Watch information flow backward through a controlled gate
How can a controlled gate change the control qubit instead of the target?
In a controlled gate, the control qubit decides whether the gate fires on the target. Normally, the target changes and the control stays the same. But if you prepare the target in a special state called an eigenstate, something surprising happens: the target stays the same and the control picks up a phase instead. This experiment shows you that effect step by step.
The CNOT gate applies an X (bit flip) to the target when the control is . If the target is in |−⟩, which is an eigenstate of X with eigenvalue −1, then flipping it just multiplies the state by −1. That −1 factor appears on the control's branch as a phase. The target does not change at all. A final Hadamard on the control converts this hidden phase into a measurable bit, proving the information flowed backward.
Put the control qubit into superposition: . This creates the two branches that will later pick up different phases. The Bloch sphere for qubit 0 moves to the equator.
Flip qubit 1 from to . This is the first step of preparing the eigenstate. The Bloch sphere for qubit 1 moves to the south pole.
H converts into |−⟩ = (−. This is the eigenstate of X with eigenvalue −1. The target is now ready for kickback. Its Bloch sphere arrow points along −X.
CNOT applies X to the target when the control is . Since X|−⟩ = −|−⟩, the target stays in |−⟩ and the control's branch picks up a −1 phase. The control is now in |−⟩ = (−. Watch carefully: the target's Bloch sphere does not move, but the control's arrow has rotated.
The final Hadamard converts the phase on the control into a visible bit. H maps |−⟩ = (− to . The control qubit is now deterministically , proving the phase kickback happened. Without the Z-phase on the control, H would have returned it to .
It is tempting to think the target qubit must change because that is where the CNOT gate applies its operation. When the target is in an eigenstate of the controlled gate, the eigenvalue appears as a phase on the control instead. The gate is doing exactly what it always does; the surprising result comes from the choice of input state.
The control qubit (q0) measures with certainty. The target qubit (q1) remains in |−⟩. Every shot returns 1 on the control, confirming that the phase kickback occurred.
This experiment demonstrates the central mechanism of the Phase Kickback lesson. That lesson explains how eigenvalues become phases, and why this backward flow of information is the engine behind Deutsch's algorithm, quantum phase estimation, and essentially every quantum speedup. Seeing it happen in the simulator makes the abstract idea tangible.
Read the full lesson →Why does the target qubit stay unchanged after the CNOT in this experiment?