Circuit quantization derives the quantum Hamiltonian from a classical superconducting circuit. Start with the circuit diagram (nodes, components), identify the generalized coordinates (node fluxes, charges), write the classical Lagrangian, then apply quantization rules: $\phi \to \hat{\phi}$ (flux operator), $Q \to \hat{Q}$ (charge operator), with commutation relations $[\hat{\phi}, \hat{Q}] = i\hbar$.

The result is a quantum Hamiltonian $\hat{H}$ whose eigenvalues and eigenstates describe the qubit's energy levels and states.

from scqubits import Transmon
 
# Transmon: quantized LC circuit with Josephson junction
# Classical circuit: inductance L_J (Josephson) + capacitance C
# Quantum Hamiltonian: H = 4E_C n^2 - E_J cos(φ)
# where E_C = e²/(2C) is charging energy
# E_J = Φ_0 I_c / (2π) is Josephson energy
 
transmon = Transmon(EJ=15.0, EC=0.3, ng=0.0, ncut=30)
# EJ, EC fully specify the transmon's Hamiltonian

Each node in the circuit becomes a degree of freedom. For a two-junction transmon (two Josephson junctions), there's one relevant flux degree of freedom (the other is frozen out by symmetry). For more complex circuits, you track multiple flux/charge coordinates.

scqubits automates circuit quantization for standard qubit geometries. For custom circuits, use the QuantumCircuit class or derive the Hamiltonian by hand.

For weakly driven qubits, the rotating wave approximation (RWA) simplifies the Hamiltonian by dropping rapidly oscillating terms. This is valid when the drive frequency is close to the qubit frequency and coupling is weak. scqubits can apply RWA automatically.

Circuit quantization is the bridge between engineering (circuit design) and physics (quantum Hamiltonian).