Capacitance Matrix (C)
======================

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

.. 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
   * - 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