How the target controls the controller.
In standard classical logic, a Control Bit dictates what happens to a target. However, in quantum mechanics, the relationship is symmetric. When the target qubit is in an eigenstate of the operator, the phase is “kicked back” to the control qubit.
|+⟩
|−⟩
|−⟩
|−⟩
CNOT CIRCUIT
Notice above: The Target qubit remains unchanged (|−⟩), but the Control qubit flips from |+⟩ to |−⟩.
STEP 1: DEFINE INITIAL STATE
|ψ0⟩ = |+⟩ ⊗ |−⟩
= (1/√2) (|0⟩ + |1⟩) ⊗ (1/√2) (|0⟩ − |1⟩)
= (1/√2) (|0⟩ + |1⟩) ⊗ (1/√2) (|0⟩ − |1⟩)
STEP 2: EXPAND TERMS
Multiplying coefficients gives 1/2:
|ψ0⟩ = ½ [ |00⟩ − |01⟩ + |10⟩ − |11⟩ ]
|ψ0⟩ = ½ [ |00⟩ − |01⟩ + |10⟩ − |11⟩ ]
STEP 3: APPLY CNOT GATE
Target flips (0↔1) only if Control is 1:
|ψ1⟩ = ½ [ |00⟩ − |01⟩ + |11⟩ − |10⟩ ]
|ψ1⟩ = ½ [ |00⟩ − |01⟩ + |11⟩ − |10⟩ ]
STEP 4: FACTOR & REARRANGE
Group terms by the control qubit:
|ψ1⟩ = ½ [ |0⟩(|0⟩ − |1⟩) − |1⟩(|0⟩ − |1⟩) ]
|ψ1⟩ = ½ [ |0⟩(|0⟩ − |1⟩) − |1⟩(|0⟩ − |1⟩) ]
∴ |ψ1⟩ = |−⟩ ⊗ |−⟩
Why is this important?
The math shows that while we applied the gate to the target, the relative phase of the control qubit changed from positive to negative. This mechanism is the foundation of quantum algorithms like Shor’s and Grover’s.
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