In the realm of quantum computation, the concept of a universal family of quantum gates holds significant importance. A universal family of gates refers to a set of quantum gates that can be used to approximate any unitary transformation to any desired degree of accuracy.
The CNOT gate and the Hadamard gate are two fundamental gates that are often included in such a universal family due to their unique properties and capabilities.
The CNOT gate, short for Controlled-NOT gate, is a two-qubit gate that performs a NOT operation (bit-flip) on the target qubit only if the control qubit is in the state |1⟩. In matrix form, the CNOT gate can be represented as:
[text{CNOT} = begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 0 & 1 \
0 & 0 & 1 & 0
end{bmatrix}
]
The Hadamard gate is a single-qubit gate that creates superposition and performs a basis change. It transforms the |0⟩ state to (|0⟩ + |1⟩) / √2 and the |1⟩ state to (|0⟩ – |1⟩) / √2. The matrix representation of the Hadamard gate is:
[H = frac{1}{sqrt{2}} begin{bmatrix}
1 & 1 \
1 & -1
end{bmatrix}
]
To form a universal family of gates, it is important to have a set of gates that can generate any unitary transformation on a quantum system. The CNOT gate is essential for entangling qubits, a key requirement for quantum computation. The Hadamard gate, on the other hand, is important for creating superposition and performing basis changes, enabling a wider range of quantum operations.
When combined with other gates such as the single-qubit phase gate, the CNOT gate and the Hadamard gate form a powerful set of 3 operations that can approximate any unitary transformation (or any other quantum gate or a set of such gates). This ability to approximate any unitary transformation is what makes them part of a universal family of gates.
The CNOT gate and the Hadamard gate are integral components of a universal family of quantum gates due to their capabilities in entangling qubits, creating superposition, and enabling a wide range of quantum operations. By combining these gates with other quantum gates (sufficiently with the single qubit phase gate), it is possible to approximate any unitary transformation, making them essential building blocks in quantum computation.
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