Tunable Coupler

Figure

Description

A tunable coupler is a superconducting coupling element — typically a frequency-tunable transmon or a flux-tunable resonator — placed between two computational qubits to enable fast, high-fidelity ON/OFF switching of qubit-qubit interactions. By modulating the coupler’s frequency via an external flux bias, the effective coupling $g_{\text{eff}}$ between the computational qubits can be tuned from a large value (ON, enabling fast entangling gates) to near zero (OFF, suppressing unwanted always-on $ZZ$ crosstalk during idle periods).

The coupler mediates the interaction between qubits through virtual excitation exchange: when the coupler frequency is far detuned from both qubits, it mediates a second-order coupling. The direct capacitive coupling between qubits and the coupler-mediated coupling have opposite signs, and at a specific coupler frequency they cancel exactly, yielding $g_{\text{eff}}=0$. This zero-coupling condition is the key innovation — it enables idling with negligible crosstalk.

Two-qubit gates (CZ, iSWAP, or $\sqrt{\text{iSWAP}}$) are implemented by briefly tuning the coupler to activate the interaction, performing a controlled phase accumulation or excitation swap, then returning to the OFF point. Gate times of 10–60 ns have been demonstrated with fidelities exceeding 99.5%.

Tunable couplers are the enabling technology behind Google’s Sycamore (2019) and Willow (2024) processors, and IBM’s Heron architecture.

Hamiltonian

The system of two qubits ($Q_1$, $Q_2$) coupled via a tunable coupler ($C$) has the Hamiltonian:

$H={\sum }_{i\in \{1,2,C\}}{\omega }_i{\hat{a}}_i^{†}{\hat{a}}_i+\frac{{\alpha }_i}{2}{\hat{a}}_i^{†2}{\hat{a}}_i^2+{\sum }_{i<j}{g}_{ij}({\hat{a}}_i^{†}{\hat{a}}_j+{\hat{a}}_i{\hat{a}}_j^{†})$

where $g_{ij}$ are the bare capacitive couplings. Adiabatic elimination of the coupler (valid when $|{\omega }_C-{\omega }_{1,2}|\gg g_{1C},g_{2C}$) gives an effective qubit-qubit coupling:

$g_{\text{eff}}=g_{12}+\frac{g_{1C}{g}_{2C}}{2}\left(\frac{1}{{\omega }_1-{\omega }_C}+\frac{1}{{\omega }_2-{\omega }_C}\right)$

The zero-coupling condition $g_{\text{eff}}=0$ is satisfied when the coupler-mediated term exactly cancels the direct coupling $g_{12}$. This occurs at a specific coupler frequency ${\omega }_C^{\text{off}}$.

The residual $ZZ$ interaction at the OFF point is:

${\zeta }_{ZZ}=\frac{2g_{\text{eff}}^2{\alpha }_1{\alpha }_2}{({\Delta }_{12}^2-{\alpha }_1^2)({\Delta }_{12}^2-{\alpha }_2^2)}+O(g_{\text{eff}}^4)$

which can be tuned below $10\text{kHz}$ at the zero-coupling point.

Motivation

Fixed-frequency transmon qubits coupled by fixed capacitors suffer from always-on $ZZ$ coupling (${\zeta }_{ZZ}\sim 50-500\text{kHz}$) that accumulates correlated phase errors during idle time. Frequency-tunable qubits can reduce this but introduce flux-noise sensitivity. The tunable coupler resolves this tradeoff: the computational qubits can remain at fixed (or nearly fixed) frequencies, preserving their coherence, while the coupler provides the dynamic control needed for entangling gates and idle isolation. This enables simultaneous high coherence, fast gates, and low crosstalk — the three requirements for scalable superconducting quantum processors.

Key Findings

• Yan et al. (2018) demonstrated the first tunable-coupler architecture with ON/OFF ratio $>1000$ and CZ gate fidelity 99.1%.
• Sung et al. (2021) achieved CZ gate fidelity of 99.76% with 60 ns gate time using a parametrically modulated coupler.
• Google Sycamore (2019) used tunable couplers for 86 qubit pairs, achieving median CZ fidelity 99.4%.
• Google Willow (2024) improved to 99.7–99.85% median CZ fidelity, enabling below-threshold surface code operation.
• IBM Heron (2024) adopted tunable couplers after transitioning from fixed-coupling cross-resonance gates, achieving 99.5% 2Q fidelity.
• Residual $ZZ$ coupling suppressed to $<10\text{kHz}$ at the OFF point, enabling high-fidelity simultaneous single-qubit operations.

Key Metrics

Metric	Value	Notes	Fidelity reference
ON/OFF coupling ratio	>1000:1	$g_{\text{eff}}^{\text{on}}/g_{\text{eff}}^{\text{off}}$	Yan et al. 2018
CZ gate fidelity	99.5–99.85%	Randomized benchmarking; Willow SOTA	Google Willow 2024
iSWAP gate fidelity	99.5–99.7%	Used in Sycamore-type circuits	Arute et al. 2019
Gate time (2Q)	10–60 ns	Coupler-activated CZ or iSWAP	Sung et al. 2021
Residual $ZZ$ at OFF	<10 kHz	At zero-coupling operating point	Yan et al. 2018
Coupler frequency range	4–8 GHz	Flux-tunable, typically above qubit frequencies	Yan et al. 2018
Coupler $T_1$	5–20 μs	Shorter than qubits; not a computational element	—
Operating temperature	10–20 mK	Dilution refrigerator	—

Linked Papers

• yan-2018-tunable-coupler
• sung-2021-parametric-coupler

Related Entries

• transmon
• circuit-qed
• fluxonium
• gmon