Basic Quantum Gates in Qiskit: A Beginner’s Guide

Table of Contents

1. Introduction
2. What Are Quantum Gates?
3. Setting Up Qiskit
4. Creating a Quantum Circuit
5. Single-Qubit Gates
   5.1. Identity Gate (I)
   5.2. Pauli Gates (X, Y, Z)
   5.3. Hadamard Gate (H)
   5.4. Phase Gates (S, T)
   5.5. Rotation Gates (RX, RY, RZ)
6. Two-Qubit Gates
   6.1. Controlled NOT Gate (CX or CNOT)
   6.2. Controlled-Z Gate (CZ)
   6.3. SWAP Gate
   6.4. iSWAP and Controlled Gates
7. Multi-Qubit Gates
   7.1. Toffoli (CCX)
   7.2. Fredkin (CSWAP)
8. Gate Syntax in Qiskit
9. Circuit Visualization
10. Running and Measuring the Circuit
11. Simulating Gate Operations
12. Exercises: Combine Gates to Create Quantum States
13. Understanding Gate Matrices
14. Gate Reversibility and Unitarity
15. Hardware Considerations
16. Using Decomposition to Build Complex Gates
17. Saving and Loading Circuits
18. Best Practices for Gate Usage
19. Advanced Reading and Next Steps
20. Conclusion

1. Introduction

Quantum gates are the fundamental operations in quantum computing. They transform qubit states in ways that reflect the unique principles of quantum mechanics. Qiskit provides an accessible way to use and understand these gates.

2. What Are Quantum Gates?

Quantum gates are unitary transformations that operate on qubits. Unlike classical logic gates, they are reversible and allow for superposition and entanglement.

3. Setting Up Qiskit

pip install qiskit

4. Creating a Quantum Circuit

from qiskit import QuantumCircuit
qc = QuantumCircuit(2)  # Two qubits

5. Single-Qubit Gates

5.1. Identity Gate (I)

qc.i(0)

Does nothing to the qubit; useful as a placeholder.

5.2. Pauli Gates

qc.x(0)  # Bit flip
qc.y(0)  # Bit + phase flip
qc.z(0)  # Phase flip

5.3. Hadamard Gate (H)

qc.h(0)

Creates superposition: \( |0
angle
ightarrow rac{1}{\sqrt{2}}(|0
angle + |1
angle) \)

5.4. Phase Gates (S, T)

qc.s(0)  # 90° phase
qc.t(0)  # 45° phase

5.5. Rotation Gates

qc.rx(theta, 0)
qc.ry(theta, 0)
qc.rz(theta, 0)

Rotate qubit around X, Y, or Z axis.

6. Two-Qubit Gates

6.1. Controlled-NOT (CX or CNOT)

qc.cx(0, 1)

Flips qubit 1 if qubit 0 is |1⟩

6.2. Controlled-Z (CZ)

qc.cz(0, 1)

6.3. SWAP

qc.swap(0, 1)

Swaps the states of two qubits.

6.4. iSWAP and Controlled Gates

qc.iswap(0, 1)
qc.cp(pi/2, 0, 1)  # Controlled phase gate

7. Multi-Qubit Gates

7.1. Toffoli (CCX)

qc.ccx(0, 1, 2)

AND-controlled NOT gate; flips qubit 2 only if 0 and 1 are |1⟩.

7.2. Fredkin (CSWAP)

qc.cswap(0, 1, 2)

Swaps qubits 1 and 2 only if qubit 0 is |1⟩.

8. Gate Syntax in Qiskit

Each gate has a standard Python function:

• qc.h(qubit)
• qc.cx(control, target)
• qc.rz(theta, qubit)

9. Circuit Visualization

qc.draw('mpl')

Visualizes circuit as a gate diagram.

10. Running and Measuring the Circuit

from qiskit import Aer, execute
qc.measure_all()
sim = Aer.get_backend('qasm_simulator')
result = execute(qc, sim, shots=1024).result()
counts = result.get_counts()
print(counts)

11. Simulating Gate Operations

Use statevector_simulator or unitary_simulator for full wavefunction or gate matrix:

from qiskit.quantum_info import Statevector
print(Statevector.from_instruction(qc))

12. Exercises: Combine Gates to Create Quantum States

• Create a Bell state with H and CX
• Apply rotations to visualize on Bloch sphere

13. Understanding Gate Matrices

All gates are unitary matrices (U†U = I). Example:
\[
X = egin{bmatrix}0 & 1 \ 1 & 0\end{bmatrix}
\quad
H = rac{1}{\sqrt{2}}egin{bmatrix}1 & 1 \ 1 & -1\end{bmatrix}
\]

14. Gate Reversibility and Unitarity

All quantum gates are reversible by definition.

15. Hardware Considerations

• CX is often noisier than single-qubit gates
• Real hardware may decompose complex gates into native basis

16. Using Decomposition to Build Complex Gates

qc.unitary(U_matrix, [0], label='U')

17. Saving and Loading Circuits

qc.qasm(filename='my_circuit.qasm')

18. Best Practices for Gate Usage

• Minimize CX and multi-qubit gates
• Optimize circuit depth
• Use transpiler for backend optimization

19. Advanced Reading and Next Steps

• Learn about SU(2) decomposition
• Explore parameterized circuits and variational gates

20. Conclusion

Mastering basic quantum gates is essential to building and understanding quantum algorithms. Qiskit provides a powerful and intuitive framework to explore them through real and simulated execution.