Inductance Matrix (L)
=====================

Inductance outputs connect geometry, kinetic inductance, mutual inductance, and
                Josephson energy used in qubit Hamiltonian extraction.

.. 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
   * - 6
     - Self-Inductance (L_ii)
     - L_ii / nH
     - Q3D magnetoquasistatic solver; pyEPR energy participation
     - 0.5 – 5 nH
     - 1 – 3 nH (target Ej/Ec ~ 50–80)
     - 0.5 – 5 nH
     - < 0.1 or > 20 nH
     - Sets Josephson energy Ej = Φ₀²/2L; directly determines qubit frequency f ≈
                        √(8EjEc)/h
     - L = L_geometric + L_kinetic; kinetic inductance is material-dependent (Al:
                        ~0.1–1 pH/sq)
   * - 7
     - Mutual Inductance (M_ij)
     - M_ij / pH
     - Q3D off-diagonal L matrix extraction
     - 1 – 50 pH
     - < 5 pH (idle qubits)
     - 5 – 50 pH
     - > 500 pH
     - Magnetically coupled flux between loops drives parasitic ZZ interaction and
                        inductive crosstalk
     - Intentional M_ij used in flux-tunable couplers; unintentional M_ij sets
                        residual ZZ floor
   * - 8
     - Geometric Inductance
     - L_geo / pH/μm
     - Q3D partial inductance extraction (PEEC)
     - 0.3 – 0.8 pH/μm
     - < 0.5 pH/μm (wide ground traces)
     - 0.5 – 1 pH/μm
     - > 2 pH/μm
     - Inductance from current path geometry adds to kinetic inductance to set
                        total L and mode freq
     - Slot cuts in ground plane drastically increase L_geo; continuous ground
                        plane is preferred
   * - 9
     - Kinetic Inductance (L_k)
     - L_k / pH/sq
     - Microwave resonator fitting; L_k = ℏ²/(π Δ e² n_s t)
     - 1–2 pH/sq (Al, 100–200 nm); 30–200 pH/sq (NbTiN)
     - < 2 pH/sq (standard Al transmon, 100–200 nm film)
     - 1 – 10 pH/sq
     - > 100 pH/sq (non-KI devices)
     - Inertia of Cooper pairs; provides non-linearity in KI qubits; adds to
                        geometric L in CPW
     - High-L_k materials (NbTiN, TiN, NbN) used deliberately for kinetic
                        inductance qubit designs
   * - 10
     - Josephson Inductance (L_J)
     - L_J / nH
     - Derived: L_J = Φ₀/(2π I_c) = Φ₀²/(2Ej); I_c from RₙA measurement
     - 5 – 15 nH
     - 8 – 12 nH (Ej/Ec ~ 50–80)
     - 5 – 20 nH
     - < 1 or > 50 nH
     - Non-linear inductance of JJ; L_J(φ) = L_J0/cos(φ) provides essential quantum
                        non-linearity
     - L_J is the ONLY non-linear element; its ratio to shunting capacitance sets
                        anharmonicity α