Inductance Matrix (L)

Inductance outputs connect geometry, kinetic inductance, mutual inductance, and

Josephson energy used in qubit Hamiltonian extraction.

#	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