Multi-Qubit Couplings

QForge provides flexible modeling for multi-qubit interactions, essential for simulating 2-qubit gates like CNOT and CZ. This page details the physical Hamiltonians used in the simulation.

Capacitive Coupling

The most common coupling for fixed-frequency transmons (e.g., in Cross-Resonance gates). The interaction is transverse.

\[H_{int} = g \left( a^\dagger b + a b^\dagger \right)\]

where: * \(a, a^\dagger\) are operators for the control qubit (Q1). * \(b, b^\dagger\) are operators for the target qubit (Q2). * \(g\) is the coupling strength in GHz.

Physics: This term represents exchange interaction. In the dispersive limit (\(|\Delta| \gg g\)), it leads to a small hybridization of the states. When driven at the target frequency (Cross-Resonance), it activates a \(ZX\) interaction essential for CNOT.

Inductive / ZZ Coupling

Often an effective model for weak dispersive interactions or residual coupling from higher levels.

\[H_{int} = g \hat{n}_1 \hat{n}_2 = g (a^\dagger a)(b^\dagger b)\]

where: * \(\hat{n}_i\) is the number operator for qubit \(i\).

Physics: This is a longitudinal coupling that shifts energy levels depending on the state of the other qubit. It naturally implements a CPHASE (CZ) evolution over time \(T = \pi/g\).

Tunable Coupler (Effective)

For tunable couplers (like g-mon or transmons with flux loops), the effective coupling \(g\) can be modulated in time. QForge models the identifying interaction Hamiltonian which is then modulated by a pulse envelope \(f(t)\).

For a tunable exchange interaction (Swap/iSwap):

\[H(t) = g_{max} f(t) \left( a^\dagger b + a b^\dagger \right)\]

For a tunable CZ gate (adiabatic or diabatic flux pulse):

\[H(t) = g_{eff}(t) |11\rangle\langle 11|\]

(Note: The exact Hamiltonian depends on the implementation details, e.g., using a third coupler element vs. direct flux tuning).