Controlled Operations: CNOT

This is where quantum computing gets interesting. The CNOT (Controlled-NOT) gate is your first two-qubit gate—it creates correlations between qubits that have no classical analog.

The CNOT Gate

The CNOT (Controlled-X) gate has two qubits:

• Control qubit: The one that decides whether to act
• Target qubit: The one that gets flipped

CNOT:  ┌───┐
control:─┤   ├──
         │ X │
target:──┤   ├──
         └───┘

The rule: **flip the target if and only if the control is \(|1\rangle\).

CNOT on Basis States

Let’s see what happens:

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

# Case 1: Control = 0, Target = 0 → No change
qc1 = QuantumCircuit(2)
qc1.cx(0, 1)  # CNOT with control=0, target=1
print("|00⟩ after CNOT:", Statevector(qc1))

# Case 2: Control = 1, Target = 0 → Flip target
qc2 = QuantumCircuit(2)
qc2.x(0)  # Set control to 1
qc2.cx(0, 1)
print("|10⟩ after CNOT:", Statevector(qc2))

# Case 3: Control = 0, Target = 1 → No change
qc3 = QuantumCircuit(2)
qc3.x(1)  # Set target to 1
qc3.cx(0, 1)
print("|01⟩ after CNOT:", Statevector(qc3))

# Case 4: Control = 1, Target = 1 → Flip target
qc4 = QuantumCircuit(2)
qc4.x([0, 1])  # Both are 1
qc4.cx(0, 1)
print("|11⟩ after CNOT:", Statevector(qc4))

Summary:

CNOT Syntax in Qiskit

qc.cx(control, target)

The first argument is the control qubit, the second is the target.

qc.cx(0, 1)  # Qubit 0 is control, qubit 1 is target
qc.cx(1, 0)  # Qubit 1 is control, qubit 0 is target

These are NOT the same! The control-target orientation matters.

CNOT on Superpositions

Here’s where it gets interesting:

# CNOT on superposition state
qc = QuantumCircuit(2)
qc.h(0)  # Put control in superposition
qc.cx(0, 1)

print("State:", Statevector(qc))

Output:

Statevector([0.70710678+0.j, 0.+0.j, 0.+0.j, 0.70710678+0.j],
            dims=(2, 2))

This is \(\frac{|00\rangle + |11\rangle}{\sqrt{2}}\)—a Bell state!

Reading the Statevector

For a 2-qubit statevector, the indices correspond to basis states:

The state is \(\frac{|00\rangle + |11\rangle}{\sqrt{2}}\).

Why This Is Special

The state \(\frac{|00\rangle + |11\rangle}{\sqrt{2}}\) has a property that classical bits can’t replicate: quantum correlation.

If you measure qubit 0:

• You see 0 with 50% probability → qubit 1 is also 0
• You see 1 with 50% probability → qubit 1 is also 1

The outcomes are perfectly correlated, even though neither outcome is predetermined!

CNOT Is Not Reversible? Wrong!

Classical XOR is not reversible: 0 XOR 0 = 0, but so does 1 XOR 1 = 0.

Quantum CNOT is reversible:

• Apply CNOT twice and you get back to the original state
• The quantum operation is unitary

# CNOT twice = identity
qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
qc.cx(0, 1)

print("CNOT twice:", Statevector(qc))

Circuit Diagrams

Qiskit prints readable circuit diagrams:

qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
print(qc)

     ┌───┐
q_0: ┤ H ├──■──
     └───┘  │
q_1: ──────■──

The ■ is the control, the ⊕ is the target (often shown as ■───⊕───).

More Complex CNOT Patterns

CNOT Chain

# Create a GHZ state (Greenberger-Horne-Zeilinger)
qc = QuantumCircuit(3)
qc.h(0)
qc.cx(0, 1)
qc.cx(1, 2)
print(Statevector(qc))

This creates \(\frac{|000\rangle + |111\rangle}{\sqrt{2}}\).

CNOT Fan-Out

# One control, multiple targets
qc = QuantumCircuit(3)
qc.h(0)
qc.cx(0, 1)
qc.cx(0, 2)
print(Statevector(qc))

This creates \(\frac{|000\rangle + |110\rangle}{\sqrt{2}}\)? No wait—verify this yourself!

Checkpoint Exercises

Exercise 1

Apply CNOT to \(|01\rangle\). What state do you get?

Exercise 2

Apply CNOT to \(|10\rangle\). What state do you get?

Exercise 3

Create the Bell state \(\frac{|01\rangle + |10\rangle}{\sqrt{2}}\). (Hint: start with the right basis state)

Exercise 4

Create the GHZ state \(\frac{|000\rangle + |111\rangle}{\sqrt{2}}\) on three qubits.

Exercise 5

Show that CNOT is its own inverse (applying it twice returns to the original state).

Common Mistakes

Summary

Key concepts:

• CNOT flips the target if control is \(|1\rangle\)
• Syntax: cx(control, target) in Qiskit
• On superposition: creates Bell states
• CNOT is reversible (its own inverse)
• Creates quantum correlations (entanglement)

The CNOT gate is the workhorse of two-qubit quantum circuits. Master it, and everything else becomes easier.

Official Documentation: The CNOT (Controlled-X) gate is documented in the Qiskit circuit library. See also IBM’s tutorial on multi-qubit circuits.

Next: Entanglement and Bell States