Two coupled transmon qubits (control, target) driven at the target's frequency realise, in the doubly-rotating frame, the effective cross-resonance Hamiltonian used to entangle real superconducting qubits (IBM's native two-qubit gate):
H_CR(t) = (Ω(t)/2) · [ a_ZX (Z⊗X) + a_IX (I⊗X) ]
a_ZX = 1 (fixed, the wanted entangling term)
a_IX = crosstalk slider (an always-on stray term)
The Z⊗X term rotates the target about X at a rate that depends on the control's Z eigenvalue (+1 for |0⟩, −1 for |1⟩) — this conditional rotation is exactly what creates entanglement. The I⊗X term rotates the target regardless of the control and does nothing useful.
This simulator integrates the true 4-dimensional Schrödinger equation (Runge-Kutta 4) for the joint 2-qubit state under a smooth raised-cosine drive envelope, no shortcuts. With Echo ON, the pulse is split into two halves of opposite sign with a real π-pulse (bit flip) on the control qubit in between:
U = X_c · exp(+i H_CR T/4) · X_c · exp(-i H_CR T/4)
Because Z⊗X and I⊗X commute, this echo exactly cancels the I⊗X crosstalk term while the Z⊗X entangling rotation survives — the same trick used to calibrate real echoed-CR gates on IBM hardware.
- Control preparation — |0⟩ or |1⟩ keeps the joint state a product state (Z⊗I is conserved, so the target stays pure); |+⟩ puts the control in superposition, so the two conditional rotations pull the target's reduced state apart and it becomes mixed — entanglement, visible as the main target arrow shrinking.
- Ghost arrows — the two faint dashed arrows on the target sphere are the exact conditional target states for control=|0⟩ (blue) and control=|1⟩ (red); their separation angle Δθ is a direct, measurable signature of the ZX interaction strength.
- Bloch length |n| — 1 = pure state, 0 = maximally mixed. It equals √(2·Tr(ρ²) − 1) for the true reduced density matrix, computed every frame from the real 2-qubit statevector.