Coupler Frequency
How a Tunable Coupler Uses Frequency to Gate Qubit Interaction
Modern superconducting processors place a third circuit element, the coupler, between every pair of nearest-neighbor data qubits. The coupler is itself a small transmon or SQUID-based resonator with a flux-tunable Josephson inductance, so a current in a nearby flux line shifts its resonant frequency over a range of several gigahertz. That tunable frequency is the lever the control system uses to turn the two-qubit interaction on and off. When the coupler is parked far above the qubit band, the qubits are effectively decoupled; when it is pulsed toward the band, they exchange energy rapidly. This architecture decouples idle qubits far better than fixed direct coupling, which is why it dominates large-scale processors.
The physics rests on two competing coupling channels. There is a small direct capacitive coupling between the two qubits, and there is a larger virtual coupling routed through the coupler that scales as the product of the two qubit-coupler couplings divided by the coupler detuning. These two terms carry opposite sign, so a particular coupler frequency makes them cancel exactly. That cancellation point is the idle, or off, frequency, and finding it is a per-device calibration: the coupler flux is swept while the residual exchange and static ZZ shift are measured by Ramsey or conditional-phase experiments.
During an entangling gate the coupler frequency is pulsed downward, breaking the cancellation and ramping the net coupling up to tens of megahertz. The flux-pulse edges must be smooth (error-function, net-zero, or Slepian shapes are common) so the coupler mode is not excited and leakage into non-computational states stays low. The same hardware also supports parametric gates, where the coupler frequency is modulated sinusoidally to bridge the qubit detuning rather than moved as a step.
Idle Point Versus Interaction Point
The two frequencies that matter most are the idle point and the interaction point. At the idle point the static ZZ rate is minimized, often to the 1 to 10 kHz level, so neighboring qubits accumulate negligible conditional phase while sitting in memory. At the interaction point, reached by a fast flux pulse, J is large enough to complete a controlled-Z in roughly 30 to 60 ns including ramps. The frequency difference between these two operating points is usually 1 to 2 GHz.
Coupler Frequency Governing Relations
fc(Φ) ≈ fc,max × √|cos(π Φ / Φ0)|
Effective exchange coupling (two paths):
J ≈ g12 + (g1 × g2 / 2) × [ 1/(Δ1c) + 1/(Δ2c) ]
Off condition (idle frequency):
g12 ≈ − (g1 × g2 / 2) × [ 1/(Δ1c) + 1/(Δ2c) ] ⇒ J ≈ 0
Where fc = coupler frequency, Φ = applied flux, Φ0 = flux quantum, g12 = direct qubit-qubit coupling, g1,g2 = qubit-coupler couplings, Δic = fqubit,i − fc = coupler detuning. Example: g1=g2≈100 MHz, Δ≈−2 GHz, g12≈5 MHz → tuning fc shifts Δ and drives J through zero.
Operating-Point Comparison
| Operating Point | Coupler Frequency | Coupler-Qubit Detuning | Effective J/2π | Static ZZ Rate | Purpose |
|---|---|---|---|---|---|
| Idle / off | 6.5 to 7.5 GHz | 1.5 to 2.5 GHz above qubits | < 0.1 MHz | 1 to 10 kHz | Memory, single-qubit gates |
| CZ interaction | 5.5 to 6.5 GHz | 0.5 to 1.5 GHz | 10 to 30 MHz | Active (pulsed) | Controlled-Z entangling gate |
| iSWAP / parametric | Modulated ±0.3 GHz | Time-averaged | 2 to 15 MHz | Drive dependent | Exchange and parametric gates |
| Sweet spot (max fc) | 7.5 to 8.5 GHz | > 2.5 GHz | ≈ 0.1 to 1 MHz residual | tens of kHz | Flux-noise-insensitive parking (above the null) |
Frequently Asked Questions
At what coupler frequency does the effective qubit-qubit coupling turn off?
The net coupling vanishes when the direct capacitive term and the coupler-mediated virtual term cancel. For a tunable transmon coupler this off point usually sits 1.5 to 2.5 GHz above the qubit band, often near 6.5 to 7.5 GHz when the data qubits run around 4.5 to 5.5 GHz. At that idle frequency the static ZZ interaction can be held below 1 to 10 kHz. The exact off frequency depends on the couplings g1, g2, the anharmonicities, and the qubit detunings, so each device is calibrated by sweeping coupler flux and measuring the residual exchange.
How fast can the coupler frequency be tuned for a CZ gate?
A flux pulse moves the coupler from its idle point down toward the qubit band by 1 to 2 GHz in 5 to 20 ns, using smoothed net-zero or error-function edges to avoid exciting the coupler mode and to limit leakage. Total CZ time is typically 30 to 60 ns including ramps, with on-state J/2π reaching 10 to 30 MHz. Faster pulses raise leakage into the coupler and higher transmon levels, so the ramp shape is optimized against gate speed.
Why must the coupler frequency stay well above both qubit frequencies?
A large detuning keeps the coupler in its ground state so it acts as a virtual photon bus rather than a populated mode. The mediated coupling scales as g1 × g2 / Δ, so a big detuning gives a small, easily nulled J at idle while reducing the detuning during a gate sharply increases it. If the coupler frequency approached a qubit it would hybridize, populate real photons, and cause leakage and frequency collisions. Designs typically park the coupler 1.5 to 2.5 GHz above the highest qubit.
What sets the achievable tuning range of the coupler frequency?
The range follows from the SQUID loop design. A symmetric SQUID coupler tunes from a maximum frequency at zero flux down toward zero as the flux approaches half a flux quantum, following fc ≈ fc,max√|cos(πΦ/Φ0)|, giving several gigahertz of range but rising flux-noise sensitivity as you move off the fc,max sweet spot. The null (off) point where J cancels does not coincide with fc,max; it sits a bit lower, where the coupler-qubit detuning lands near the value that balances the direct and virtual terms. Asymmetric SQUIDs raise the lower frequency floor to keep fc,max finite and reduce flux-noise sensitivity. Practical couplers expose a usable 2 to 3 GHz span between the parking region near fc,max and the interaction frequency near the qubit band.