Conductance Matrix (G)
======================

Conductance outputs describe leakage paths, dielectric loss, and shunt conductance
                that can reduce resonator Q and qubit coherence.

.. list-table::
   :header-rows: 1

   * - #
     - 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