Capacitance Matrix (C)

Capacitance outputs are the main Q3D bridge into transmon energy, coupling, detuning, and readout design.

#	Parameter	Symbol / Unit	Extraction Method	Typical Q3D Value	Ideal / Optimal	Good Range	Worst Case	Why It Matters	Key Design Note
15	Qubit Self-Capacitance (C_Sigma)	C_Sigma / fF	Q3D electrostatic solve; Maxwell capacitance matrix	60 – 100 fF	60 – 100 fF (transmon shunting)	40 – 200 fF	< 10 fF or > 500 fF	Sets charging energy Ec = e²/2C_Σ; large C_Sigma → transmon regime → exponentially reduced charge noise	C_Sigma = Σ|C_ij| from Maxwell matrix; target Ej/Ec = 50–80 for optimal transmon performance
16	Readout Resonator Capacitance (C_r)	C_r / fF	Q3D capacitance matrix + HFSS eigenmode simulation	200 – 500 fF	200 – 500 fF (λ/4 CPW)	100 – 600 fF	< 50 or > 1 pF	Resonator mode capacitance sets frequency: ω_r = 1/√(L_r C_r); must target 6.5–8 GHz window	Combined with Q_ext sets readout bandwidth κ = ω_r/Q_ext; trade-off between speed and SNR
17	Qubit–Resonator Coupling Cap (C_g)	C_g / fF	Q3D Maxwell matrix off-diagonal C_12 between qubit island and resonator	1 – 10 fF	1 – 10 fF (dispersive limit)	0.5 – 15 fF	< 0.1 or > 50 fF	Sets coupling g = C_g/(2C_Σ)·√(ω_q ω_r/L_r C_r); must stay dispersive (g ≪ qubit–resonator detuning)	g / 2pi target 50–150 MHz; too large → strong coupling regime; Purcell decay ∝ (g/Δ)² × κ
18	Qubit–Qubit Coupling Cap (C_J)	C_J / fF	Q3D full capacitance matrix between qubit islands	0.5 – 5 fF	0.5 – 5 fF (tunable coupler)	0.2 – 10 fF	< 0.05 or > 30 fF	Drives direct transverse coupling J; residual C_J causes always-on ZZ unless tunable coupler used	Modern heavy-hex lattice uses tunable couplers to cancel residual ZZ to < 10 kHz
19	Pad-to-Ground Parasitic Cap	C_pad / fF	Q3D with ground plane mesh	< 5 fF per pad	< 5 fF (small footprint)	1 – 20 fF	> 50 fF	Unintended pad-to-ground capacitance shifts qubit frequency from design target	Each 1 fF of parasitic shifts f_qubit by ~10–30 MHz; critical to include in Hamiltonian model
20	Trace Mutual Capacitance (C_ij)	C_ij / fF	Q3D electrostatic solve off-diagonal extraction	< 1 fF (separated lines)	< 1 fF	1 – 5 fF	> 20 fF	Capacitive coupling between control lines causes microwave crosstalk in drive and readout paths	Overlapping traces on adjacent layers is the primary source; add ground shield layer to suppress