In standard classical logic, a control bit dictates what happens to a target. In quantum mechanics the relationship is symmetric. When the target qubit is in an eigenstate of the gate operator, the eigenvalue phase is kicked back onto the control qubit, leaving the target unchanged while flipping the relative phase of the control. This post derives that result from first principles using the CNOT gate on |+⟩ ⊗ |−⟩.
1 · The CNOT Circuit and the Kickback Setup
The circuit places the control qubit in superposition |+⟩ and the target qubit in |−⟩. Since |−⟩ is an eigenstate of X with eigenvalue −1, the kickback occurs.
control:
|+⟩
|−⟩
target:
|−⟩
+
|−⟩
Key observation: The target qubit is unchanged after the CNOT. The control qubit flips from |+⟩ to |−⟩. The phase was kicked back to the control, not forward to the target.
2 · Full Derivation: |+⟩ ⊗ |−⟩ through CNOT
Step 1 — Define the initial state
|ψ₀⟩ = |+⟩ ⊗ |−⟩
= (1/√2)(|0⟩ + |1⟩) ⊗ (1/√2)(|0⟩ − |1⟩)
Step 2 — Expand the tensor product
|ψ₀⟩ = (1/2)[ |00⟩ − |01⟩ + |10⟩ − |11⟩ ]
Step 3 — Apply the CNOT gate
|ψ₁⟩ = (1/2)[ |00⟩ − |01⟩ + |11⟩ − |10⟩ ]
Step 4 — Factor and identify the result
|ψ₁⟩ = (1/2)[ |0⟩(|0⟩ − |1⟩) − |1⟩(|0⟩ − |1⟩) ]
= (1/√2)(|0⟩ − |1⟩) ⊗ (1/√2)(|0⟩ − |1⟩)
= |−⟩ ⊗ |−⟩
The target qubit is still |−⟩ — unchanged by the CNOT. The control qubit changed from |+⟩ to |−⟩. The −1 eigenvalue of the target has been kicked back as a relative phase onto the control.
3 · Why Phase Kickback Matters
The math shows that while we applied a gate to the target, the relative phase of the control qubit changed from positive to negative. This is a structural property of controlled unitaries acting on their eigenstates, and it appears at the core of every major quantum algorithm.
| Algorithm | How phase kickback is used |
|---|---|
| Deutsch’s Algorithm | The oracle kicks a −1 phase onto the control qubit to encode whether f is constant or balanced, extractable with a single H gate measurement. |
| Grover’s Algorithm | The oracle flips the sign of the target state’s amplitude by kicking a −1 phase back to the control register, enabling amplitude amplification. |
| Shor’s Algorithm | Quantum Phase Estimation relies entirely on phase kickback to transfer eigenvalue information from the target register to the control register. |
The general rule: for any unitary U with eigenstate |u⟩ (so U|u⟩ = eⁱᵖ|u⟩), a controlled-U gate with the control in superposition kicks the phase eⁱᵖ back to the control qubit. The target is unchanged.
Part of the Quantum Series 2026. Originally published on Techucation.
