Couplings and Interactions

Couplings describe how multiple qubits interact. Common coupling mechanisms:

• Capacitive coupling: two qubits share a capacitor, Hamiltonian contains $\propto Q_1 Q_2$ terms
• Inductive coupling: qubits share an inductor, Hamiltonian contains $\propto \phi_1 \phi_2$ terms
• Charge coupling: gate voltage on one qubit affects another
• Tunable coupling: flux-dependent or controllable

from scqubits import Transmon, HilbertSpace, InteractionTerm
 
qubit1 = Transmon(EJ=15.0, EC=0.3, ncut=30)
qubit2 = Transmon(EJ=15.0, EC=0.3, ncut=30)
 
hilbert = HilbertSpace([qubit1, qubit2])
 
# Capacitive coupling: g * n_1 * n_2
g = 0.01  # Coupling strength (GHz)
interaction = InteractionTerm(
    g=g,
    op1=qubit1.n_operator(),
    op2=qubit2.n_operator()
)
hilbert.add_interaction(interaction)
 
# Compute coupled spectrum
evals_coupled = hilbert.hamiltonian().eigenenergies()

Coupling strength $g$ sets the two-qubit gate speed: larger $g$ enables faster gates. But strong coupling increases error sensitivity and causes unwanted shifts in single-qubit frequencies (dispersive shifts).

Typical two-qubit couplings: $g \sim 1$ MHz to $100$ MHz. Gate times: $\sim 10$ ns to $100$ ns.