More Two-Qubit Gates: CZ, SWAP

Beyond CNOT, Qiskit provides several useful two-qubit gates. We’ll cover the most important ones: CZ (Controlled-Z) and SWAP.

The CZ Gate (Controlled-Z)

The CZ gate applies a Z operation when the control is \(|1\rangle\):

\[CZ|ij\rangle = (-1)^{ij}|ij\rangle\]

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

# CZ on |11⟩ gives -|11⟩
qc = QuantumCircuit(2)
qc.x([0, 1])  # Prepare |11⟩
qc.cz(0, 1)

print("CZ|11⟩:", Statevector(qc))

Output: \(-|11\rangle\)

CZ Creates Phase Entanglement

On superposition states, CZ creates phase correlations:

# Create superposition then apply CZ
qc = QuantumCircuit(2)
qc.h(0)
qc.h(1)
qc.cz(0, 1)

print("CZ on superposition:", Statevector(qc))

The \(|11\rangle\) component picks up a minus sign.

CZ vs CNOT

The SWAP Gate

The SWAP gate exchanges the states of two qubits:

\[SWAP|ij\rangle = |ji\rangle\]

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

# SWAP |01⟩ → |10⟩
qc = QuantumCircuit(2)
qc.x(1)  # |01⟩
qc.swap(0, 1)
print("SWAP|01⟩:", Statevector(qc))

Implementing SWAP with CNOTs

SWAP can be decomposed into three CNOTs:

def swap_via_cnot(qc, a, b):
    """Swap qubits a and b using three CNOTs."""
    qc.cx(a, b)
    qc.cx(b, a)
    qc.cx(a, b)

# Verify it's equivalent to SWAP
qc1 = QuantumCircuit(2)
qc1.swap(0, 1)

qc2 = QuantumCircuit(2)
swap_via_cnot(qc2, 0, 1)

from qiskit.quantum_info import Operator
print("SWAP == 3×CNOT:", Operator(qc1) == Operator(qc2))

Other Two-Qubit Gates

iSWAP

Performs a swap with an additional phase:

\[iSWAP|01\rangle = i|10\rangle\] \[iSWAP|10\rangle = i|01\rangle\]

qc = QuantumCircuit(2)
qc.iswap(0, 1)

RZZ (Controlled-RZ)

A controlled rotation around the Z axis:

from math import pi
qc = QuantumCircuit(2)
qc.rzz(pi/4, 0, 1)

Controlled-H

Hadamard controlled on one qubit:

qc = QuantumCircuit(2)
qc.ch(0, 1)  # Controlled-H

Gate Decompositions

Many “native” gates are actually decomposed into simpler gates on real hardware:

# CZ can be made from CNOT + H on target
qc = QuantumCircuit(2)
qc.h(1)  # H on target
qc.cx(0, 1)
qc.h(1)  # H on target

# Verify: CZ == H-CNOT-H
from qiskit.quantum_info import Operator
print("H-CNOT-H == CZ:", Operator(qc) == Operator(QuantumCircuit(2, circuit=qc).compose(QuantumCircuit(2))))

Actually, let’s verify this correctly:

cz = QuantumCircuit(2)
cz.cz(0, 1)

h_cnot_h = QuantumCircuit(2)
h_cnot_h.h(1)
h_cnot_h.cx(0, 1)
h_cnot_h.h(1)

print("CZ == H-CNOT-H:", Operator(cz) == Operator(h_cnot_h))

Summary of Two-Qubit Gates

Checkpoint Exercises

Exercise 1

Create a circuit that swaps \(|10\rangle\) to \(|01\rangle\).

Exercise 2

Show that CZ on \(|++\rangle\) (both qubits in plus state) gives a phase difference.

Exercise 3

Verify that SWAP = CNOT(0,1) CNOT(1,0) CNOT(0,1).

Exercise 4

Use CZ to create a phase-entangled state starting from \(|++\rangle\).

Exercise 5

Create a circuit that applies CZ only when both qubits are in \(|1\rangle\).

Summary

Key concepts:

• CZ adds phase when both qubits are \(|1\rangle\)
• SWAP exchanges qubit states
• Many gates are equivalent under basis transformations
• Gate decomposition matters for hardware implementation

With two-qubit gates mastered, you’re ready to build complex circuits and patterns.

Official Documentation: The CZ and SWAP gates are documented in the Qiskit circuit library and SWAPGate.

Try These On QCoder

• QPC002 B3: SWAP Qubits

Next: State Preparation Patterns