Status

Original proposal — Koch et al. (2007):

Predicted exponentially suppressed charge dispersion in the large- $E_{J} / E_{C}$ regime while retaining usable anharmonicity
Developed the circuit-QED coupling and dispersive-readout theory for the new design
Established compatibility with fixed-frequency or flux-tunable split-junction variants

Scaling milestones:

Schreier et al. (2008): First experimental demonstration, $T_{1} \sim 7 μ s$ in 2D CPW geometry
Paik et al. (2011): 3D transmon in aluminum cavity, $T_{1} > 60 μ s$
Barends et al. (2014): Google Xmon variant, gate fidelities >99.4% (1Q) and >99.0% (2Q)
Arute et al. (2019): Google Sycamore, 53-qubit quantum computational advantage demonstration
Place et al. (2021): Tantalum-based transmon, $T_{1} > 300 μ s$ in planar geometry
Google Quantum AI (online 2024; archival 2025): Willow-class 105-qubit processor with below-threshold surface-code error correction and median 2Q CZ errors of 0.15–0.3%
Tuokkola et al. (2025): Aalto University, first transmon to break the millisecond coherence barrier — $T_{2, echo} = 1.06 ms$ (record), median $T_{1} = 502 μ s$ , quality factor $\sim 8 \times 1 0^{6}$ ; planar geometry fabricated in academic cleanrooms
Bland et al. (2025): Princeton, tantalum-on-high-resistivity-silicon 2D transmon — $T_{1}$ up to $1.68 ms$ , time-averaged $Q$ up to $1.5 \times 1 0^{7}$ ; 1Q gate fidelity 99.994%; mean $T_{1} = 0.45 \pm 0.18 ms$ across 45 qubits on 9 chips
Marxer et al. (2026): Peer-reviewed simultaneous-operation benchmark on an IQM transmon device — 99.93% CZ fidelity averaged over 40 hours, simultaneous 1Q fidelities above 99.98%, and readout fidelities above 99.94%; PALEA was used to calibrate coherent CZ error and leakage
Gao et al. (2026): 6.8 ns $X$ and $X /2$ gates above 99.98% fidelity with leakage below $2 \times 1 0^{- 5}$ , using recursive DRAG corrections that suppress higher-order multiphoton transitions

State-of-the-art snapshot (October 2026): planar tantalum-on-silicon transmons have reached $T_{1} = 1.68 ms$ and 99.994% 1Q fidelity (Bland et al. 2025), while the longest cited echo dephasing result is $T_{2, echo} = 1.06 ms$ (Tuokkola et al. 2025). For balanced device operation rather than a cherry-picked maximum, Marxer et al. report a 40-hour-averaged CZ fidelity of 99.93% alongside >99.98% simultaneous 1Q and >99.94% readout fidelities. Gao et al. set a complementary speed benchmark of >99.98% fidelity at 6.8 ns.

Key Metrics

Metric	Value	Notes	Fidelity reference
$T_{1}$	100 μs – 1.68 ms	Typical planar: ~100–500 μs; SOTA planar (Ta on Si): 1.68 ms	Bland et al. 2025
$T_{2}$ (echo)	100 μs – 1.06 ms	Often $T_{2} \approx 2 T_{1}$ with echo; SOTA 1.06 ms	Tuokkola et al. 2025
Anharmonicity $α /2 π$	−200 to −350 MHz	$\approx - E_{C} / h$	Koch et al. 2007
$E_{J} / E_{C}$	20–100	Typical operating regime	Koch et al. 2007
1Q gate fidelity	99.9–99.994%	RB; best cited value is 99.994% on Ta-on-Si	Bland et al. 2025
2Q CZ fidelity	99.0–99.93%	99.93% averaged over 40 hours on a tunable-coupler device	Marxer et al. 2026
Readout fidelity	97–99.94%	Dispersive readout with Purcell filtering and near-quantum-limited amplification	Marxer et al. 2026
Gate time (1Q)	6.8–50 ns	6.8 ns $X$ / $X /2$ gates demonstrated above 99.98% fidelity	Gao et al. 2026
Gate time (2Q CZ)	33–400 ns	33 ns total duration for the 99.93% PALEA-calibrated CZ benchmark	Marxer et al. 2026
Transition frequency	4–6 GHz	Representative audited devices operated at 4.80–5.32 GHz	Marxer et al. 2026; Gao et al. 2026
Operating temperature	~10 mK	Dilution-refrigerator mixing-chamber stage	Marxer et al. 2026

