What is universal gate?

Q.) What is universal gate?

 Subject: Digital Circuit and System Design

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Question: What is a universal gate?

Answer:

In quantum computing, a universal gate is a quantum logic gate that can be used to construct any other quantum circuit. This means that, given a universal gate, it is possible to perform any quantum computation.

There are many different universal gates, but the most common one is the CNOT gate. The CNOT gate is a two-qubit gate that flips the target qubit if the control qubit is in the state $|1\rangle$.

The CNOT gate can be used to construct other quantum gates. For example, the Hadamard gate can be constructed using two CNOT gates and a single-qubit rotation gate. The Toffoli gate can be constructed using three CNOT gates.

The fact that there exist universal gates is a very important result in quantum computing. It means that it is possible to build a quantum computer that can perform any quantum computation. This is in contrast to classical computers, which are limited to performing certain types of computations.

Examples of Universal Gates:

  1. CNOT Gate:

    • Controlled-NOT gate is the most common universal quantum gate.
    • It takes two qubits as input and flips the target qubit if the control qubit is $|1\rangle$.
    • It can be represented by the matrix: $$CNOT = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$

  2. Hadamard Gate:

    • It is a single-qubit gate that puts a qubit into an equal superposition of $|0\rangle$ and $|1\rangle$ states.
    • It can be represented by the matrix: $$H = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$

  3. Toffoli Gate:

    • It is a three-qubit gate that acts as a controlled-controlled-NOT gate.
    • It flips the target qubit if both the control qubits are $|1\rangle$.
    • It can be represented by the matrix: $$Toffoli = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}$$

Universality is a crucial concept in quantum computing, as it implies that a quantum computer with a finite number of universal gates can simulate any classical or quantum algorithm. This property is what gives quantum computers their potential for solving problems that are intractable for classical computers.

In summary, a universal gate is a quantum logic gate that can be used to construct any other quantum circuit. This means that, given a universal gate, it is possible to perform any quantum computation. The most common universal gate is the CNOT gate, which can be used to construct other gates such as the Hadamard gate and the Toffoli gate.