Lesson: Building Quantum Gates from Universal Sets of Gates

Introduction

Quantum gates are fundamental building blocks for quantum computing. They allow us to manipulate quantum bits (qubits) and perform complex calculations. In this lesson, we will explore how to construct quantum gates from a small set of universal gates.

What is a Universal Set of Gates?

A universal set of gates is a set of gates that can be used to construct any other quantum gate. The most common universal set is the CNOT gate and the single-qubit gates X, Y, and Z.

• CNOT gate: The CNOT gate (controlled-NOT gate) flips the target qubit if the control qubit is in the |1⟩ state.
• X gate: The X gate (NOT gate) flips the state of a qubit.
• Y gate: The Y gate is a rotation gate that transforms |0⟩ to |+⟩ and |1⟩ to |-⟩.
• Z gate: The Z gate is a phase gate that flips the phase of a qubit.

Example: Building a Hadamard Gate

The Hadamard gate (H gate) is a unitary gate that transforms a qubit into a superposition of |0⟩ and |1⟩. It can be constructed using the CNOT and single-qubit gates as follows:

H = CNOT(1, 0) * S(0) * CNOT(0, 1) * T(0)

where:

• S(0) is a single-qubit phase shift gate.
• T(0) is a single-qubit π/4 rotation gate.

Building Arbitrary Gates

Any arbitrary quantum gate can be constructed using a universal set of gates. The method for constructing a gate is called quantum circuit synthesis.

Learning Resources

• Quantum Gates and Circuits
• Quantum Control and Gates
• Quantum Circuit Synthesis

Assessment

1. List the gates in the universal set.
2. Describe the action of the CNOT gate.
3. Explain how to construct a Hadamard gate using the universal set.
4. Discuss the importance of quantum gates in quantum computing.