Scaling Considerations

Frequency crowding: as qubit count grows, avoiding spectral collisions between qubits, readout resonators, and two-level system (TLS) defects becomes increasingly difficult. This is a primary challenge for large-scale fixed-frequency transmon arrays.
Low anharmonicity: the $∣ α ∣ \sim 200 - 350 MHz$ anharmonicity, while sufficient for current gate speeds, causes persistent leakage to $∣2 ⟩$ and higher levels at faster gate rates. Competitors like fluxonium offer $\sim 10 \times$ higher anharmonicity.
TLS defect losses: two-level system defects at material interfaces (metal–substrate, metal–air, substrate–air) are the dominant source of $T_{1}$ variability and the primary obstacle to uniform coherence across large arrays.
Quasiparticle poisoning: nonequilibrium quasiparticles limit $T_{1}$ and cause parity switching events. Mitigation strategies (normal-metal traps, phonon engineering) are active areas of research.
Residual ZZ coupling: always-on parasitic $ZZ$ interaction between neighboring transmons is a significant source of correlated errors in multi-qubit systems, requiring careful frequency allocation or dynamical decoupling.
Cryogenic I/O overhead: dedicated microwave control, flux-bias, and readout wiring carries heat and consumes routing volume at the mixing-chamber stage. Large-scale systems therefore require aggressive multiplexing, cryogenic control electronics, or alternative interconnects.

References

Original proposal

J. Koch et al., “Charge-insensitive qubit design derived from the Cooper pair box,” PRA 76, 042319 (2007) — arXiv:cond-mat/0703002

Key experimental milestones

J. A. Schreier et al., “Suppressing charge noise decoherence in superconducting charge qubits,” PRB 77, 180502(R) (2008) — first experimental transmon
H. Paik et al., “Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture,” PRL 107, 240501 (2011) — 3D transmon
R. Barends et al., “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500 (2014) — Xmon, surface code threshold
F. Arute et al., “Quantum supremacy using a programmable superconducting processor,” Nature 574, 505 (2019) — Sycamore, 53 qubits
A. P. M. Place et al., “New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds,” Nature Commun. 12, 1779 (2021) — tantalum transmon
Google Quantum AI and Collaborators, “Quantum error correction below the surface code threshold,” Nature 638, 920 (2025) — arXiv:2408.13687 — Willow-class 105-qubit processor
M. Tuokkola et al., “Methods to achieve near-millisecond energy relaxation and dephasing times for a superconducting transmon qubit,” Nature Commun. 16, 5421 (2025) — arXiv:2407.18778 — first ms-scale T₂ in transmon
M. P. Bland et al., “Millisecond lifetimes and coherence times in 2D transmon qubits,” Nature 647, 343 (2025) — arXiv:2503.14798 — Ta-on-Si, T₁ up to 1.68 ms

Gate optimization

F. Motzoi et al., “Simple pulses for elimination of leakage in weakly nonlinear qubits,” PRL 103, 110501 (2009) — DRAG pulse
Z. Li et al., “Error per single-qubit gate below 10⁻⁴ in a superconducting qubit,” npj Quantum Info. 9, 111 (2023) — arXiv:2302.08690 — SOTA 1Q error rate 7.42 × 10⁻⁵ (fidelity 99.9926%)
F. Marxer et al., “Above 99.9% Fidelity Single-Qubit Gates, Two-Qubit Gates, and Readout in a Single Superconducting Quantum Device,” PRX Quantum 7 (2026) — arXiv:2508.16437 — simultaneous high-fidelity 1Q, CZ, and readout benchmark
Y. Gao et al., “Ultrafast single-qubit gates through multi-photon transition removal,” Nature Commun. 17, 10874 (2026) — arXiv:2511.22365 — 6.8 ns low-leakage transmon gates

Linked Papers

Evergreen context

charge-noise-in-superconducting-qubits — why the transmon exists at all
charge-noise-sweet-spot — the design pattern behind large-E_J/E_C operation
josephson-junction-as-nonlinear-element — the circuit element that provides the weak anharmonicity
coherence-time-hierarchy — how to interpret the T_1, T_2, and echo numbers in the metrics table
jaynes-cummings-in-circuits — the transmon usually enters circuit QED through the same one-qubit, one-mode interaction that later gets reduced into dispersive readout and cavity-mediated coupling pictures
dispersive-readout-mechanism — the cQED readout architecture that let the transmon replace destructive tunneling-style superconducting measurements
purcell-protection-via-detuning — why transmons are usually parked far enough from their readout resonators, and then paired with Purcell filters, to keep fast measurement from quietly shortening $T_{1}$

cooper-pair-box-charge-qubit — ancestor (charge-sensitive limit)
circuit-qed — readout and coupling architecture
fluxonium — alternative SC qubit with higher anharmonicity
xmon — planar transmon variant (Google)
gmon — tunable-coupler transmon variant (Google)
dual-rail-superconducting-qubit — encoded transmon-pair qubit
tunable-coupler — suppresses idle coupling while enabling fast entangling gates
qubit-readout — dispersive measurement, Purcell filtering, and amplifier chain

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Description

Hamiltonian

Motivation

Experimental Status

Key Metrics

Scaling Considerations

References

Original proposal

Key experimental milestones

Gate optimization

Linked Papers

Evergreen context

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Transmon

Figure

Description

Hamiltonian

Motivation

Experimental Status

Key Metrics

Scaling Considerations

References

Original proposal

Key experimental milestones

Gate optimization

Linked Papers

Evergreen context

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Table of Contents

Backlinks