Resonator & Coupling

Resonator & Coupling Parameters: Readout and bus resonator characteristics plus qubit-resonator coupling extracted via EPR

1. Resonator Properties

Parameter	Symbol	Unit	Description	Optimal / Best Value	Good Range	Acceptable Range	Poor / Worst Value	Physical Significance
Resonator Frequency	omega_r / 2pi	GHz	Bare resonator frequency; should be detuned from qubit to remain in dispersive regime.	6.5 – 8.5 GHz	5 – 10 GHz	3 – 5 GHz	< 2 GHz	Must satisfy |Δ| = |ω_q - ω_r| > > g for dispersive approximation; EPR gives bare frequency.
Resonator Internal Q	Q_int	dimensionless	Internal (material-limited) quality factor of the readout resonator.	> 10⁵	10⁴ – 10⁵	10³ – 10⁴	< 500	Low Q_int adds photon loss increasing measurement back-action and reducing readout SNR.
Resonator External Q	Q_ext	dimensionless	External quality factor set by coupling to transmission line; determines measurement bandwidth.	10³ – 10⁴ (fast readout)	500 – 2×10⁴	50 – 499	< 50	Too high → slow readout; too low → Purcell-enhanced qubit decay. Optimized with Purcell filter.
Resonator–Qubit Detuning |Δ|	|Δ|/2π	GHz	Frequency detuning between qubit and resonator; must be large compared to coupling g.	1.0 – 3.0 GHz	0.5 – 4.0 GHz	0.1 – 0.5 GHz	< 0.05 GHz	Small detuning violates dispersive approximation; EPR hybridization factor chi tracks this.
Purcell Decay Rate	γ_P/2π	kHz	Qubit decay rate via resonator Purcell channel; κ * (g/Δ)². Must be < < 1/T1_target.	< 1 kHz	1 – 10 kHz	10 – 100 kHz	> 500 kHz	Dominant T_1 limit in many designs without Purcell filter; directly predicted by EPR p_J^res.

2. Coupling Strengths

Parameter	Symbol	Unit	Description	Optimal / Best Value	Good Range	Acceptable Range	Poor / Worst Value	Physical Significance
Transverse Coupling g / 2pi	g / 2pi	MHz	Qubit-resonator coupling strength (Jaynes-Cummings); extracted from EPR as g = √(p_J^res * ω_r * ω_q / 2).	50 – 150 MHz	20 – 200 MHz	5 – 19 MHz	< 2 MHz	Sets χ and readout speed; g/|Δ| < 0.1 required for dispersive regime.
Dispersive Coupling g/Δ ratio	g/|Δ|	dimensionless	Dimensionless ratio quantifying proximity to strong-coupling limit; must be < < 1 for dispersive readout.	0.01 – 0.05	0.005 – 0.09	0.09 – 0.15	> 0.20	Ratio > 0.1 causes photon-number-dependent qubit dephasing and higher-order dispersive corrections.
Bus Coupler Coupling (2Q)	J / 2pi	MHz	Exchange coupling between two qubits via bus resonator or direct capacitance.	5 – 20 MHz	2 – 30 MHz	0.5 – 1.9 MHz	< 0.1 MHz	Too weak → slow two-qubit gates; too strong → residual ZZ. EPR predicts J from geometry.
Residual ZZ (static)	zeta_ZZ / 2pi	kHz	Always-on longitudinal (ZZ) qubit–qubit interaction; source of conditional phase errors.	< 10 kHz	< 100 kHz	100 – 500 kHz	> 1 MHz	Limits two-qubit gate fidelity via conditional phase accumulation; minimized by tunable couplers.
Stray Coupling (nearest-neighbor)	J_stray / 2pi	kHz	Unintended coupling between non-adjacent qubits; extracted from full-chip EPR simulation.	< 10 kHz	< 50 kHz	50 – 200 kHz	> 500 kHz	Degrades multi-qubit gate fidelity; EPR full-chip simulation essential to identify stray modes.