Multi-Qubit Gates

Multi-qubit gates can create entangled states between two qubits. They have two inputs and two outputs, and they are reversible, unlike classical gates, which cannot entangle two bits. The main two-qubit gates are:

Controlled-NOT gate (CNOT)

• This gate flips the state of the target qubit if and only if the control qubit is in the |1⟩ state.

SWAP gate

• This gate swaps the states of the two qubits.

Toffoli gate

• This gate is a controlled-controlled-NOT gate, which means that it flips the state of the target qubit if and only if both the control qubits are in the |1⟩ state.

These gates are essential for many quantum algorithms, such as Shor's algorithm for factoring large numbers and Grover's algorithm for searching unsorted databases.

Controlled-NOT Gate

The controlled-NOT (CNOT) gate is a two-qubit gate that flips the state of the target qubit if and only if the control qubit is in the |1⟩ state. The control qubit itself remains unchanged.

The CNOT gate is a universal gate, which means that it can be used to implement any other quantum gate. It is also a reversible gate, which means that it can be undone.

The CNOT gate is a fundamental building block for many quantum algorithms, such as Shor's algorithm for factoring large numbers.

Input (controlled qubit, target qubit)	Output (controlled qubit, target qubit)
00	00
01	01
10	11
11	10

Table: Truth Table for controlled-NOT gate.

In matrix form, this gate is shown as:

Fig. Working of CNOT gate where controlled qubit is shown by filled circle on the wire and target qubit is shown by empty circle.

·         SWAP Gate:

The SWAP gate is a two-qubit gate that swaps the states of the two input qubits. For example, if the input qubits are in the states |1⟩ and |0⟩, the output qubits will be in the states |0⟩ and |1⟩.

The SWAP gate can be implemented using three CNOT gates. The first CNOT gate swaps the states of the two qubits, the second CNOT gate swaps the states of the two qubits again, and the third CNOT gate swaps the states of the two qubits one final time.

In matrix form, the SWAP gate is shown as:

Fig. Working of SWAP Gate and truth Table.

Fig. Swapping of two input qubit states by using three CNOT Gates.

Toffoli Gate

The Toffoli gate is a three-qubit gate that only changes the state of the target qubit if and only if both of the control qubits are in the |1⟩ state. If the control qubits are not in the |1⟩ state, the target qubit remains unchanged.

The Toffoli gate is also known as the controlled-controlled-NOT gate or the CCNOT gate. It is a universal gate, which means that it can be used to implement any other quantum gate.

The Toffoli gate is a fundamental building block for many quantum algorithms, such as the Deutsch-Jozsa algorithm and the Simon's algorithm.

In matrix form, this gate is represented as:

Fig. Truth table and quantum circuit for Toffoli gate.

Fredkin Gate

The Fredkin gate is a three-qubit gate that is also known as the controlled-SWAP gate or the CSWAP gate. It swaps the states of the two target qubits if and only if the first control qubit is in the |1⟩ state. If the first control qubit is not in the |1⟩ state, the two target qubits remain unchanged.

The Fredkin gate is a universal gate, which means that it can be used to implement any other quantum gate. It is also a reversible gate, which means that it can be undone.

The Fredkin gate is a fundamental building block for many quantum algorithms, such as the Deutsch-Jozsa algorithm and the Simon's algorithm.

In matrix form, this gate is shown as:

Table: Truth table for Fredkin gate.

Fig. Working of quantum circuit for Fredkin gate.