Conductance Matrix (G)

Conductance outputs describe leakage paths, dielectric loss, and shunt conductance

that can reduce resonator Q and qubit coherence.

#	Parameter	Symbol / Unit	Extraction Method	Typical Q3D Value	Ideal / Optimal	Good Range	Worst Case	Why It Matters	Key Design Note
11	Self-Conductance (G_ii)	G_ii / nS	Q3D DC conductance solve	< 0.1 nS	< 0.1 nS (high-R substrate)	0.1 – 1 nS	> 10 nS	Leakage current to ground through substrate limits resonator quality factor directly	G_ii = 1/R_leak; appears as parallel resistance Q = ω_r C_r / G in resonator model
12	Substrate Bulk Conductance	G_bulk / nS	Resistivity measurement + Q3D G matrix	< 0.01 nS	< 0.01 nS (> 10 kΩcm Si at 4K)	0.01 – 0.5 nS	> 5 nS	Low-resistivity Si drastically degrades resonator Q; bulk conductance is the dominant channel	Use high-resistivity Si (> 10 kΩcm) or sapphire; resistivity increases >100× when cooled to 4K (carrier freeze-out; magnitude is strongly doping-dependent)
13	Surface / Interface Conductance	G_surf / nS/μm	Surface participation ratio + loss tangent fitting	< 0.001 nS/μm	< 0.001 nS/μm (clean passivated)	0.001 – 0.05 nS/μm	> 0.1 nS/μm	Conductance along metal-air or metal-substrate interfaces is a major T₁ limitation via TLS	Adsorbed water and organic residues increase G_surf; HF dip and N₂ purge before cooldown helps
14	Mutual Conductance (G_ij)	G_ij / pS	Q3D off-diagonal G matrix extraction	< 1 pS	< 1 pS (isolated conductors)	1 – 50 pS	> 500 pS	Leakage coupling between signal conductors via substrate surface conduction channels	Non-zero G_ij in presence of surface water or conductive substrate; fixed by surface clean or guard rings