Quantum Gates
Introduction
Quantum gates, or quantum logic gates, are fundamental building blocks in the field of quantum computing. They operate on a small number of qubits, and are the quantum equivalent of classical logic gates. Unlike their classical counterparts, quantum gates can perform operations that seem counter-intuitive from a classical perspective, such as creating superposition states and entanglement.
Basic Quantum Gates
Quantum gates manipulate qubits through operations represented by unitary matrices. The most basic quantum gates operate on single qubits, and include the Pauli gates, the Hadamard gate, and the phase shift gates.
Pauli Gates
The Pauli gates are a set of three quantum gates known as X, Y, and Z, named after the Pauli matrices from which they are derived. The X gate is equivalent to the classical NOT gate, flipping the state of a qubit from |0⟩ to |1⟩ and vice versa. The Y and Z gates perform more complex operations, introducing a phase shift of π to the qubit state.
Hadamard Gate
The Hadamard gate (H) is another fundamental single-qubit gate. It performs a rotation of the Bloch sphere, a geometric representation of qubit states, that maps the basis state |0⟩ to a superposition state (|0⟩ + |1⟩)/√2, and |1⟩ to (|0⟩ - |1⟩)/√2. This gate is crucial for creating superposition states in quantum computing.
Phase Shift Gates
Phase shift gates introduce a phase shift to the qubit state. The most common phase shift gate is the S gate, which introduces a phase shift of π/2. The T gate, another common phase shift gate, introduces a phase shift of π/4.
Multi-Qubit Gates
Multi-qubit gates operate on two or more qubits. They are essential for creating entangled states and performing complex quantum computations. The most common multi-qubit gates are the CNOT gate, the SWAP gate, and the Toffoli gate.
CNOT Gate
The Controlled NOT (CNOT) gate operates on two qubits: a control qubit and a target qubit. If the control qubit is in state |1⟩, the target qubit is flipped. If the control qubit is in state |0⟩, the target qubit remains unchanged. This gate is crucial for creating entangled states.
SWAP Gate
The SWAP gate exchanges the states of two qubits. If one qubit is in state |0⟩ and the other is in state |1⟩, the SWAP gate will result in the first qubit being in state |1⟩ and the second in state |0⟩.
Toffoli Gate
The Toffoli gate, or the controlled-controlled NOT (CCNOT) gate, operates on three qubits. It flips the state of the third qubit if the first two qubits are both in state |1⟩. The Toffoli gate is universal for classical computation, meaning any classical logic operation can be constructed from a combination of Toffoli gates.
Quantum Gate Operations
Quantum gate operations are represented by unitary matrices. The action of a quantum gate on a qubit is represented by the multiplication of the qubit's state vector by the gate's matrix. This operation preserves the length of the state vector, a requirement for quantum operations known as unitarity.
Quantum Circuits
Quantum gates are typically represented graphically in a quantum circuit. Each qubit is represented by a horizontal line, or wire, and gates are represented by symbols on these wires. The sequence of gates represents the computation performed on the qubits.